Oct 15, 2018 - is made possible by the University Libraries, University of Arizona. ... Download date ... Dissertation Director: Bruce R. Barrett ..... three open quantum systems, which particles are free to enter and leave via .... of the Hamiltonian H; it is the operator that generates translations in time [21]. ...... [9] G. D. Mahan.
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Cardamone, David Michael
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Dephasing and Decoherence in Open Quantum Systems: A Dyson’s Equation Approach
by David Michael Cardamone
A Dissertation Submitted to the Faculty of the DEPARTMENT OF PHYSICS In Partial Fulfillment of the Requirements For the Degree of DOCTOR OF PHILOSOPHY In the Graduate College THE UNIVERSITY OF ARIZONA
2005
3
THE UNIVERSITY OF ARIZONA GRADUATE COLLEGE As members of the Dissertation Committee, we certify that we have read the dissertation prepared by David M. Cardamone entitled Dephasing and Decoherence in Open Quantum Systems: A Dyson’s Equation Approach and recommend that it be accepted as fulfilling the dissertation requirement for the Degree of Doctor of Philosophy Date: 08/04/05 Bruce R. Barrett Date: 08/04/05 Charles A. Stafford Date: 08/04/05 Sumitendra Mazumdar Date: 08/04/05 Michael A. Shupe Date: 08/04/05 Koen Visscher Final approval and acceptance of this dissertation is contingent upon the candidate’s submission of the final copies of the dissertation to the Graduate College. We hereby certify that we have read this dissertation prepared under our direction and recommend that it be accepted as fulfilling the dissertation requirement. Date: 08/04/05 Dissertation Director: Bruce R. Barrett Date: 08/04/05 Dissertation Director: Charles A. Stafford
4
STATEMENT BY AUTHOR This dissertation has been submitted in partial fulfillment of requirements for an advanced degree at The University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library. Brief quotations from this dissertation are allowable without special permission, provided that accurate acknowledgment of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his or her judgment the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author.
SIGNED:
David Michael Cardamone
5
ACKNOWLEDGEMENTS From the bottom of my heart, the greatest thanks I have must go to my wife, Martha. Through the years, she has been an unparalleled source of love, hope, advice, support, and friendship. Her wisdom has informed each choice I have made, and her example has inspired me. Thank you, Martha. I must also thank my parents and grandparents, who provided me with a safe childhood, allowed me the luxury of exploring my own interests, and, above all, gave me their confidence. By their example, they taught me a sense of personal responsibility, ethics, and morality, which shaped the person I have become. On a final personal note, I do not want to forget my friends in Tucson who have helped me in numerous ways over the years. I was fortunate to travel professionally quite a bit during my grad student years, and James Little, Geoff Schmidt, and Jeremy Jones facilitated this in all those important ways that friends do. I also thank Tucson Kendo Kai for helping me find the determination, courage, and character necessary to a grad student’s lifestyle. On the professional side, I could not have been more honored or fortunate to work under the tutelage and supervision of Profs. Bruce Barrett and Charles Stafford. They, too, gave me their confidence. Much more useful than teaching me physics (although they did that as well), they showed me how to learn physics. I shall never forget their tireless efforts to guide me on the long journey from inexperienced student to practicing physicist. An additional very special thanks is due to Prof. Sumit Mazumdar, with whom I have also had the privilege of collaborating the last two years. Although Sumit had many answers, equally valuable in this collaboration were his questions. He took the time to give excellent and thoughtful career advice, which was key in getting me where I am today. Indeed, the entire community of the University of Arizona Department of Physics have been welcoming and helpful to me during my time here. Over the years, my thesis committee, including those mentioned above as well as Profs. Mike Shupe, Koen Visscher, and Srin Manne, have always found time to help me with advice or encouragement. So too have the other faculty of the department, including especially Keith Dienes, Fulvio Melia, Jan Refelski, Bob Thews, Bira van Kolck, J. D. Garcia, and Carlos Bertulani. Among all the helpful staff, Mike Eklund and Phil Goisman always went above and beyond the call of duty without complaint, for which I wish to express my appreciation and admiration. My understanding of the scientific issues discussed in this work has benefitted enormously from numerous engaging discussions over the years. In particular, thanks are due to Chang-hua Zhang, Jerome B¨ urki, Jeremie Korta, Ned Wingreen, Peter von Brentano, Micah Johnson, Ryoji Okamoto, Dan Stein, Anna Wilson, Paul Davidson, Mahir Hussein, Adam Sargeant, and George Kirczenow. The pleasure of discussion and collaboration with such outstanding physicists is one I hope will continue for many years.
6
For Martha, meae vitae.
7
TABLE OF CONTENTS LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
CHAPTER 1: INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
1.1
Green Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
1.1.1
General theory of Green functions . . . . . . . . . . . . . . . . . .
16
1.1.2
Electrostatic Green functions . . . . . . . . . . . . . . . . . . . .
18
1.1.3
Quantum mechanical Green functions . . . . . . . . . . . . . . .
19
Physical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
1.2.1
Coupled quantum dots . . . . . . . . . . . . . . . . . . . . . . . .
21
1.2.2
Decay of superdeformed nuclei . . . . . . . . . . . . . . . . . . . .
21
1.2.3
Molecular electronics . . . . . . . . . . . . . . . . . . . . . . . . .
22
CHAPTER 2: DYSON’S EQUATION . . . . . . . . . . . . . . . . . . . . . . . . .
23
1.2
2.1
Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
2.1.1
S-matrix expansion . . . . . . . . . . . . . . . . . . . . . . . . . .
23
2.1.2
Diagrammatic approach . . . . . . . . . . . . . . . . . . . . . . .
26
Self-Energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
2.2.1
Hybridization: adding a second level . . . . . . . . . . . . . . . .
30
2.2.2
Decoherence: a single continuum . . . . . . . . . . . . . . . . . .
31
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
CHAPTER 3: COUPLED QUANTUM DOTS . . . . . . . . . . . . . . . . . . . .
34
2.2
2.3
3.1
3.2
Quantum Dots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
3.1.1
History and fabrication . . . . . . . . . . . . . . . . . . . . . . . .
36
3.1.2
Experimental studies . . . . . . . . . . . . . . . . . . . . . . . . .
37
Two-Level Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
3.2.1
39
Realm of Applicability to Quantum Dots . . . . . . . . . . . . . .
8
TABLE OF CONTENTS –Continued 3.2.2
Hamiltonian of the coupled dot system . . . . . . . . . . . . . . .
39
3.2.3
Spin-boson analogy . . . . . . . . . . . . . . . . . . . . . . . . . .
41
Green Function Treatment . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
3.3.1
Without leads . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
3.3.2
Including the leads . . . . . . . . . . . . . . . . . . . . . . . . . .
45
Limiting Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
3.4.1
Identical dots . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
3.4.2
Identical lead couplings . . . . . . . . . . . . . . . . . . . . . . . .
50
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
CHAPTER 4: DECAY OF SUPERDEFORMED NUCLEI . . . . . . . . . . . . .
52
3.3
3.4
3.5
4.1
4.2
4.3
4.4
4.5
Nuclear Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
4.1.1
Normal deformation . . . . . . . . . . . . . . . . . . . . . . . . .
53
4.1.2
Superdeformation . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
4.1.3
Experimental signatures of deformation . . . . . . . . . . . . . .
57
Decay Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
4.2.1
Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
4.2.2
Double-well paradigm . . . . . . . . . . . . . . . . . . . . . . . .
63
Two-State Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
65
4.3.1
Two-state Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . .
66
4.3.2
Energy broadenings . . . . . . . . . . . . . . . . . . . . . . . . . .
67
4.3.3
Green function treatment . . . . . . . . . . . . . . . . . . . . . .
70
4.3.4
Branching ratios . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
Tunneling Width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
72
4.4.1
Relation between branching ratios and tunneling width . . . . . .
72
4.4.2
Measurement of the tunneling width . . . . . . . . . . . . . . . .
73
4.4.3
Limits of the tunneling width . . . . . . . . . . . . . . . . . . . .
75
Statistical Theory of Tunneling . . . . . . . . . . . . . . . . . . . . . . . .
76
4.5.1
Gaussian orthogonal ensemble . . . . . . . . . . . . . . . . . . . .
76
4.5.2
Implications for tunneling . . . . . . . . . . . . . . . . . . . . . .
77
9
TABLE OF CONTENTS –Continued 4.6
Adding More Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
80
4.6.1
Three-state model
. . . . . . . . . . . . . . . . . . . . . . . . . .
80
4.6.2
Infinite-band approximation . . . . . . . . . . . . . . . . . . . . .
84
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
CHAPTER 5: MOLECULAR ELECTRONICS . . . . . . . . . . . . . . . . . . . .
88
4.7
5.1
5.2
5.3
5.4
Fabrication of Single-Molecular Systems . . . . . . . . . . . . . . . . . . .
88
5.1.1
Scanning-tunneling microscopic techniques . . . . . . . . . . . . .
89
5.1.2
Mechanically controllable break junction . . . . . . . . . . . . . .
90
5.1.3
Other techniques . . . . . . . . . . . . . . . . . . . . . . . . . . .
92
Modeling Molecular Electronics . . . . . . . . . . . . . . . . . . . . . . . .
92
5.2.1
Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
5.2.2
Hartree-Fock approximation . . . . . . . . . . . . . . . . . . . . .
95
5.2.3
Non-equilibrium Green function theory . . . . . . . . . . . . . . .
95
5.2.4
Equal-time correlation functions . . . . . . . . . . . . . . . . . . .
98
5.2.5
Landauer-B¨ uttiker formalism . . . . . . . . . . . . . . . . . . . .
99
Quantum Interference Effect Transistor . . . . . . . . . . . . . . . . . . . 101 5.3.1
Tunable conductance suppression . . . . . . . . . . . . . . . . . . 101
5.3.2
Finite voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
CHAPTER 6: DISCUSSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
10
LIST OF FIGURES 2.1
Dyson’s equation expansion of the full retarded self-energy Σ ? . . . . . . .
28
3.1
Electron micrograph images of experimental quantum dots
. . . . . . . .
36
3.2
Experimental spectra and addition energies of quantum dots . . . . . . .
38
3.3
Schematic diagram of the double quantum dot system . . . . . . . . . . .
40
3.4
Hybridization of energy levels in the double-dot system without leads
. .
43
3.5
Coherent Rabi oscillations in the double-dot system without leads . . . .
45
3.6
Mixture of coherent and incoherent behavior in the full system . . . . . .
47
4.1
Evidence for shell closures in the first excited state of even-even nuclei . .
54
4.2
Evidence for shell closures in nuclear separation energies . . . . . . . . . .
54
4.3
Normal deformation and superdeformation on the table of nuclides . . . .
56
4.4
Superdeformation from a harmonic oscillator potential . . . . . . . . . . .
58
152 Dy
4.5
Decay spectrum of superdeformed
. . . . . . . . . . . . . . . . . . .
61
4.6
Universality in the decay of superdeformed nuclei of A ≈ 190 . . . . . . .
62
4.7
Diagram of the superdeformed decay process . . . . . . . . . . . . . . . .
63
4.8
Types of potentials historically used to model superdeformed decay . . . .
64
4.9
Two-level model of superdeformed decay . . . . . . . . . . . . . . . . . . .
65
4.10 Gaussian orthogonal ensemble probability distributions for the energies of the two levels on either side of the decaying superdeformed level . . . . .
78
4.11 Branching ratios in the two- and three-level model . . . . . . . . . . . . .
83
4.12 Branching ratios in the two- and inifite-level models . . . . . . . . . . . .
86
5.1
Artist’s conception of the Quantum Interference Effect Transistor, a singlemolecular device . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2
Scanning tunneling microscope approach to creating single-molecule junctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3
89 91
Mechanically controllable break junction approach to creating single-molecule junctions
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
5.4
The Keldysh contour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
97
5.5
Schematic diagrams of Quantum Interference Effect Transistors . . . . . . 102
11
LIST OF FIGURES –Continued 5.6
Cancellation of paths in a QuIET, and lifting of that effect by the introduction of a third lead . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.7
Transmission probabilities for various Quantum Interference Effect Transistors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.8
Lead configurations for a Quantum Interference Effect Transistor based on [18]-annulene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.9
Possible placements for a third led in the benzene QuIET . . . . . . . . . 106
5.10 I − V characteristic of a Quantum Interference Effect Transistor . . . . . 107
12
LIST OF TABLES 3.1
Free-electron densities of states for nanoscale systems of different dimensionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1
35
Inputs and results of the two-level model for specific superdeformed decays 69
13
ABSTRACT In this work, the Dyson’s equation formalism is outlined and applied to several open quantum systems. These systems are composed of a core, quantum-mechanical set of discrete states and several continua, representing macroscopic systems. The macroscopic systems introduce decoherence, as well as allowing the total particle number in the system to change. Dyson’s equation, an expansion in terms of proper self-energy terms, is derived. The hybridization of two quantum levels is reproduced in this formalism, and it is shown that decoherence follows naturally when one of the levels is replaced by a continuum. The work considers three physical systems in detail. The first, quantum dots coupled in series with two leads, is presented in a realistic two-level model. Dyson’s equation is used to account for the leads exactly to all orders in perturbation theory, and the time dynamics of a single electron in the dots is calculated. It is shown that decoherence from the leads damps the coherent Rabi oscillations of the electron. Several regimes of physical interest are considered, and it is shown that the difference in couplings of the two leads plays a central role in the decoherence processes. The second system relates to the decay-out of superdeformed nuclei. In this case, decoherence is provided by coupling to the electromagnetic field. Two, three, and infinite-level models are considered within the discrete system. It is shown that the two-level model is usually sufficient to describe decay-out for the classic regions of nuclear superdeformation. Furthermore, a statistical model for the normal-deformed states allows extraction of parameters of interest to nuclear structure from the two-level model. An explanation for the universality of decay profiles is also given in that model. The final system is a proposed small molecular transistor. The Quantum Interference Effect Transistor is based on a single monocyclic aromatic annulene molecule, with two leads arranged in the meta configuration. This device is shown to be completely opaque to charge carriers, due to destructive interference. This coherence effect can be tunably broken by introducing new paths with a real or imaginary self-energy, and an excellent molecular transistor is the result.
14
CHAPTER 1 INTRODUCTION In the last century, the study of physics experienced an unprecedented Renaissance. With the advent of quantum mechanics and relativity, a Kuhnian paradigm shift [1] occurred in the discipline. These theories allowed an understanding of experimental systems far beyond the realm of everyday experience, but the advance came at a price: the body of knowledge became compartmentalized, and it was now necessary to categorize physical systems according to their length scales, energy scales, or even the forces involved. It is not always immediately clear what should happen in systems that lie at the interface of these different theoretical regimes. This problem is to be distinguished from that of limits, which were in all cases understood at the founding of the new theories. The present work, for example, is concerned with the interface between quantum-mechanical and classical systems. The appropriate limit, of course, is well known as the Bohr correspondence principle [2], which states that a classical system is obtained if a quantum system is taken in the limit ~ → 0, where h = 2π~ is Planck’s constant. The question of interface between the classical and quantum-mechanical regimes, by contrast, asks an entirely different question: what happens to a system that is neither wholly quantum-mechanical, nor wholly classical? Especially in the case of quantum mechanics, an understanding of interfacial systems is of paramount importance, for the simple reason that no system of physical interest is purely quantum-mechanical. This is not mere pedantry: a system of physical interest is, by definition, limited to the set of those upon which measurements are, or will be, performed. As such, these systems always contain non-quantum components. When the interplay between the classical and quantum regimes is significant, the system is referred to as mesoscopic [3]. In general, mesoscopic systems have coherence lengths comparable to their size, so that they are primarily quantum, but non-quantum effects are also important in them. Such effects invariably arise from interaction with a classical, macroscopic system [4]. The additional system can generate dephasing, in which particles accrue additional phase individually, as well as decoherence, in which phase coherent effects are utterly destroyed [5–7]. A mesoscopic system can be of any
15 length scale on which quantum coherence can exist: this thesis deals with nanoscale, molecular, and femtoscale (nuclear) mesoscopic systems. This is not to say that quantum mechanics is useless without reference to an external theory, for we emphasize that an investigation of the interface between two theoretical frameworks need only make reference to the more general one. In this case, the Bohr correspondence principle tells us that quantum mechanics contains all the information of classical mechanics. Our task is therefore not to seek “new physics”, for the quantum theory is, in this sense, complete. Rather, we wish to understand more fully what we already have. As we study each mesoscopic system, the classical components will be built up from discrete and fully quantum-mechanical sets of states. The purpose of this thesis is to explore the role of dephasing and decoherence in three open quantum systems, which particles are free to enter and leave via macroscopic continua. To study these, we make use of the Dyson’s equation formalism [8], based on equilibrium and non-equilibrium Green functions [4, 9]. The goals will be to determine the effect of phase-breaking processes in these systems, and to develop a rigorous and intuitive approach, which will be applicable to the study of future such systems, as well.
1.1
Green Functions
Classical phenomena are characterized by continua of states, in which a variable may take on any value within a particular range, whereas quantum-mechanical systems generally have discrete sets of allowable values for certain variables. Indeed, the name of the theory was determined by this remarkable and wholly new phenomenon: we refer to variables which are restricted to a discrete set of values as quantized. We can expect, then, that while the quantum mechanical parts of the systems under study are characterized by discrete states, the classical components consist of continua. A formalism is required, therefore, which deals well with such large numbers of states in a rigorous, quantum-mechanically exact manner. Such a formalism is found in the Dyson’s equation approach to many-body quantum theory, which centers on treatment of the quantity known as the Green function [10]. It is remarkable that many-body theory provides such an apt solution to the problem, as we are not primarily interested in many-body effects. In fact, throughout this work,
16 interactions between the quantum system and classical environment are treated as entirely single-particle effects. The reason the Green function formalism performs so well in such circumstances is that many-body theory is, at its essence, a theory of many states, which is precisely the item of current interest.
1.1.1
General theory of Green functions
The Green function approach was first put forward by its namesake, George Green, as a method to find the electrostatic potential of a charge configuration with an arbitrary set of boundary conditions [11]. Green’s solution to this problem is, in fact, quite general, not only to the case of the Poisson equation so described, but to any linear differential equation [12]: Dψ(x) = −ρ(x),
(1.1)
where D is a general linear differential operator in x, which is an ordered set of co¨ordinates. The Green function is defined by the related equation D1 G(x2 , x1 ) = −δ(x1 − x2 ),
(1.2)
where the subscript i on D indicates to which set of co¨ordinates x i it applies, and δ is the Dirac delta. Using G, a solution to Eq. (1.1) can be constructed, given a set of boundary conditions. The secular equation for D is Dφn (x) = λn φn (x),
(1.3)
where φn and λn are D’s eigenfunctions and eigenvalues, respectively. The eigenfunctions are a complete, orthonormal set of states: ∞ X
n=−∞
φn (x1 )φ∗n (x2 )
= δ(x1 − x2 ),
Z
dτ φn (x)φ∗m (x) = δnm ,
(1.4)
where dτ is the volume element corresponding to the basis x, δ nm is the Kronecker delta, and the integral is to be taken over the entire space of x. This property allows an expansion of the Green function in terms of the eigenstates: G(x2 , x1 ) =
∞ X
n=0
an (x1 )φn (x2 ).
(1.5)
17 Substituting this result into Eq. (1.2) yields D1
∞ X
an (x1 )φn (x2 ) =
∞ X
n=0
n=0
an (x1 )λn φn (x2 ) = −δ(x1 − x2 ).
(1.6)
It follows that Z Z ∞ X dτ2 an (x1 )λn φn (x2 )φ∗m (x2 ) = − dτ2 δ(x1 − x2 )φ∗m (x2 ) = −φ∗m (x1 ) = am (x1 )λm , n=0
(1.7)
where dτi is the volume element corresponding to the basis x i . The Green function is thus seen to be [12] G(x2 , x1 ) = −
∞ X φ∗ (x1 )φn (x2 ) n
λn
n=0
.
(1.8)
This expansion for G(x2 , x1 ) is closely related to the charge fluctuation resonance expansion, which has seen a great deal of success in the nanoscopic literature [13–18]. The paramount utility of the Green function follows directly from the eigenfunction expansion (1.8). Multiplying Eq. (1.1) by G(x 2 , x1 ) and integrating, we find Z
dτ1
∞ X φ∗ (x1 )φn (x2 ) n
λn
n=0
D1 ψ(x1 ) =
Z
dτ1 G(x2 , x1 )ρ(x1 ).
(1.9)
Writing the solution ψ(x) as an expansion in the eigenstates and eigenvalues of D, ψ(x) =
∞ X
bn φn (x),
(1.10)
n=0
we arrive at [12] Z
dτ1 G(x2 , x1 )ρ(x1 ) =
Z
dτ1
∞ ∞ X X φ∗n (x1 )φn (x2 ) bn φn (x2 ) = ψ(x2 ). λm bm φm (x1 ) = λ n n=0 n=0
m=0
(1.11)
This is an extraordinary result, and the central conclusion that underlies all work using Green functions. Equation (1.11) says that any linear differential equation may be solved by applying an integral transform to the inhomogeneous term ρ(x). The kernel of this transform is the Green function, defined by Eq. (1.2), solution of which is generally a much simpler task than the original equation (1.1). In essence, Eq. (1.11) gives the inverse of D.
18
1.1.2
Electrostatic Green functions
Green introduced this method of treating linear differential equations to solve the Poisson equation [11] ∇2 ψ(r) = −ρ(r),
(1.12)
a problem of central importance at the time due to its application in electrostatics, where ρ/4π is the charge density and ψ(r) the electrostatic potential. Since it is also, nearly invariably, the first use of Green functions encountered in a physicist’s education [19], it may be of some value to briefly review the solution. From the identity ∇2
1 = −4πδ(r1 − r2 ), |r1 − r2 |
(1.13)
it is apparent that the Green function corresponding to D = ∇ 2 is [19] G(r2 , r1 ) =
1 + F(r2 , r1 ), 4π|r1 − r2 |
(1.14)
where F(r2 , r1 ) can be any function which satisfies the Laplace equation, ∇2 F(r2 , r1 ) = 0.
(1.15)
The choice of F(r2 , r1 ) depends on the boundary conditions of the system. Green derived a simple theorem for any two fields A and B [11], I Z I Z dA dB 2 2 dτ B∇ A + dσB = , dτ A∇ B + dσA dn dn V V
(1.16)
which follows from Gauss’s theorem. The surface integrals are taken over the boundary of the volume V , and
d dn
represents differentiation with respect to a unit vector normal
to the boundary. Performing the integration of Eq. (1.11), we find Z 1 ψ(r2 ) = dτ1 + F(r2 , r1 ) ρ(r1 ). 4π|r1 − r2 | By Eqs. (1.15) and (1.12), Equation (1.16) implies Z I dψ(r1 ) dF(r2 , r1 ) − ψ(r1 ) , dτ1 F(r2 , r1 )ρ(r1 ) = dσ1 F(r2 , r1 ) dn1 dn1 V
(1.17)
(1.18)
where the subscripts on dσ1 and n1 indicates they correspond to r1 . If V includes the entire region containing charge, we find I Z ρ(r1 ) dψ(r1 ) dF(r2 , r1 ) − ψ(r1 ) . dτ1 + dσ1 F(r2 , r1 ) ψ(r2 ) = 4π|r1 − r2 | dn1 dn1 V
(1.19)
19 Since it appears only in the surface integral with ψ(r 1 ), it is manifest that F(r2 , r1 ) is fixed by the boundary conditions of a particular physical system.
1.1.3
Quantum mechanical Green functions
Next, we consider a Green function approach to the quantum mechanical process of time evolution. The formalism which results forms the foundation of the current work, allowing as it does the complete solution of all time dynamics for a particular quantum system. We begin with the quantum-mechanical initial-value problem. In the Schr¨odinger picture, state kets time evolve while the operators remain stationary. For the moment assuming time-translational symmetry [20], a state ket is given by |ψ(t)i = e−iHt/~ |ψ0 i,
(1.20)
where |ψ0 i is the ket at time t = 0. Equation (1.20) follows from the defining property of the Hamiltonian H; it is the operator that generates translations in time [21]. H has real eigenvalues; thus it is Hermitian. The time-evolution operator e −iHt/~ is consequently unitary:
e−iHt/~
−1
† = e−iHt/~ = eiHt/~ .
(1.21)
This operator time evolves backwards by an amount t. Acting from the left on Eq. (1.20), then, we arrive at the linear differential equation eiHt/~ |ψ(t)i = |ψ0 i.
(1.22)
This is the equation to be solved by the Green function approach. Equation (1.22) is included for completeness, but, of course, we already know the solution: Equation (1.20) corresponds exactly to Equation (1.11), with the usual quantum mechanical understanding of operators in place of integrals. The Green function (operator) is thus G(t) = −e−iHt/~ .
(1.23)
In the Green function approach, it is almost always easier to consider the energy domain. We construct the Fourier transform: Z ∞ dt i(E−H)t/~ e . G(E) = − −∞ ~
(1.24)
20 The result is poorly defined, but this should not trouble us, since we have not specified boundary conditions. For example, if we wish to solve an initial-value problem, we should construct the retarded Green function [9], G(E) ≡ iG
(t≥0)
−i (E + i0 ) = ~ +
Z
∞
dt ei(E−H+i0
0
+ )t/~
=
1 , E − H + i0+
(1.25)
where the superscript (t ≥ 0) denotes that the integral of Eq. (1.24) should be evaluated
only for nonnegative t. The notation 0 + is used throughout this work to denote a quantity which is to be taken to zero from the right when the calculation is finished. The restriction to nonnegative times is the crucial characteristic of the retarded Green function. The i0+ is added to make integrals over positive time converge, and the overall factor of i is an unobservable notational convenience, equivalent to giving the same phase rotation to all states. The final result (1.25) is of central importance, and clearly highlights the inverse relationship between Green function and Hamiltonian. In addition to the retarded Green function (1.25), we shall occasionally make use of the advanced Green function G† as well. This operator, the Hermitian adjoint of G, corresponds to boundary conditions for which the final state of the system is known.
1.2
Physical Systems
In the next chapter, we develop the formalism of Dyson’s equation [8], which allows extrapolation from a system with understood dynamics to a new one. By partitioning the Hamiltonian into solved and unsolved terms, Dyson’s equation develops a controlled expansion in terms of a quantity Σ, called the retarded proper self-energy. In general, there is no reason for this expansion to converge quickly. If the selfenergy is complicated, an arbitrary number of terms may be necessary to ensure even a qualitatively accurate description of the physics [9]. In this work, however, we take the tactic of choosing physical systems that lend themselves to reasonable approximations which reduce Dyson’s equation to an exactly soluble form. Then, the sum can be done exactly to all orders in the perturbation Σ. While we restrict ourselves to the study of three systems in detail, this is by no means the limit of the approach. The broadly general nature of the formalism allows for many other applications, as well.
21
1.2.1
Coupled quantum dots
Examination of particular physical systems begins in Chapter 3 with the problem of two coupled quantum dots. Since quantum dots, nanoscale electron systems, are often described as “artifical atoms” [22–24], this system can be thought of, in some sense, as an artifical diatomic molecule [25, 26]. We shall explore all coupling regimes, corresponding to both covalent and ionic bonding. Coupled quantum dot systems have generated a plethora of interesting experimental and theoretical results [26–28]. It has further been suggested that such a system could be used as a logic gate [29, 30] or a qubit, the building block of a quantum computer [31–34]. Naturally, questions of environmental effects are central to such possibilities, especially the latter. A purely isolated quantum system is of no use as a qubit, and yet coherence effects must be well preserved if the device is to be useful. As the first physical system we shall investigate, a simple approach is taken to the physics of the coupled quantum dot. The system is well approximated by a two-level model, with each dot coupled to its own macroscopic lead [15, 35–37]. An electron is injected into one dot, and its time behavior is calculated exactly via Dyson’s Equation. The results are contrasted with those for the same system neglecting lead effects. The interplay between classical leads and quantum mechanical dots is seen to damp the Rabi oscillations characteristic to the isolated two-state system, opening a new expanse of interesting physical regimes for experimental exploration.
1.2.2
Decay of superdeformed nuclei
In Chapter 4, the two-level model is expanded to treat the decay-out process of superdeformed bands in nuclear physics. These bands consist of axially symmetric, ellipsoidal states with major-to-minor axis ratios of about 2:1, an important prediction of the shell model [38]. Intraband decay occurs as the nucleus loses angular momentum to the environment through coupling to the electromagnetic continuum. Superdeformed bands abruptly lose their strength to less deformed bands through a statistical decay process. We demonstrate conclusively that this phenomenon is well understood within a simple two-level model, exactly solvable in the Dyson’s Equation approach. The decoherent effects of electromagnetic decay processes are included on the
22 same footing as the coherent effects, which mix levels of different deformation. Via a statistical model, parameters of direct consequence to nuclear structure theory are extracted from experimental results. Furthermore, the consistency of the two-level approximation is verified via the addition of first one, then an infinite number of extra levels.
1.2.3
Molecular electronics
Chapter 5 continues the investigations by moving to a treatment of molecular electronic systems. These experimentally realized systems consist of small molcules, whose electron dynamics are governed by a discrete set of states, in contact with macroscopic metallic leads. Theoretical modeling of such a system at finite voltages presents several new challenges. The model of previous chapters is expanded to non-equilibrium, interacting systems. A comprehensive picture for modeling molecular electronics is shown to follow directly from the techniques of this work. Moreover, the intuitive understanding granted by our study of the previous systems motivates proposal of a small molecular transistor based on the interplay of coherent and decoherent effects. The Quantum Interference Effect Transistor, as it is called, operates based on tunable breaking of a coherent current suppression caused by perfect destructive interference of paths through the molecule. The results, which mimic the current-voltage characteristics of classical transistors in all important regards, are valid for a large class of molecules and lead arrangements.
23
CHAPTER 2 DYSON’S EQUATION In this chapter, we shall derive and begin to use the most central result of Green function theory, Dyson’s equation. This formalism provides a perturbative expansion, whereby a Hamiltonian is partitioned into solved and unsolved parts, and the effect of the the unsolved part is included systematically. In this work, we shall focus on methods of summing the series to all orders, so that the results of Dyson’s equation remain exact. In the latter part of this chapter, we begin this process with two simple examples. A single quantum level is placed in contact with both another single level, and an infinite continuum of states representing a classical system. In both these cases, Dyson’s equation can be summed exactly to all orders.
2.1
Derivation
Derivation of Dyson’s equation [8, 39] centers on the development of a diagrammatic expansion of the quantity known as the S-matrix, closely related to the Green function. The essential problem is to describe the time dynamics of a system whose Hamiltonian H = H 0 + HI
(2.1)
consists of a part H0 whose eigenvalues and eigenstates are known, and an additional part HI . It is not immediately clear how the addition of H I impacts the behavior of the system for the general case, in which the commutator [H 0 , HI ] 6= 0. It is assumed that H0 and HI can be written in a second-quantized form [39], that is, in terms of creation and annihilation operators.
2.1.1
S-matrix expansion
The first step in constructing Dyson’s equation is to link the expectation values of observables to the known eigenstates of H 0 . The secular equation of H0 is H0 |φn i = En |φn i,
(2.2)
24 which pertains to an exact, possibly many-body, solution. It is assumed that in the absence of the perturbation HI , the system lies in the ground state |φ 0 i, and that each piece is separately Hermitian. Since we are interested in finite times only, it is permissible to rewrite the full Hamiltonian (2.1) so that for infinite times t → ±∞, it reduces to H 0 : H(t) = H0 + lim HI e−η|t| , η→0
(2.3)
an approach which is known as adiabatic “switching on” [39]. We note that the derivation of Sec. 1.1.3 no longer strictly applies, as the Hamiltonian is now time-dependant. The defining property of H is that it generates instantaneous time translations [21]: iH(t)dt |ψ(t + dt)i = 1 − |ψ(t)i, ~
(2.4)
which implies the time-dependant Sch¨odinger equation. It follows that, in the general case [39], |ψ(t)i = T e
− ~i
Rt
t0
dt0 H(t0 )
|ψ(t0 )i = −G(t, t0 )|ψ(t0 )i.
(2.5)
The exponential function is to be interpreted, as usual, as a Taylor expansion. The symbol T represents a time-ordering of each term in the series, so that operators evaluated at earlier times are on the right. To differentiate the effects of H0 from those of HI , it is customary to work within the interaction picture of quantum mechanics [2, 21, 39]. Kets time-evolve as |ψI (t)i = T e
− ~i
Rt
t0
dt0 HI (t0 )
|ψI (t0 )i = GI (t, t0 )|ψI (t0 )i.
(2.6)
Expectation values are maintained by defining operators as i
i
OI (t) = e ~ H0 t Oe− ~ H0 t ,
(2.7)
where O is the Schr¨odinger-picture operator. The interaction picture is neither the rest frame of state kets, nor of operators. We note, therefore, that H I = HI (t) itself, which governs the time development of kets, rotates with the frequency H 0 /~, as described by Eq. (2.7). The advantage of Eq. (2.3) is that it allows us to make contact between expectation values and the unperturbed ground state. The expectation value of O(t) is hO(t)i =
hφ0 |GI (−∞, t)OI (t)GI (t, −∞)|φ0 i hψI (t)|OI (t)|ψI (t)i = . hψI (t)|ψI (t)i hφ0 |GI (−∞, t)GI (t, −∞)|φ0 i
(2.8)
25
In this and the next two chapters, we shall deal with physical systems strictly in equilibrium. These systems possess time reversal symmetry, so that GI (−∞, t) = GI (∞, t).
(2.9)
We can thus rewrite Eq. (2.8) to read hO(t)i =
hφ0 |GI (∞, t)OI (t)GI (t, −∞)|φ0 i . hφ0 |GI (∞, t)GI (t, −∞)|φ0 i
(2.10)
The numerator of this equation can be read as a prescription for evaluating an expectation value at time t. First, begin at the unperturbed ground state |φ o i. Then, slowly turn on HI , and evolve the state to time t. Operate on this state with the relevant operator, OI (t). Now, to get back to the state |φ0 i, time evolve the state forward further to t = ∞, switching off HI again. Since the system is time-reversal symmetric, you are guaranteed to end up right back where you started. Making use of the time-ordering operator T , we can write Eq. (2.10) in a more compact form [39]: hO(t)i =
hφo |T [OI (t)S] |φo i , hφ0 |S|φo i
(2.11)
where the S-matrix is defined as i
S ≡ GI (∞, −∞) = T e− ~
R∞
−∞
dt0 HI (t0 )
.
(2.12)
We can write the exponential expansions explicitly: ∞ X i n − T [OI (t)S] = ~ n=0 ∞ X i n − S = ~ n=0
1 n! 1 n!
Z
∞
−∞ Z ∞ −∞
dt1 · · · dt1 · · ·
Z
∞
−∞ Z ∞ −∞
dtn T [OI (t)HI (t1 ) · · · HI (tn )]
(2.13)
dtn T [HI (t1 ) · · · HI (tn )]
The picture we have is thus of all possible numbers of operations of the perturbation HI , taking place both before and after evaluation of the operator O I (t). Note that time evolution by H0 , from t = −∞ to t = ∞, is included in the definition (2.7) of the interaction-picture operator OI (t), due again to time-reversal symmetry.
26
2.1.2
Diagrammatic approach
The S-matrix expansion detailed in the previous section allows time evolution for the full Hamiltonian (2.1) to be computed without knowing its eigenstates, but instead using the ground state of H0 . Obviously, a formalism developed from this result has the potential to be of great use in the study of a number of systems. The first important step toward this goal is to build a diagrammatic expansion, known as Feynman-Dyson perturbation theory [8, 39–42]. To bridge the gap between equations and diagrams, we turn to the result known as Wick’s theorem [39, 43]. The theorem states that time-ordered products of creation and annihilation operators, such as those found in the numerator and denominator of Eq. (2.11), can be written in terms of two new operations, called contractions and normalordering: T(ABC · · · Z) =
X
N(all products with n contractions).
(2.14)
n contractions
That is, the time ordering of operators is equal to the normal ordering of those operators, plus all normal orderings with one contraction, plus those with two, and so on. The normal ordering N is a permuting operator similar to T, but before applying it, the operators of its argument must be broken down into their constituent creation and annihilation operators. Then, the normal-ordering operator places all creation operators to the left of the annihilation operators, accruing the usual factor of −1 each time a fermion operator passes through another. The difference between the time ordering and normal ordering of two operators is called a contraction: contraction(AB) ≡ C(AB) ≡ T(AB) − N(AB).
(2.15)
The two operators must be brought next to each other before the contraction is evaluated. After evaluation, the result is to be removed from further ordering operators. Of particular interest are contractions of the form h i h i C an (t2 )a†m (t1 ) = Θ (t2 − t1 ) an (t2 ), a†m (t1 ) , ±
(2.16)
since all other types of contractions in the S-matrix expansion (2.13) vanish. [· · · ] ± indicates the anticommutator or commutator, as appropriate to the algebra of the operators.
27 As is customary, we choose the vacuum, defined as the state which vanishes when acted upon by any annihilation operator, to be the ground state of the unperturbed system |φ0 i. It is easily verified that the expectation value of any normal ordering in such a state vanishes, unless all operators to be ordered are contracted. This is a very important result, as it simplifies use of Wick’s theorem (2.14) considerably. The only expectation values which remain in Eq. (2.11), after application of Wick’s theorem, are those which look like [39] h i † Θ(t2 − t1 ) φ0 an (t2 ), am (t1 ) φ0 = iG0nm (t2 , t1 ), ±
(2.17)
where G0nm (t2 , t1 ) is the retarded Green function of time [9], as defined in Sec. 1.1.3. In this case, having defined a vacuum state, we have explicitly considered both particle and hole excitations, which is the role of the (anti)commutator. The superscript 0 denotes that this Green function is related to H 0 , which is contained within the interaction-picture operators, rather than the full Hamiltonian. A complete picture of the diagrammatic approach to the S-matrix expansion (2.11) now emerges. HI and O must be written in terms of second quantization operators, and Wick’s theorem applied, after which two types of functions remain. Retarded Green functions (2.17), resulting from the movement of the interaction-picture operators O I and HI , are commonly represented by single-line arrows [39]. The other type of term which remains is the c-number part of the interaction, which may act on a single state and time, or may act on several. Such interactions are represented in the diagrammatic expansion by wavy or dashed lines, depending on their physical origin. For term n in the expansions (2.13), each diagram has nN op /2 vertices, where Nop is the number of creation and annihilation operators which compose H I . The nature of the vertices themselves depends on the character of H I . Two-body forces, for example, create vertices of four Green function lines. All topological possibilities, with appropriate prefactors, must be summed to calculate a particular term of the expansions. The presence of OI (t) in the numerator gives each diagram an “incoming” and “outgoing” line, which stretch to t = ±∞, while the denominator has no such Green functions. The denominator can be eliminated from consideration, therefore, by factoring all such “disconnected” pieces from the numerator, and canceling [39, 44]. That is, by restricting
28
Σ Σ Σ?
=
Σ + Σ
+ Σ + ··· Σ
Figure 2.1: Schematic diagram showing the Dyson’s equation expansion of the full retarded self-energy Σ? . The proper self-energy Σ is the sum of the minimal set of self-energy pieces from which Σ? can be built. our consideration to diagrams which consist, topologically, of only one piece, we automatically include all diagrams of the numerator and denominator. In general then, after this cancellation, all diagrams have a single incoming and outgoing line, in between which lie n vertices, connected to each other and the exterior Green functions by Green function lines and interactions; that is, each diagram with n > 0 can be represented schematically by Fig. 2.1. Clearly, the full S-matrix is given by [39, 40] S = G000 (∞, −∞) +
XZ kk 0
∞ −∞
dt1
Z
∞ −∞
dt2 G00k0 (∞, t2 )Σ?k0 k (t2 , t1 )G0k0 (t1 , −∞),
(2.18)
where the quantity Σ? , known as the full retarded self-energy, is simply the sum of all things that can happen to a particle that starts and finishes in |φ 0 i. While Eq. (2.18) may almost seem to be a tautology, it is extremely important. It tells us that if we can understand the full self-energy, we know the S-matrix, and, by extension, the full time dynamics of the system’s observables. The key observation here is that certain terms of Σ ? can be singled out as “proper” self-energy contributions. This subset of self-energy contributions can be defined as the minimal set from which all others can be built. That is, at no point in a proper selfenergy diagram, except, of course, for the beginning and the end, does the system return to |φ0 i. This allows an iterative expansion for Σ ? to be built up [39], as in Fig. 2.1.
29 Mathematically, the iterative equation is simplified by working in the energy domain G(E) = G0 (E) + G0 (E)Σ(E)G0 (E) + G0 (E)Σ(E)G0 (E)Σ(E)G0 (E) + · · · (2.19) = G0 (E) 1 + Σ(E)G0 (E) + Σ(E)G0 (E)Σ(E)G0 (E) + · · · (2.20) = G0 (E) [1 + Σ(E)G(E)]
(2.21)
= G0 (E) + G0 (E)Σ(E)G(E) = G0 (E) + G(E)Σ(E)G0 (E).
(2.22)
The same is true in the time domain, but intermediate times must be integrated over. Furthermore, a direct algebraic consequence of Eq. (2.22) is −1 G−1 (E) = G0 (E) − Σ(E),
(2.23)
which is also known as Dyson’s equation. Since G0 (E) is simply given by G0 (E) =
1 , E − H0 + i0+
(2.24)
a physical problem is entirely solved if the proper retarded self-energy Σ is known. For many problems, this remains an intractable requirement [9]. In studying the basic properties of coherence and decoherence, however, we shall require only the simplest of one-body forces in HI . As such, Equation (2.23) provides a full, exact solution.
2.2
Self-Energies
In this work, two types of self-energy will be key. The first is the most basic possible, that from a single level. The second is the self-energy due to a full, infinite continuum of states. As a classical system, the continuum gives rise to decoherence. In each case, we begin with a single, isolated level, which forms the solved part of the Hamiltonian: H0 = ε0 |0ih0|.
(2.25)
The retarded Green function of energy arising from this Hamiltonian is G0 (E) =
1 |0ih0| = . + E − H0 + i0 E − ε0 + i0+
(2.26)
30 As implied by Eq. (1.25), taking the real part of the pole of G(E) yields the single energy in the system. Time evolution is given by G0 (t) =
∞
Z
−∞
dE e−iEt/~ |0ih0| = e−iε0 t/~ |0ih0|, 2π E − ε0 + i0+
(2.27)
as is to be expected. Our goal for the remainder of this chapter is to gain intuition about how addition of different types of terms to this Hamiltonian changes the Green function.
2.2.1
Hybridization: adding a second level
H0 generates time evolution within a single quantum level. We can ask what effect the inclusion of a second level has on the time dynamics. We consider an addition to the Hamiltonian HI = ε1 |1ih1| + V |1ih0| + V ∗ |0ih1|,
(2.28)
which depends on the energy ε1 of the new state |1i, as well as the tunneling matrix element V , which takes the system from state |0i to the new state. We must now consider the proper retarded self-energy due to addition of the new level. It consists of only one diagram: the system leaves |0i with the interaction V , propagates
within the |1i state, and finally returns with interaction V ∗ . All diagrams in the full self-energy are simply iterations of this process for this simple system. The proper retarded self-energy is thus Σ00 =
|V |2 VV∗ = . E − ε1 + i0+ E − ε1 + i0+
(2.29)
Dyson’s equation (2.23) now yields the full Green function: + G−1 00 (E) = E − ε0 + i0 −
|V |2 (E − ε0 + i0+ ) (E − ε1 + i0+ ) − |V |2 = ; (2.30) E − ε1 + i0+ (E − ε0 + i0+ ) (E − ε1 + i0+ )
G00 (E) =
(E − ε0 + i0+ ) (E − ε1 + i0+ ) . (E − ε0 + i0+ ) (E − ε1 + i0+ ) − |V |2
(2.31)
It should be noted that this is an exact result, which includes the proper self-energy to all orders. This is the power of Dyson’s equation (2.22). In passing, we note that the Green function now has two poles, ε± =
ε0 + ε 1 1 p (ε1 − ε0 )2 + 4|V |2 , ± 2 2
(2.32)
31 which give the two hybridized energy levels of the problem. It is a general consequence of Eq. (1.25) that the real part of the poles of the green function are always the effective energy levels in a system. The effect of the second level on the coherent time dynamics of the system is clear. Due to the presence of the level |1i, the system can leave |0i, propagate for an arbitrary amount of time, and then return. During this time, it accrues a phase based, not only on the time spent, but on the energy and tunneling matrix parameters introduced to the system with the extra level. Equation (2.31) provides an exact, mathematical description of the full time dynamics which result when this process is included.
2.2.2
Decoherence: a single continuum
Now that all the building blocks are in place, the next logical topic to consider is the most basic system which exists at the interface of quantum mechanics and the classical world. Instead of for a single level, we construct the self-energy due to a full continuum of states. This schematic problem will serve as the prototype for consideration of more complicated open quantum systems. The Hamiltonian is H = H0 +
X k
εk |kihk| + Vk |kih0| + Vk∗ |0ihk|,
(2.33)
where k labels the states of the continuum. The levels of the continuum only communicate through the original |0i state, so proper self-energy terms are merely variations on Eq. (2.29), one for each of the states in the continuum. The result is Σ00 (E) =
X k
|Vk |2 . E − εk + i0+
To shed light on this result, we appeal to the Dirac identity 1 1 − iπδ(x), lim ≡P + x η→0 x + iη
(2.34)
(2.35)
where P(x) denotes the Cauchy principle value of x, to arrive at Σ00 (E) = −iπ
X k
i |Vk |2 δ(E − εk ) = − Γ(E), 2
(2.36)
where Γ(E) = 2π
X k
|Vk |2 δ(E − εk ).
(2.37)
32 Equation (2.37) happens also to be the result of Fermi’s Golden Rule in this system. This is simply due to the fact that the proper self-energy Σ is second-order in V k and diagonal. In such a case, Dyson’s equation provides a prescription for iterating the processes of second-order perturbation theory to all orders exactly. For a more detailed discussion of the applicability of Fermi’s Golden Rule to exact results, the reader is excouraged to examine Sec. 4.4.1. Equation (2.37) is an exact result, even in cases without an ideal continuum. It is easy to see, however, that in the limit of an ideal continuum, whose states all couple equally to |0i, it produces a constant function of energy. Taking this limit, as we shall often do in this work, is analogous to making use of the correspondence principle to bring a quantum system to its classical limit. Now that we know the self-energy, we can sum its effects to all orders exactly using Dyson’s equation (2.23): G00 (E) =
1 . E − ε0 + iΓ/2
(2.38)
The density operator is defined [21] by summing over the Hamiltonian’s diagonal basis |`i: ρ(E) =
X `
δ(E − ε` )|`ih`|,
(2.39)
where ε` are the eigenenergies. ρ(E) is thus directly related to the imaginary part of the Green function of energy by the Dirac identity: G(E) =
X `
X |`ih`| = −iπδ(E − ε` )|`ih`|. + E − ε` + i0
(2.40)
`
It follows that 1 ρ(E) = − Im[G(E)], π
(2.41)
where the density of states is the trace of this quantity. In the current system, this equals ρ0 (E) =
1 Γ/2 , π (E − ε0 )2 + (Γ/2)2
(2.42)
a Lorentzian. Due to its contact with the continuum, the state |0i has been broadened into a Breit-Wigner resonance. Its rate of decay is Γ/~. By particle-hole symmetry, this rate is also the rate of filling the level in the event that |0i is empty and the continuum is filled with non-interacting particles.
33
2.3
Summary
The Dyson’s equation approach derived in this chapter has been of fundamental importance to the theoretical solution of countless systems. Perhaps just as importantly, the simple form of the result (2.23) often provides an intuitive insight into the physics of a system, which might be lost in an unnecessarily complex treatment. In this work, we specifically explore the use of Dyson’s equation in situations where the self-energy is known, and the summation (2.22) can be carried out exactly to all orders in the perturbation of Σ. The results of Sec. 2.2 suggest that a quantum system can be affected in two ways by the addition of a single-particle self-energy, such as those important to the study of open quantum systems. First, if the state |0i becomes vacant, a particle from an external state may refill it. Since the new particle’s phase would then be completely uncorrelated to the old electron’s, the net result is a total loss of phase coherence, or decoherence. The other possibility is that the same particle may propagate for a time in the external system, and then re-enter the original set of levels. Such a process contributes an arbitrary phase to the process, and so is called dephasing. It is worth noting that, in the case of an infinite set of levels, as for a macroscopic continuum, dephasing and decoherence are indistinguishable. In the next chapter, we begin to apply the results found thus far to the study of a nanoscale system, two coupled quantum dots. The two-state model used is a synthesis of the results of Secs. 2.2.1 and 2.2.2, in that both states are coupled to a macroscopic continuum. A similar model is used in Chapter 4 to examine the decay-out of superdeformed nuclear bands. Thus, even the most basic results of the Green function approach are seen to generate solutions in a striking variety of problems.
34
CHAPTER 3 COUPLED QUANTUM DOTS We now turn our attention to a particular system: a series arrangement of two quantum dots, coupled to each other, and each to a macroscopic lead. An important goal of this analysis is to lay a firm pedagogical foundation for the more complex treatments of later chapters, and as such, we shall endeavor to maintain as simple and straightforward an approach as can be implemented, and yet arrive at interesting physics. The focus will remain on equilibrium systems, and the treatment is largely restricted to a noninteracting, two-state model of time dynamics. Fortunately, the nature of the physical system is such that these assumptions retain a great deal of physical merit over large regions of experimental interest. Over the last two decades, quantum dots have been found to be a rich and varied physical system. Part of their appeal comes from parallels to natural systems [28, 45]: quantum dots can be seen as “artificial atoms”, nanoscale analogues to nuclei, and test systems for theories of quantum chaos, to name only a few. Whereas natural systems are generally limited to specific examples, however, the quantum dot is a remarkably tunable structure, so that an entire range of parameters can be explored. Quantum dots are thus often the ultimate proving ground for theories of mesoscopic fermion systems. It was perhaps natural that, once the community had begun to come to terms with the existence of the “artificial atom”, study of multiple-dot systems began to thrive. Theorists and experimentalists alike have found lattices of quantum dots a subject of great interest [18, 46–53], as well as smaller “artificial molecules” involving just a few dots [15, 17, 18, 25, 26, 28, 36, 37, 52, 54–80]. Besides their inherent interest, many important technological applications of coupled quantum dot systems have been proposed [29–34, 72, 81]. Great strides which have been made toward understanding the internal electron dynamics of double-dot systems, but it is a tendency in the field to consider coupling to the macroscopic leads as, at best, a source of experimental error to be minimized, or a generic source of uniform level broadening [26]. Nevertheless, experimental systems run the gamut from weak to very strong lead coupling [45], and it seems appropriate to
35
Table 3.1: Dimensionalities d available to nanoscale quantum systems, along with their common names and free-electron densities of states. m ∗ is the electron’s effective mass. Ei , Eij , and Eijk are the discrete energies which results from confinement in 1, 2, or 3 directions, respectively. Θ is the Heaviside step function. d
Common Name
3
bulk
2
quantum well
1
quantum wire
0
quantum dot
Free-electron density of states √ m∗ 2m∗ E 2 3 π ~ m∗ X Θ (E − Ei ) π~2 r i 1 m∗ X Θ (E − Eij ) p π 2~2 E − Eij ij X δ (E − Eijk ) ijk
consider the decoherent systems in the leads as sources of intriguing physics in their own right. Furthermore, the intuitive Dyson’s equation approach presented in the previous chapters provides an ideal tool for the study, since it automatically treats both coherence and decoherence on an equal footing, without the need for prejudicial assumption.
3.1
Quantum Dots
Quantum dots are zero-dimensional nanoscale systems of confined electrons. In them, we find an experimental realization of a textbook quantum mechanics problem, i. e. the particle in a box. More than that, however, the quantum dot represents a rich and varied “sandbox” of mesoscopic and many-body physics. The experimental and theoretical mastery of these systems has opened up countless frontiers full of engrossing problems for the physicist to study. From a theoretical perspective, a quantum dot is simply the logical continuation in a progression of mesoscopic devices defined by the dimensionality of their confinement (see Table 3.1). The exact meaning of “confinement” can be made rigorous from microscopic considerations: essentially the electrons should be restricted to a length scale ` which is much less than their spacing [82, 83]. For our purposes, however, it is sufficient to require that a particular experimental system possess a discrete density of states, the unmistakable hallmark of full three-dimensional confinement.
36
Figure 3.1: Electron micrograph images of experimental quantum dot systems. (a) An array of vertical quantum dots. The horizontal bars are 5µm. The main picture is from Ref. [84], while the inset diagram of a single dot was added in Ref. [28]. (b) A lateral double quantum dot system from Ref. [26]. The darker grey is the quantum well structure, and the lighter is the electrodes, which define the two quantum dots in the center tunably. The area available to the left dot is a 320×320nm 2 , while the region of the right dot is 280×280nm2 .
3.1.1
History and fabrication
The first quantum dots [84–86] were an outgrowth of earlier experiments into fabrication of quantum wells and developments in the technology of lithography. These “vertical” quantum dots (see Fig. 3.1a) began life as semiconductor herterostructures, usually GaAsAlGaAs or GaAs-InGaAs. With proper doping, mobile electrons are trapped in a twodimensional region near the interface of the heterostructure, forming a quantum well system. To form a quantum dot, the surrounding heterostructure material was removed by lithography, thus providing the additional two dimensions of electron confinement. Vertical quantum dots have been constructed in the series arrangement we discuss in this work. Another, more common, possibility for this type of experiment, however, is the lateral quantum dot (see Fig. 3.1b). This style of device also makes use of a semiconductor heterostructure to form a two-dimensional system. The difference lies in the method used to provide lateral confinement. Instead of etching, metallic electrodes are deposited on the surface of the quantum well device [87–89]. These are gated to tunably control all
37 matrix elements of the system’s Hamiltonian. Lateral quantum dots have become even more common in recent years, as they take full advantage of the tunability of quantum dot systems. Semiconductor quantum dots are not the only nanoscale systems that have been demonstrated to posses discrete electronic energy levels. Metallic nanograins [90, 91], selfassembling islands [92, 93], quantum “corrals” made via scanning tunneling microscope [94], and macromolecular systems [95, 96] have all been fabricated with physics similar to the original heterostructure systems. While the series arrangement we shall discuss here may be more difficult to contrive in some of these systems than in others, the theory we develop is general enough to apply to any.
3.1.2
Experimental studies
There exists a large variety of experimental techniques available to the study of quantum dot systems. Rather than attempt a review of the experimental literature, we choose to focus on the techniques and results that are the most relevant for the case of quantum dots coupled in series. For a comprehensive introduction to the varied types of experiments on quantum dots, and their implications, see Ref. [27] Experiments characterizing the electronic spectra of dots (see Fig. 3.2a) are of the most fundamental interest. The observation of discrete energy levels [84–86], with its implications for electronic confinement, has already been mentioned as a result of paramount importance. Vindicating analogies to atomic and nuclear systems, a further result of great import was the observation of shell structure in quantum dots [97]. The presence of shell closures and “magic numbers” (see Fig. 3.2b) has significant benefit to the applicability of the two-level model we shall use. A second, related class of experiments center on electron transport through and among dots. Charging and discharging of such a small nanostructure is dominated by the quantized nature of the electric charge. This Coulomb blockade effect leads to oscillations in the conductance of quantum dots [14]: only when points of charge degeneracy in parameter space are neared is current allowed to flow (see Fig. 3.2a). This, in turn, allows one to count how many electrons are on a dot by emptying it, and then slowly refilling while counting current peaks [98]. By this method, an experimental system can be “tailored”
38
(a)
(b)
Figure 3.2: Results of an experimental study of the spectra of vertical quantum dots, from Ref. [97]. (a) Current vs. gate voltage in a dot of diameter .5µm. This result demonstrates the discrete nature of the density of states, as well as Coulomb blockade oscillations. (b) Electron addition energies for two dots of diameter D. The peaks indicate especially stable electron numbers, implying shell closures. The magic numbers of these dots are 2, 6, and 12. The inset shows a diagram of the dots used in the study. to have whatever number of particles is desired. Studies of devices such as single-electron boxes [99], turnstiles [100], pumps [101], and transistors [102] have demonstrated the viability of this idea. Many other types of experiments fall outside the scope of this work. Measurements of the optical, magnetic, and even phonon properties of quantum dots have been made [27]. One prominent regime of quantum dot physics that will be intentionally absent from the present study is the Kondo effect in quantum dots [103, 104]. For the purposes of this work, we assume that the system is far away from the regime for which such lead-lead or lead-dot correlations are relevant.
3.2
Two-Level Model
We assume that electrons in each dot are limited to only one energy level. As long as the dots remain within their ground states, this is known to be a rather good approximation [26]. Previous theoretical studies applying this model to coupled quantum dots have met with much success [15, 35–37]. For a discussion of how to move beyond the two-level
39 model, the reader is encouraged to examine Sec. 4.6 of the present work.
3.2.1
Realm of Applicability to Quantum Dots
The two-level approximation amounts to an assumption that interactions with those electrons of less energy in each dot can be treated as constant, and that states with higher energy can be neglected. In a non-interacting picture, this is permissible so long as any tunneling matrix elements connecting our two states to other states are much less than the energy difference with those states [37]. When this is true, such extra states will play very little role in the dynamics of the system. These conditions are especially likely to be fulfilled if the dots are filled exactly to shell closure. Then, the extra electron we inject is forced to begin a new shell, and interacts mainly in a mean-field sense with the lower electrons. This situation is reminiscent of the “core” approach to the shell model studies of nuclear physics. Due to the artificial and tunable nature of quantum dot systems, and especially the ability to controllably add and remove particles from systems, such a situation is quite realizable in practice. A further consideration is that the temperature of the system must not cause excitations between levels. Thanks to remarkable efforts by experimentalists, quantum dot systems usually have effective electron temperatures of around 10–15mK [26]. This corresponds to a level spacing of .86–1.29µeV, comfortably below those common in quantum dots. Of course, the two-level model will be most accurate when we need not ignore electrons of lower energy levels, because there is only one electron on the dot. Such single-electron devices have indeed been fabricated [99–102]. They provide the best opportunity for experimental verification of theoretical studies, such as ours, that rely solely on singleparticle effects.
3.2.2
Hamiltonian of the coupled dot system
For the rest of this discussion, we assume the conditions of the preceeding argument are met, so that we are justified in working within the two-level model. Each dot is coupled to its own lead: this series arrangements of dots and leads is shown in Fig. 3.3.
40
Figure 3.3: Schematic diagram of the double quantum dot system. Two quantum dots are coupled to each other, and each to its own macroscopic lead. This situation is also frequently known as a triple-barrier system. In the two-level model, the two dots are completely described by their energy levels ε 1 and ε2 , which are defined in the absence of both leads and the tunneling between the two dots V . Γ 1 /~ and Γ2 /~ give the rates for electrons to enter and leave the double-dot system through each lead. From Ref. [37]. The Hamiltonian of the system can be written as the sum of three terms [37]: H = Hdots + Hleads + Htun ,
(3.1)
where Hdots generates the time evolution of the two-level system in the absence of the leads, Hleads is likewise the Hamiltonian of the isolated leads, and H tun gives the coupling between them. The isolated dot Hamiltonian in the two-level model is Hdots = ε1 d†1 d1 + ε2 d†2 d2 − V d†1 d2 − V ∗ d†2 d1 .
(3.2)
Here εn is the isolated energy level of dot n, V parameterizes the hopping from dot 2 to dot 1, and dn is the operator which annihilates an electron in the state of dot n. Gauge invariance allows us to choose exactly one phase in the problem: we use this freedom to set V real and positive. Equation (3.2) neglects interdot interactions. We shall extend a similar model to fully include the Coulomb interaction in Chapter 5. In general, we assume the leads are ideal Fermi gasses, devoid of the complications of many-body correlations such as Kondo physics, superconductivity, etc. They posses a diagonal representation, and in that basis their Hamiltonian can be written Hleads =
2 X X
α=1 k∈α
k c†k ck ,
(3.3)
41 where k is the energy of a particular state k in lead α, and c kσ annihilates an electron in state k. Coupling between the leads and dots is provided by the third term of the Hamiltonian: Htun =
X X
hnαi k∈α
Vnk d†n ck + H.c. .
(3.4)
Here Vnk parameterizes the tunneling between dot n and state k of lead α. The notation hnαi reminds us that Vnk 6= 0 only if it would connect a dot to its own lead. The source of decoherence, dephasing, and dissipation in this system is clear from Eq. (3.4). Through it, electrons can leave the system. New electrons can enter the system as well, and, having random phase, they can only contribute to the dynamics in incoherent ways. A combination of these two results is also possible: an old electron may be replaced by a different one from the leads, thus causing decoherence. Similarly, an electron may tunnel into a lead, propagate via Eq. (3.3) for arbitrary time, and itself return later with a new phase. Equation (3.2) describes a well understood quantum-mechanical problem. The new and interesting physics is introduced by the leads in the terms (3.3) and (3.4). These contributions require an infinite-dimensional Fock space. It is important to realize that one does not really care what happens to electrons in the leads, only in the dots. The beauty of the Dyson’s equation approach is that it allows us to incorporate the effects of the leads exactly, as they appear from inside the dots.
3.2.3
Spin-boson analogy
Having constructed the two-state Hamiltonian of our system, we arrive at an appropriate juncture to draw analogy to a very well studied problem, that of the spin-boson Hamiltonian [105]: HSB
X 1 1 = − ~∆SB σx + εSB σz + 2 2 ν
p2 1 mν ων x2ν + α 2 2mν
X 1 + q0 σz Cν xν . 2 ν
(3.5)
Here σx and σz are the Pauli matrices for spin- 12 systems. This Hamiltonian is often used to describe two-state systems, especially spin or isospin systems, in contact with a bath of harmonic oscillators. The oscillators usually represent bosonic states, such as phonons or photons.
42 The first two terms are to be taken in analogy to our H dots ; that is, they define the closed, well understood two-level system. The Pauli matrices play the role of the quadratic operators in Eq. (3.2). ~∆ SB and εSB should be taken in analogy to our V and ε1 − ε2 , respectively. The harmonic oscillators, labeled by the index ν, usually represent a bath of bosonic states, such as phonons or photons, which provide dissipation to the spin states. The internal Hamiltonian of the oscillators themselves is given by the third term in Eq. (3.5), comparable to bosonic excitations of the lead system described by Eq. (3.3). The oscillators, with position and momentum operators x ν and pν , respectively, are defined by their mass and frequency parameters mν and ων . The final term in the spin-boson Hamiltonian (3.5) plays a similar role to H tun of Eq. (3.4); its purpose it to provide the coupling between the spin and bosonic systems. Although physical meaning may be ascribed in certain systems, in the general case q 0 Cν simply parameterizes the coupling of each oscillator to the two-level system [105]. The spin-boson formalism is itself a powerful tool, and many parallels can be drawn with the Dyson’s equation approach. The most central conclusion of the theory surrounding its use is that the effect of the boson system on the spin part is entirely contained within the spectral function of the coupling [105]: J(ω) =
π X Cα2 δ(ω − ωα ). 2 α mα ωα
(3.6)
The spirit of this result should seem quite familiar: we are interested only in the oscillators’ effect on the spin system, and so we “trace out” the extra degrees of freedom, and replace them with J(ω). Equation (3.6) should be compared with the functions Γ(E), for example Eq. (2.37), which arise in the Dyson’s equation approach. Although Γ(E) is not, strictly speaking, a spectral function, its derivation from the retarded self-energy Σ is reminiscent of the method used to extract a spectral function from a Green function [9].
3.3
Green Function Treatment
Having constructed the Hamiltonian of the complete system in Sec. 3.2.2, we are ready to attempt a solution of the problem using Dyson’s equation. Before doing that, however, we take the opportunity to re-examine the textbook problem of the closed system’s
43
(a)
ε+
ε1
(b)
ε2
ω ε−
Figure 3.4: Hybridization of energy levels in the double-dot system without leads. (a) The energy levels in the absence of V , ε1 and ε2 , are combined by their tunneling matrix √ 2 element to give two new levels, ε+ and ε− , as given by Eq. (3.7). ω = ∆ + 4V 2 is the detuning of the two new orbitals. (b) The graph of ε±V−ε (solid lines) vs. ∆/V shows the anticrossing which results when two coupled levels approach each other. The dashed ε −ε lines give 1,2V . After Ref. [26]. Hamiltonian, given by Eq. (3.2). This will allow us to compare and contrast with the results of the full problem, the better to determine the effect of the leads.
3.3.1
Without leads
The first step in understanding the full problem will be to remind ourselves of some results for the two-level system without coupling to an environment. Diagonalizing the Hamiltonian is a simple problem in elementary quantum mechanics. The eigenenergies are simply the results for two hybridized levels:
ε± = ε ±
s
∆ 2
2
+V2
(3.7)
where ε ≡ (ε1 + ε2 )/2 is the mean of the two isolated energy levels, and ∆ ≡ ε 2 − ε1 is their detuning. The physical meaning of Eq. (3.7) is demonstrated in Fig. 3.4. The eigenstates themselves are the well known antibonding and bonding solutions to the twostate problem: ψ± = r
1+
1
ε± −ε1 V
2
ε± − ε 1 ψ1 + ψ2 , V
where ψn are the wavevectors of each isolated dot.
(3.8)
44 As a simple exercise, let us see if we can arrive at this result within the Green function formalism. We work within the basis of the individual dots. The retarded Green function of the system is given by
Gdots (E) = E − Hdots + i0
+ −1
=
E − ε1 + i0+ V
V E − ε2 + i0+
−1
E − ε2 + i0+ −V 1 . (3.9) = (E − ε1 + i0+ )(E − ε2 + i0+ ) − V 2 −V E − ε1 + i0+
As expected from Eq. (1.25), the real part of the poles of the Green function gives us the energies of the eigenstates. Solving for the poles recovers Eq. (3.7).
The retarded Green function Gdots (E) is characterized by the same eigenfunctions as the Hamiltonian Hdots . The secular equation is Gdots (E)ψ ± =
1 ψ±, E − ε± + i0+
(3.10)
where the prefactor on the right is the eigenvalues of the retarded Green function. Equation (3.8) follows directly from Eq. (3.10). It is, of course, natural and necessary that the eigenvalues and eigenstates of H dots , the generator of time translations for the system, should be contained within G dots (E), since that quantity determines the time dynamics of the problem. We raise the point here not only to serve as an instructional example, but also to emphasize a key element of the theory: the Green function and Hamiltonian formulations of quantum mechanics are completely interchangeable. The correspondence is exactly that between a differential (Schr¨odinger equation) and integral (Green function) approach, rigorous within linear inverse theory [12]. For a generic state, the time evolution is most easily seen from the time-domain Green function Gdots (t) =
Z
∞
−∞
dE Gdots (E)e−iEt . 2π
(3.11)
If a single electron is localized in dot 1 at time 0, the probability of finding it in the same dot at a later time t is [26, 37] 1 P1 (t) = |G11 (t)| = 2 ω 2
ωt ωt + ω 2 cos2 ∆ sin 2 2 2
2
,
(3.12)
45
Figure 3.5: Example of the coherent Rabi oscillations described by Eqs. (3.12) and (3.13). In this graph, ∆ = V . In the absence of dissipation from the leads, the coherent oscillations continue forever. while P2 (t) = |G21 (t)|2 = 1 − P1 (t) =
4V 2 ωt sin2 . ω2 2
(3.13)
Equations (3.12) and (3.13) describe the coherent Rabi oscillations of a two-level system [21]. The frequency of oscillation is given by the detuning between the bonding and antibonding orbitals: ω=
p
. ∆2 + 4V 2 ~.
(3.14)
An example of these coherent oscillations is shown in Fig. 3.5.
3.3.2
Including the leads
We proceed now to include the leads in our description of the system. The motivation for this procedure is evident from the results (3.12) and (3.13) of the previous subsection: since the Rabi oscillations continue forever, without decoherence, dephasing, or dissipation, such an approach clearly neglects an important part of the physics. Since there are no processes which allow an electron to tunnel from dot 1 into a lead, propagate within the lead, and then return by tunneling into dot 2, or visa versa, the tunneling rates of leads 1 and 2 are uncorrelated. This is an essential point, as it tells us
46 the retarded self-energy is diagonal, and that we are dealing with two separate broadening phenomena. At this stage, we may refer ourselves to the solution of the single level in contact with a macroscopic continuum of states (Sec. 2.2.2), and simply write down the solution. The retarded self-energy has the form i Σnm (E) = − δnm Γn (E), 2
(3.15)
where the tunneling rate into and out of dot n is given by the Fermi’s Golden Rule result (2.37) Γn (E) = 2π
XX
hnαi k∈α
|Vnk |2 δ(E − ε0k ).
(3.16)
Equation (3.15) contains all time evolution within the leads. Dyson’s equation (2.23) allows us to use it to solve for the effect of the leads exactly to all orders in perturbation theory. The full Green function is thus −1 −1 −1 −V E − ε1 + 2i Γ1 (E) . = G(E) = Gdots (E) − Σ(E) −V E − ε2 + 2i Γ2 (E)
(3.17)
Inverting this result we find [37] G(E) =
i E − ε1 + Γ1 (E) 2
−1 i 2 E − ε2 + Γ2 (E) − V 2 i V E − ε2 + 2 Γ1 (E) , (3.18) × V E − ε1 + 2i Γ2 (E)
which should be compared with Eq. (3.9).
We now make an assumption, known as the broad-band approximation and often utilized in the literature, that the energies of the dots lie well within continua of the leads. If this is the case, the densities of states in the leads can be effectively replaced with a constant function, and the tunneling matrix elements V nk are uniform. The Γn (E) may then be replaced by constant parameters. If the leads are indeed good metallic conductors, this approximation is very well justified. Further, since the focus of this work is the interplay between quantum and macroscopic systems, the case of uniform continua in the leads is of primary interest here.
47
(a)
(b)
Figure 3.6: Examples of the mixture of coherent and incoherent behavior found in the full system of dots and leads, as given by Eqs. (3.19) and (3.20). In both, ∆ = V . (a) An example of the underdamped case, where decoherence from the leads is relatively weak. Γ = 3Γ0 = .3V . (b) An example of the overdamped case. Γ = 3Γ 0 = 3V . Let us again consider the case that our single electron is localized in dot 1 at time t = 0. After performing the Fourier transform of Eq. (3.18) by contour integration, we are able to compute the probabilities [37] V 2 −Γt/~ ωi + Γ0 ωi t Γ0 − ωi −ωi t iωr + Γ0 iωr t iωr − Γ0 −iωr t e e + e + e + e P1 (t) = |ω|2 Γ0 − ω i ωi + Γ 0 iωr − Γ0 iωr + Γ0 (3.19) and P2 (t) = Here Γ≡
2V 2 −Γt/~ e (cosh ωi t − cos ωr t) . |ω|2
(3.20)
Γ1 + Γ 2 , 2
(3.21)
Γ0 ≡
Γ2 − Γ 1 , 2
and ωr and ωi are the real and imaginary parts of the complex Rabi frequency q 2 ω ≡ ωr + iωi ≡ 4V 2 + (∆ − iΓ0 ) ~.
(3.22)
Figure 3.6 shows two examples of behaviors which can result from this solution. Taking the limits Γ1 , Γ2 → 0+ of Eqs (3.19) and (3.20) yields Eqs. (3.12) and (3.13). We note, however, that, except in this limit, the identity P 1 (t)+P2 (t) = 1 no longer holds:
48 the open nature of the system allows particles to both enter and leave the system. The incoherent physics introduced by the leads is reflected in the real exponential (hyperbolic) functions in Eqs. (3.19) and (3.20). Whereas ω r governs the coherent Rabi oscillations of the system, ωi describes the decoherence processes due to the leads. For long times, the decoherence dominates, as we should expect. As t → ∞, the probability in both dots
falls as e−Γt/~ due to the infinite bath of states into which electrons can escape.
We can understand our general result more by comparing Eq. (3.22) with its counterpart Eq. (3.7) in the isolated system. Whereas in the isolated case, we considered the energies of the Hamiltonian, which are observable and hence real, in the lead-coupled situation we generalize the discussion to consideration of the complex poles of the Green function ε˜± =
ε˜1 + ε˜2 ~ω ± , 2 2
(3.23)
where it is to be understood that ε˜i = εi − iΓi /2 are the poles of the Green function for dots isolated from each other, i.e. in the limit V → 0. Equation (3.23) shows that ~ω remains the displacement in the complex plane between the poles of the full Green function. Since it determines time dynamics of the system, the retarded Green function, together with the lead couplings Γi , naturally contains all linear-response conductance information. The transmission probability between leads 1 and 2 for charge carriers of a given energy is given by the multi-terminal current formula (see Sec. 5.2.5): T12 (E) = Γ1 Γ2 |G12 (E)|2 .
(3.24)
In the linear response, all charge carriers have energy equal to the Fermi energy ε F of the leads. Thus, the conductance of the double-dot device is equal to [4] 1 2e2 = T12 (εF ), R h
(3.25)
where the prefactor is simply the conductance quantum.
3.4
Limiting Cases
Eqs. (3.19) and (3.20) represent a complete solution of our model, in as much as they contain all information regarding the time evolution of the experimental observables in
49 the problem. Greater understanding of the physical meaning of this result can be gained through examination of the limiting cases of identical dots and identical leads.
3.4.1
Identical dots
We first consider the case of identical dots: ε 1 = ε2 . When this is true, the complex Rabi frequency (3.22) simplifies to ω=
. p 4V 2 − Γ02 ~.
(3.26)
This quantity is either purely real or purely imaginary, depending on the relative values of 2|V | and |Γ0 |.
In the case that 2|V | > |Γ0 |, the Rabi frequency is purely real. Working with the
algebraically simpler probability P 2 (t), we find [37] P2 (t) =
4V 2 −Γt/~ 2 ωt e sin . ω2 2
(3.27)
In this limit, incoherent interference has vanished entirely, and the solution exhibits only the coherent behavior characteristic of Rabi oscillations. In fact, Equation (3.27) is nearly identical to the result (3.13) for an isolated double-dot system. It differs only in the presence of the exponential envelope function e −Γt/~ , which appears equally in both P1 (t) and P2 (t), signifying the statistical escape of electrons from the discrete system to the infinite-dimensional Fock space of the leads. In the case of identical dots but 2|V | < |Γ 0 |, we find a purely imaginary ω. Then [37], P2 (t) =
4V 2 −Γt/~ |ω|t e sinh2 . 2 |ω| 2
(3.28)
The lack of circular functions denotes an absence of coherent Rabi oscillations in this regime. In this limit, transport between the two dots is totally incoherent. These results possess a particularly intriguing characteristic. V is the tunneling matrix 2 element of interdot tunneling, while 2 Γ1 +Γ is approximately the frequency of tunneling ~
events coupling the dot system to the macroscopic world. One might have supposed that a competition between these two quantities would determine the extent to which the system exhibits coherent or incoherent behavior. Instead, we find it is the difference of Γn ’s that is to be compared with 4|V |. We explore this surprising result further by examining the case of identical leads, next.
50
3.4.2
Identical lead couplings
Motivated by the results of the previous subsection, we turn now to a consideration of identical lead couplings, as opposed to identical dots. Rather than require ε 1 = ε2 , we set Γ1 = Γ2 . In this case [37], ω=
. p 4V 2 + ∆2 ~
(3.29)
is again purely real, regardless of the energies of the individual dots. Fully coherent Rabi oscillation is thus recovered, and equation (3.27) once again describes the time dynamics. We can understand this surprising result, and the results of Sec. 3.4.1, in terms of measurement theory. Incoherent transport between the two dots corresponds to measurement of the two-dot system by the macroscopic bath of states. If Γ 1 = Γ2 , the presence of an electron in either dot 1 or dot 2 is seen as the same by the macroscopic environment, and so no measurement can take place regarding where the electron is. In this case, therefore, transport can only be coherent. Moreover, the Green function of time factorizes [15, 106] to G(t) = e−Γt/~ Gdots (t), so the effect of the leads is merely to contribute the decay term. As Γ0 is increased, the distinction the environment makes between the dots comes into competition with V , which tends to mix the two dot states.
3.5
Summary
Using the Green function formalism, we have solved the two-level model for the case of quantum dots in equilibrium with a macroscopic reservoir. Dyson’s equation (2.22) allowed us to exactly sum all diagrams which included interaction with the leads. Unlike most studies present in the literature, this treatment lets us consider regimes where decoherence may play a central role. Indeed, the result is that a competition between the hopping matrix element V and the difference in coupling of the two dots to the environment 2Γ0 determines the mixture of coherent and incoherent behavior, a result well understood in the context of measurement theory. We have deliberately kept the treatment simple so as not to obscure the important physics of coherence and decoherence in this system. The most significant ways we might improve the model of this chapter are the inclusion of other levels in the dots and the inclusion of interdot interactions. Although the treatment is for different physical systems,
51 the reader interested in such further steps is encouraged to examine the later chapters of this work, where both such issues are addressed more fully. Presently, we turn our attention to a natural system which, while physically quite different, nevertheless exhibits much of the same physics of the two-dot model of this chapter. The wide scope of problems that can be treated with the Dyson’s equation approach is a central aspect of its utility and power. Chapter 4 deals with the decay-out process of superdeformed nuclear bands.
52
CHAPTER 4 DECAY OF SUPERDEFORMED NUCLEI Our next topic is the decay of superdeformed nuclei. While, on the surface, the interdisciplinary leap from the time-dynamics of quantum dots may seem great, it will become clear that the two problems are, in reality, not at all dissimilar in their underlying physics. This, in itself, illustrates one of the central reasons for choosing a Green function approach: it illuminates, rather than obscures, the underlying physical principles of a problem, and so theoretical insights gained for one system are not lost when we move to the next. Superdeformed nuclei are a striking example of a counterintuitive phenomenon encountered in the study of a many-body, quantum mechanical system. If one pictures the nucleus as a rotating drop of fluid, some deformation, i.e. departure from a spherical shape to an ellipsoidal one, may be expected on purely classical grounds. The strength of the internal forces of the nucleus, however, indicate that one should expect this result to be very slight, as indeed it usually is. For certain highly excited, high angular momentum nuclei, however, a set of states is observed with approximately 2:1 major-to-minor-axis ratios. These superdeformed (SD) states, like their normally deformed (ND) cousins, form rotational bands, which are observed to persist over many states, and then suddenly decay into ND bands of lower energy. An understanding of this process becomes our current goal.
4.1
Nuclear Deformation
The departure of nuclei from a spherical shape is of enduring interest in nuclear structure theory. As a topic, it lies at the intersection of two important, extraordinarily successful, pictures of nuclear behavior, the shell model of Mayer and Jensen [107] and Bohr and Mottelson’s collective motion model [108]. Moreover, the phenomenon cannot be properly understood without taking a first step beyond these essentially single-particle models, to a picture which includes the pairing force, a “residual” many-body interaction [109]. The ground states of most nuclei are found experimentally to be somewhat ellipsoidal, simply because many-body effects make such a configuration of nucleons energetically favorable. The phenomenon can be thought of as small admixtures of spherical harmonics
53 beyond the monopole entering into the mean-field potential for nucleons, and creating a lower-energy state. This is to be distinguished from the larger deformations that fall under the heading of superdeformation: in these cases, the entire shell structure of the nucleus is different. Ground states and low-angular-momentum states are never superdeformed.
4.1.1
Normal deformation
Many nuclei, known as normally deformed, are found experimentally to possess a majorto-minor axis of about 1.3:1. In general, a nucleus with filled neutron and proton shells is spherical. As a nucleon shell begins to be filled, nuclei tend to become prolate, with one axis larger than the other two. Once the point of shell half-filling is passed, deformation is generally oblate, with one axis smaller than the other two. The starting point to understanding deformation of nuclei is the shell model, which, like the atomic shell model, centers on a mean-field picture of stability, in which the single-particle spectrum tends to clump into close-lying groups of levels called shells [107]. Nuclear shells exist for both the nucleon and proton systems, and those nuclei for which both species of shell are closed are the most stable. Shell closures occur at the “magic numbers”: for neutrons, these are 2, 8, 20, 28, 50, 82, and 126; and for protons, 2, 8, 20, 40, and 82. Experimental signals of a major shell closure include large excitation energies (see Fig. 4.1) and large proton and neutron separation energies (see Fig. 4.2) [38]. We recall from study of a quantum mechanical particle in a central, spherical potential that shells arise from the degeneracy of levels of the same principal quantum number n. Neglecting magnetic interactions, for example, the energies of the electron in a hydrogen atom are En =
−13.6eV , n2
(4.1)
while the orbital quantum number l, azimuthal quantum number m, and spin s are left undetermined by a generic measurement of energy. This leads to a shell of 2n 2 levels, degenerate until lifted by fine and hyperfine splitting. In order to examine deformation, it is advisable to decompose the spatial wavefunctions of the Hamiltonian into a spherical basis: ψk (r) =
X nlm
(k)
Cnlm Rn (r)Ylm (θ, φ),
(4.2)
54
Figure 4.1: Energies of the first excited state of nuclei with even proton number (Z) and neutron number (N ), multiplied by A 1/3 = (Z + N )1/3 , which is approximately proportional to the nuclear radius. Nuclei near shell closures in one or both fermion systems have higher excitation energies, since the ground state is so stable. From Ref. [38], based on data from Ref. [110].
Figure 4.2: Separation energies, calculated from experimental binding energies, of nucleons and α-particles from stable nuclei. The small oscillations with period 2 are due to the pairing interactions, and the larger steps are due to shell closures. From Ref. [38]
55 where the label k identifies a particular wavefunction, R n (r) is the radial wavefunction appropriate to the central potential, and Y lm (θ, φ) are the spherical harmonics. In the case that the mean-field is spherically symmetric, only one term in Eq. (4.2) will play a role, in analogy to the hydrogen atom. The occupation probability of a state k is then Pk =
Z
2 Z X X (k) 2 (k) 2 2 m Cnlm Rn (r)Yl (θ, φ) = d3 r d r Cnlm |Rn (r)| |Ylm (θ, φ)| , 3
(4.3)
nlm
nlm
where the final step makes use of the fact that the Y lm (θ, φ) are orthogonal. For a mean field axially symmetric about the z axis, the angular momentum projection on that axis ˆ z commutes with the Hamiltonian, and thus the eigenstates ψ k (r) do not mix different L (k)
m values. Thus, the only m-dependence of the C nlm can be to determine whether a state is filled or not: (k)
Cnlm
0, state unfilled . = (k) C , state filled nl
(4.4)
An important property of the spherical harmonics is [2] l X
m=−l
|Ylm (θ, φ)|2 =
2l + 1 . 4π
It is clear now that in the case of a filled shell, Z ∞ X (k) 2 Pk = (2l + 1) dr Cnl |Rn (r)|2 . nl
(4.5)
(4.6)
0
The integral over angular coordinates has vanished, and so we may read Eq. (4.6) as
follows: knowledge of how to mix the various spherical harmonics is not required to compute the probability of finding a nucleon in any single-particle eigenstate. Since the spherical harmonics are the generators of a general deformation in spherical coordinates, the state k cannot be found to be deformed by any measurement. Clearly, the spherical symmetry of Eq. (4.6) holds for filled shells, in which the second line of Eq. (4.4) is always used. If the intra-shell ordering of levels is such that nucleons fill up closed subshells of a given orbital angular momentum l before full shell closure is reached, it will result in further examples of spherical nuclei. If, however, the ordering of levels is such that groups of states with the same l do not fill easily, spherical nuclei will
56 (a)
(b)
Figure 4.3: (a) Table of nuclides showing experimentally observed regions of ground state normal deformation, from Ref. [38]. The large shaded area shows the stable nuclides, while the smaller islands are the regions of deformation. Note that regions of spherical nuclei do not begin mid-shell. (b) Portion of the table of nuclei, showing calculated value of the lowest angular momentum quantum number for which SD states occur, from Ref. [111]. In contrast with (a), SD states occur for every nuclide, but never as the ground state. The letter denoted on the chart gives the mixture of standard quadrupole superdeformation ε with a cross-axial γ mode for the lowest-energy SD state. be exclusively found near shell closures. The presence or absence of normal deformation is thus seen to be closely related to the fine structure of nuclear shells. In fact, the latter is the case, due to a many-body effect called the pairing interaction [109]. Since the shell model is a mean-field model, corrections, called residual interactions, must be included to compensate for the absence of many-body effects. The pairing interaction is one such: a strong, short-range, attractive force, so that two nucleons starting a new shell find it energetically favorable to overlap their wavefunctions as much as possible. This leads them to form pairs of azimuthal angular momentum ±|m|, with |m| as large as possible. m = 0 states, necessary to close any group of specific orbital angular momentum l, thus tend to fill last in any shell. As Fig. 4.3a shows, the conclusion that spherical nuclei occur only near major shell closures agrees well with experimental evidence.
4.1.2
Superdeformation
The phenomenon of superdeformation, like that of normal deformation, has its roots in the shell structure of nuclear ground states. In this case, it is related to new sets of magic
57 numbers and shell closures, which form as deformation is applied to the shell model. In fact, sets of shell closures called hyperdeformed states are posited to exist for even greater deformations than are currently available to experiment [112]. As an example, we consider the elliptical harmonic oscillator, which can be taken as a lowest-order approximation to an axially symmetric mean field. The Hamiltonian of such a system is HEHO =
~2 m 2 2 2 x21 + x22 , ω3 x3 + ω ⊥ + 2m 2
(4.7)
where m is the effective mass, xi is the ith body-fixed co¨ordinate, and ωi is the frequency for oscillation in the ith axis. The symmetry implies ω1 = ω2 = ω⊥ . The eigenenergies of HEHO are EnEHO 3 n⊥
=
1 n3 + 2
~ω3 + (n⊥ + 1) ~ω⊥ ,
(4.8)
where the two quantum numbers n3 and n⊥ are nonnegative integers. Figure 4.4 shows how these levels move as a function of deformation. New sets of shell closure appear at axes which are in the ratios of small integers, and are most strong when one of those integers is 1. Superdeformation, in particular, corresponds to the shell closures at axis ratios 2:1 and 1:2. In fact, superdeformation is a general prediction, not only of nuclear shell models, but of shell models in general. New sets of shell closures and magic numbers corresponding to large elliptical deformations are known in quantum dot systems [28], and they have have been predicted for systems of cylindrical symmetry, such as nanowires [113], as well. SD states have been observed across a wide variety of nuclei. They were first observed in
152 Dy
[114], and the mass region near A ≈ 150 has seen considerable experimental
interest [114–126]. Even more study has been dedicated to the A ≈ 190 region, for example Refs. [117, 127–160]. Since the discovery of these regions, lighter regions of SD nuclei have also been identified [161–169]. Figure 4.3b shows the results of calculations for the onset of superdeformation across the table of nuclides.
4.1.3
Experimental signatures of deformation
The typical SD decay experiment involves collision of two heavy ions [170]. At high angular momenta, an SD state is often yrast, meaning it is the lowest-energy state for that angular momentum. When large amounts of angular momentum stay within the system,
58
Figure 4.4: Eigenenergies of the elliptical harmonic oscillator Hamiltonian (4.7), as a function of a deformation parameter δ osc = ω⊥ω−ω3 , where the average oscillator frequency ω = 31 (2ω⊥ + ω3 ). Small-integer values of axis ratios are noted. In particular, note the new sets of shell closures and magic numbers which occur for prolate (2:1) and oblate (1:2) superdeformation. At the axis ratio 3:1, another set of hyperdeformed shell closures appears. From Ref. [108]. SD nuclei are thus often produced from such reactions. In general, two experimental signatures are usable to detect these nuclei. An intrinsic method of detecting an SD state is related to its quadrupole electric moment. We recall, from electrostatics, that in the multipole expansion of the electric field due to a charge distribution, the quadrupole moment about the body-fixed 3-axis is [19]: Q=
r
16π 5
Z
3
d r
Y20∗ (θ, φ)r 2 ρ(r)
=
Z
d3 r 3x23 − r 2 ρ(r).
(4.9)
This quantity provides a measure of a nucleus’s ellipsoidal deformation from a spherical charge distribution. Making an assumption that the nucleus is an ellipsoid of revolution, with charge inside
59 it distributed evenly, the charge density is q 4 Ze2 , r⊥ ≤ R⊥ 1 − πR⊥ R3 3 q ρ(r) = 0, r ⊥ > R⊥ 1 −
x23 R3 x23 R3
,
(4.10)
where R⊥ and R3 are the maximum extent of the nucleus along the two symmetric axes p and the third axis, respectively, and r ⊥ = x21 + x22 is the distance of the point r from
the x3 -axis. The result for the quadrupole moment is Q=
2 2 . Ze R32 − R⊥ 5
(4.11)
Note that, since the x3 -axis is defined as the body-fixed axis that is not equal to the other two in length, prolate nuclei have a positive Q, and oblate nuclei a negative one. According to Eq. (4.11), the quadrupole moment of an SD nucleus is roughly three times that of an ND state. Experiments have succeeded in using this property to identify SD bands [124, 128, 130, 134, 146–148, 152, 155, 156, 168]. It is not always convenient to measure the electric quadrupole moment of a decaying nucleus. Fortunately, superdeformed nuclei possess another experimental signature: their rotational spectrum. When a state breaks spherical symmetry, as the SD and ND states ˆ no longer commutes with the Hamiltonian. do, the orbital angular momentum operator L Thus, time evolution causes transitions between states of differing angular momentum in the body-fixed frame. These occur via electromagnetic decay, so the transition energies can be measured. The energy levels of a quantum mechanical rotor are well known [2]: EIrotor =
I(I + 1)~2 + E0rotor , 2I
(4.12)
where E0rotor is the energy of the zero-orbital-angular-momentum bandhead state, I is the orbital angular momentum quantum number about the body-fixed axis of rotation, and I is the moment of inertia about the axis of rotation. In general, I may be a function of angular momentum, so that the spectrum is distorted from the constant-I “rigid rotor” case. ND nuclei, for example, experience a phenomenon called “centrifugal stretching”, which causes the moment of inertia increases with I, so that the spectrum is compressed relative to the rigid rotor [108]. By contrast, the SD states are much more rigid against this effect [111, 171]. In this case, I may be approximated by a constant.
60 Because of their strong quadrupole deformation, SD nuclear decay is dominated by coupling to the quadrupole electric field, called E2 decay. The symmetry of E2 matrix elements is such that two units of angular of momentum are lost with each event. This, together with the nearly constant I , provides a unique character to SD rotational bands. The total energy of the photons emitted in an E2 decay from a state with angular momentum ~I is ∆E(I) =
I(I + 1) − (I − 2)(I − 1)~2 2(2I − 1)~2 = . 2I 2I
(4.13)
The peak spacing observed in a decay experiment, or “double difference”, is thus δE ≡ ∆E(I) − ∆E(I − 2) =
2(2I − 1) − 2[2(I − 2) − 1]~2 4~2 = , 2I I
(4.14)
a constant. In addition to an exceptionally large quadrupole moment, then, a second experimental signature of SD nuclei is their nearly uniform “picket fence” rotational decay spectrum (see Fig. 4.5).
4.2
Decay Process
We turn our attention now to the decay processes of SD nuclei, a subject of much experimental [118, 119, 126, 133, 134, 138, 140, 143, 146, 147, 149–151, 153–155, 157, 159, 160, 172] and theoretical interest [106, 111, 169, 170, 173–189]. While the process of decay is intriguing in itself, a thorough understanding of this phenomenon is also regarded as a promising avenue to explore aspects of microscopic nuclear structure [111].
4.2.1
Experiments
Superdeformed bands are typically populated in the laboratory by heavy-ion reactions. A collision, for example of
48 Ca
and
108 Pd
[114], is used to produce highly excited, high-
angular-momentum nuclei, which then decay to lower energies by shedding photons and nucleons. If little angular momentum is lost in the decay processes, the metastable state so resulting is likely to be superdeformed, if possible, since at high angular momenta SD states are yrast. From there, the nuclei decay down the SD rotational band, losing two units of angular momentum by E2 decay at each step [171]. These decay events are readily observed, and,
61
Figure 4.5: Example of the “picket fence” decay spectrum of SD bands, from Ref. [170]’s study of 152 Dy. Angular momentum values I are noted in units of ~. Also noteworthy is the extremely sharp loss of strength from the band, another hallmark of SD decay. with proper data analysis, the unique “picket fence” decay spectrum can be extracted. The strength of this signal is proportional to the number of nuclei in the experimental ensemble which have remained in the SD band. Nuclei may exit the band through decays into other SD bands, or into ND states which exist at the same angular momentum. In practice, SD nuclei generally travel quite easily down a rotational band for some time. These bands are observed to retain their strength for many states, with negligible losses to others. Quite suddenly, however, the band decays (see Fig. 4.6a): sometime after it has ceased to be yrast, it loses almost all of its strength over just one or two states. After a series of statistical decays through unrelated states, the nuclei then continue via decays dominated by E1 (electric dipole) transitions down the ND rotational band [171]. This whole process is outlined by Fig. 4.7 Since decay-out occurs while the nucleus is still well above the SD bandhead, it is
62
in−band SD intensity (%)
100
(a)
(b)
190
Hg Hg 194 Hg 192 Pb 194 Pb 196 Pb 192
50
0
6
10
14
18
22
spin of initial level Figure 4.6: (a) Decay profiles of several SD bands in the A ≈ 190 mass region. Note how suddenly each decay occurs. (b) The profiles of (a), but shifted in angular momentum so that the leftmost point, the last point in which the SD band is experimentally observed to retain any strength, are aligned. The profiles exhibit a universal behavior. Both graphs are from Ref. [189].
a subject of great interest. The SD state still has very high energy when it leaves the rotational band, so the density of states to which it decays is nearly constant. In short, there seems to be no reasonable explanation, a priori, why SD rotational bands lose their strength so quickly. The decay-out process becomes even more shocking when the results from different SD bands within the same mass region are compared. When corrected for the different angular momenta at which decay occurs, many decay profiles overlap almost perfectly (see Fig. 4.6b) [189]. This universal behavior suggests a strikingly elegant and simple physical processes lies behind the decay of SD bands.
Energy
63
E2
SD
ND E1 Angular Momentum
Figure 4.7: Schematic diagram of the SD decay process. The nucleus enters an SD band at a very high angular momentum, for which the state is yrast. It then decays via E2 transitions beyond the angular momentum for which ND states lie lower in energy. Suddenly, over the course of only a few states, the SD band loses its strength, via statistical transitions, to the lower-lying ND band. Finally, the nucleus continues to decay down the ND rotational band via E1 transitions.
4.2.2
Double-well paradigm
Without exception, theoretical efforts to describe the decay-out of superdeformed nuclei model the transition via a potential function of both the deformation ε and angular momentum quantum number I [189]. It was noted early in the theoretical study of the decay processes [175, 176] that a double-well potential models the phenomenon much more accurately than a traditional fission-style decay, in which the state of a single well decays through a barrier into a continuum of states (see Fig. 4.8). The most appropriate picture for SD decay is thus found to be two sets of discrete states, separated by a potential barrier, and each state broadened by a different coupling to the electromagnetic
(b)
Potential
(a)
Potential
64
ND SD
SD
Deformation
Deformation
ND Figure 4.8: Schematic examples of potentials historically used to model SD decay, as a function of deformation. The potentials change with angular momentum, as well. (a) Double-well potential, which accurately reflects the physical process [175, 176]. The states of each well, SD or ND, are broadened according to their coupling with the electromagnetic field. (b) Potentials appropriate to describe nuclear fission, but not the SD decay process. The ND states are represented by a continuum. In the case of the double-humped (dashed) barrier, the bound states of the secondary well may play a perturbative role, with their importance depending on the energy of the decaying SD state, but they are strongly broadened by the ND continuum [190]. (b) can be read as the result of taking a continuum (infinite ND well depth) approximation of (a). field. The continuum of the electromagnetic spectrum allows each state to irreversibly decay to lower-energy states. In addition to explaining the specifics of the decay process, a central goal in modeling SD decay is the extraction of the shape of the barrier. Of special interest is the behavior of the barrier as a function of angular momentum. A phenomenological understanding of how, or indeed if, the barrier height changes with I would yield significant insight into the underlying, microscopic roots of collective nuclear phenomena [178]. The double-well picture, then, as contrasted with a single well plus a continuum of ND states, has been shown to be essential to understanding how SD nuclei decay [175, 176]. Unfortunately, this insight is often trivialized due to an inclination in the community to draw analogies to incoherent fission processes. Many attempts have been made to consider SD decay as an analogue to a single- or double-humped fission barrier, failing to consider the important role coherence effects, such as the Rabi oscillations, play in the decay process. This framework has motivated continuum and many-level
65
εN
ΓN
V
ΓS
εS
SD
ND Figure 4.9: Schematic diagram of SD decay at a particular angular momentum, under the two-state approximation. V is the tunneling matrix element connecting the two states. Electromagnetic decay to lower-lying states gives each state its finite width, Γ N or ΓS . εN and εS are the energies of the two levels in the absence of V . From Ref. [187]. approximations for the ND well, which are algebraically identical to the single-well decay problem already known to be insufficient. Furthermore, such approximations are not justified experimentally. One strength of the Dyson’s equation approach is that it can include both coherent and incoherent effects on the same footing, and, in fact, determines automatically the regimes in which either, or both, are important. Thus, we are motivated to attempt a Green function solution to the two-well decay of SD nuclei. Once the general solution is reached, it will become clear that, in cases of experimental interest, the coherent effects play a significant role.
4.3
Two-State Model
The two-state model of superdeformed nuclear decay [106] was first put forward by Stafford and Barrett in 1999. The basic assumption is that only one state of the ND well, the nearest in energy to the SD state, plays an important role in the decay process. Other states are neglected. We are thus left with the system shown in Fig. 4.9 This assumption is similar to the two-level approximation made in Sec. 3.2 of this work. If the tunneling matrix elements which connect the SD state to all other ND are much less than the energy difference of those states from the SD level, the approximation is good. In this case, however, we deal with a collective mode of the nucleus, and so
66 interactions and shell closures are already included.
4.3.1
Two-state Hamiltonian
We again divide the full Hamiltonian into three separate parts: H = Hnuc + HEM + Hc ,
(4.15)
where the first term gives the dynamics of the bare nucleus without considering the electromagnetic field [106]: Hnuc = εS c†S cS + εN c†N cN + V c†S cN + c†N cS .
(4.16)
Here, V is the tunneling matrix element which takes the nucleus through the barrier from the SD state to the single ND one, εS and εN are the energies of the nucleus in the isolated (i.e., in the absence of V ) SD and ND wells, respectively, and c S and cN annhilate the nucleus from the appropriate state. As in Chapter 3, we take V to be real and positive without loss of generality. In this system, HEM and Hc , which are the Hamiltonian of the environmental electromagnetic field and its coupling with the nucleus, respectively, are the sources of decoherence. The electromagnetic field is clearly a continuum of photonic states, similar to the spin-boson Hamiltonian (3.5). We may thus denote it by HEM =
X X
n=S,N (n)
where εν
(n)† (n) ε(n) ν aν aν ,
(4.17)
(n)
annihilates a photon in that
v
is the energy of the ν th oscillator state, and aν
state. The superscript n keeps track of the state of the nucleus: even after the nucleus has left the two-well system by electromagnetic decay, it has a definite deformation, which affects the orthogonal modes of the dressed elctromagnetic vacuum. The coupling term is similarly Hc =
Xh ν
i Xh i 0 0 0 0 Vν(S ) c†S 0 cS aν(S )† + H.c. + Vν(N ) c†N 0 cN aν(N )† + H.c. ,
(4.18)
νN 0
where S 0 denotes the lower-angular-momentum SD state to which decay can occur, and similarly N 0 runs over all lower-angular-momentum ND states to which the nucleus can (n)
decay. Vν
are the matrix elements characterizing each process.
67
4.3.2
Energy broadenings
Analogous to the approach of Chapter 3, we shall use Dyson’s equation to exactly include the effects of HEM and Hc . Section 3.3.2 demonstrated how broadening functions are linked to the Hamiltonian, and there is little to be gained by reiteration in this case. Instead, we note that SD decay is a nearly perfect idealization of the broad-band approximation introduced in that section: the infinite, degenerate nature of photonic excitations provides almost exactly constant functions Γ n (E) in this case. In fact, the decay rates of the SD and ND levels are Γ S /~ and ΓN /~, respectively, and, as such, these quantities can be extracted from experiment. Γ S , in particular, is well determined by current results. When they are measured, it can be obtained directly from the quadrupole moments for the E2 transition down the SD band. In other cases, methods such as recoil-distance and Doppler-shift attenuation have been used to extract the lifetime of SD states from SD rotational band decay spectra [172]. The experimental uncertainty in ΓS is typically of the order of 10% [187]. ΓN is less well known, usually only to within a factor of 2 or more [187]. The standard method of its extraction is to assume a Fermi-gas, cranking model density of levels [191] √ √ π −5/4 2 aU U e , (4.19) ρ(U ) = 48a1/4 where a is a parameter to be fit to experiment, and U is the excitation energy above yrast. The use of a density of levels of this form is the main source of the high uncertainty in estimates of ΓN , since real nuclear spectra are often much more complicated. Parenthetically, we note that 1/ρ(U ), evaluated at the energy of the ND state, is the average level spacing in the isolated ND well, D N , which enters as a parameter into many statistical efforts to model SD decay. Like Γ N , this quantity should be treated as a theoretical estimate, albeit based on experimental data. The choice of density of levels gives a strong model-dependence to values of both Γ N and DN . Statistical E1 decay, which dominates decay from the ND well, is characterized by the well known giant dipole resonance (GDR) [38]. The mean-square transition matrix element for absorption of the GDR is 2 2 Eγ Γ2GDR 2 NZ 1 ~ e 2 , hβ|M|αi = 2 2m πΓGDR A Eγ2 − E02 + Eγ2 Γ2GDR ρ (Uβ )
(4.20)
68 where the width ΓGDR and centroid E0 are fixed by nuclear data, Uβ(α) is the energy of the destination (initial) state β(α), and E γ is the incident photon energy. Based on the results (4.19) and (4.20), Døssing and Vigezzi derived the width due to the inverse, decay process [192]: ΓE1 (U ) ≈ 4!
4 e2 1 ΓGDR N Z 3π ~c mc2 E04 A
U a
5/2
,
(4.21)
which is then evaluated at the estimated excitation energy of the ND state. Finally, the decay width of an ND state is estimated by the simple association ΓN ≈ ΓE1 .
(4.22)
U is generally renormalized to U −2∆ p , where 2∆p , known as the backshift parameter, takes the pairing interaction into account. Recently, however, detailed analyses [153, 172] have suggested that the magnitude of ∆ p should be much smaller than usually used, or neglected entirely. This, then, is an additional factor which adds uncertainty to current knowledge of ΓN . Table 4.1 gives experimental results for Γ S , as well as the Fermi-gas estimates of DN and ΓN . For a particular decay, only a single number is generally estimated in the literature for either characteristics of the ND well, Γ N or DN . Since the two-level model considers only a single level, well defined in energy, in the ND well, its use is especially appropriate.
Table 4.1: Experimental inputs and results of the two-level model, for decaying SD states, with the angular momentum quantum number I given in parentheses. Where different models or experiments have generated differing inputs, all are shown to give the fullest possible picture of the two-level model. References given in the right-most column correspond to experimental measurements of FN and ΓS , as well as measurements and estimations related to determination of Γ N and DN , as discussed in Sec. 4.3.2. The tunneling width Γ ↓ is then determined by Eq. (4.39), either directly, or in the cases of 152 Dy(26) and 192 Pb(10), statistically, as explained in Sec. 4.4.2. The Gaussian orthogonal ensemble average of the tunneling matrix element, hV i, is determined from ΓS , ΓN , DN , and Γ↓ , as given by Eq. (4.61) FN
152 Dy(28)
0.40 0.81 0.26 0.92 0.75 0.42 >0.91 0.40 0.97 0.40 ≥0.95 0.10 0.16 >0.93 0.10 0.38 >0.91 0.96
152 Dy(26) 192 Hg(12) 192 Hg(10) 192 Pb(16) 192 Pb(14) 192 Pb(12) 192 Pb(10) 192 Pb(8) 194 Hg(12) 194 Hg(10) 194 Hg(12) 194 Hg(10) 194 Hg(12) 194 Hg(10) 194 Hg(15) 194 Hg(13) 194 Hg(11) 194 Pb(10) 194 Pb(8) 194 Pb(6) 194 Pb(12) 194 Pb(10) 194 Pb(8) 194 Pb(6)
ΓS (meV) 10.0 7.0 0.128 0.050 0.487 0.266 0.132 0.048 0.016 0.097 0.039 0.108 0.046 0.086 0.033 0.230 0.110 0.048 0.045 0.014 0.003 0.125 0.045 0.014 0.003
ΓN (meV) 17. 17. 0.613 0.733 0.192 0.201 0.200 0.188 0.169 4.8 4.1 21. 20. 1.345 1.487 4.0 4.5 6.4 0.08 0.50 0.65 0.476 0.470 0.445 0.405
†Calculated statistically, as explained in Sec. 4.4.2.
DN (eV) 220. 194. 135. 89. 1,362. 1,258. 1,272. 1,410. 1,681. 16.3 26.2 344. 493. 19. 14. 26.5 19.9 7.2 21,700. 2,200. 1,400. 236. 244. 273. 333.
Γ↓ (meV) 11. 140.† 0.049 0.37 0.067 0.071 >0.44 0.072 1.6 0.060 ≥1.1 0.026 0.021 >0.71 0.0053 0.0087 >0.032 0.088
hV i (eV) 35. 120. 8.7 15. 250. 0.49 >2.1 5.0 35. 0.97 ≥3.0 0.52 0.34 >0.60 1100. 72. >77. 39.
hV i/DN 0.16 0.62 0.064 0.17 < 0.021 0.027 0.13 0.71 >0.15 0.030 >0.080 0.015 0.071 0.051 ≥0.21 0.020 0.017 >0.083 0.051 0.031 >0.055 0.12
Refs. [126] [126] [154, 189] [154, 189] [157, 172] [157, 172] [157, 172] [157, 172] [157, 172] [138, 146, 148, [138, 146, 148, [126] [126] [148, 189] [148, 189] [148, 183] [148, 183] [148, 183] [140, 149, 183, [140, 149, 183, [140, 149, 183, [172, 183] [172, 183] [172, 183] [172, 183]
183] 183]
193] 193] 193]
69
nucleus(I)
70
4.3.3
Green function treatment
Our approach to Dyson’s Equation differs slightly from the case of quantum dots (Sec. 3.3.2), for the sake clarity. Since Γ S and ΓN are a priori constants, it is quite natural to think of the isolated SD and ND wells as the components of H 0 , each with its own bath of bosonic states [188]. The energy levels ε S and εN are broadened, as is usual with states of finite lifetime, into Lorentzian Breit-Wigner resonances. In the Green function formalism, this is equivalent to a shifting of the poles below the real axis by an amount Γn 2 :
1 E−εS +i Γ2S
G0 (E) =
0 1 Γ E−εN +i 2N
0
.
(4.23)
Note that, in this approach, the two wells are completely unmixed by H 0 . Each state in Eq. (4.23) propagates and decays independently, without interaction with the other. Of the Hamiltonian (4.15), the only physics left unincluded in Eq. (4.23) is the third term of Eq. (4.16), that which allows tunneling through the barrier. It can be described by the simple retarded self-energy
Vˆ =
0
V
V
0
.
(4.24)
Dyson’s equation (2.22) then allows us to include its effects to all orders [106]:
h
ˆ G(E) = G−1 0 −V
i−1
E − εS + i Γ2S = −V
−V
i Γ2N
−1
E − εN + E − εN + i Γ2N 1 = V E − εS + i Γ2S E − εN + i Γ2N − V 2
V E − εS + i Γ2S
, (4.25)
a result which should be compared with Eq. (3.18).
4.3.4
Branching ratios
Rather than the probabilities Pn (t), which are as expected from Eqs. (3.19) and (3.20), the quantities measured by experimental studies are the fractions of nuclei which decay down either band, called the branching ratios. Table 4.1 gives the ND branching ratios
71 FN measuerd in decay experiments. The probability of a decay from a specific well n in a time dt is Γn Pn (t)dt. ~
dPndecay (t) =
(4.26)
Thus, the branching ratios are [106] Fn =
Z
∞
0
Γn = ~
dPndecay (t) X
In the open system, it is the identity
n
particle conservation.
Z
∞
dtPn (t).
(4.27)
0
Fn ≡ 1 that replaces
X
Pn (t) as a reflection of
n
Since the probabilities Pn (t) are not readily accessible by experiment, clarity is enhanced by eliminating them from our formulas. This can be readily accomplished via Parseval’s Theorem, ∞
Z
−∞
dt|F (t)|2 ≡
Z
∞
−∞
dE ˜ |F (E)|2 , 2π
(4.28)
which relates the continuous function F (t) to its Fourier transform F˜ (E). The branching ratios are thus shown to be [106] Γn Fn = ~
Z
∞
Γn dt|GnS (t)| = 2π~ −∞ 2
Z
∞
dE|GnS (E)|2 ,
(4.29)
−∞
where we have made use of the experimental fact that the nucleus is initially localized in the SD well. Either Equation (4.27) or (4.29) may be used to compute the branching ratios of the two-state model. The integral can be done analytically by the usual contour approach. The results are [106] FN =
(1 + ΓN /ΓS ) V 2 2
∆2 + Γ (1 + 4V 2 /ΓN ΓS )
(4.30)
for the ND branching ratio, and 2
FS = 1 − F N =
∆2 + Γ
1 + 4V 2 /ΓN ΓS − (1 + ΓN /ΓS ) V 2 2
∆2 + Γ (1 + 4V 2 /ΓN ΓS )
(4.31)
for the fraction of nuclei which continue down the SD band. In analogy with Chapter 3, ∆ is here defined as εN − εS , and Γ ≡
ΓN +ΓS . 2
72
4.4
Tunneling Width
We now introduce the concept of the tunneling width, Γ ↓ , which gives the net rate at which nuclei tunnel between the two wells. Fermi’s Golden Rule can be used to calculate this quantity: ↓
Γ = 2π
Z
∞
dEρS (E)V 2 ρN (E),
(4.32)
−∞
where ρn (E) =
Γn /2π (E − εn )2 + (Γn /2)2
(4.33)
is the density of states in well n. In this case, ρ n are simply the usual Breit-Wigner resonances associated with width-broadened discrete energy levels. The result for Γ ↓ is [106] Γ↓ =
2ΓV 2
2.
(4.34)
∆2 + Γ
Despite its derivation from Fermi’s Golden Rule, Equation (4.34) has meaning in an exact sense, as we shall shortly see. For this reason, we can simply take it as the definition of Γ↓ .
4.4.1
Relation between branching ratios and tunneling width
With the definition (4.34) of Γ↓ , Equation (4.30) can be rewritten into a more transparent form [187]: FN
ΓN Γ↓ / ΓN + Γ ↓ = . ΓS + ΓN Γ↓ / (ΓN + Γ↓ )
(4.35)
Likewise, Equation (4.31) becomes FS =
ΓS . ΓS + ΓN Γ↓ / (ΓN + Γ↓ )
(4.36)
Equations (4.35) and (4.36) have a clear physical interpretation. They are precisely the results we expect to find in the case of a two-step decay process: in order to leave the SD well, the nucleus must first tunnel out of the SD well, and then decay down the ND band. The problem, therefore, is, in its essence, one of series-added rates. The rate to leave the SD band is thus Γout =
1 1 + ↓ Γ ΓN
−1
=
ΓN Γ↓ , ΓN + Γ ↓
(4.37)
73 which should be compared with the rate associated with remaining within the SD band, ΓS . Together with the identity FS + FN = 1, the ratio of these two rates define the branching ratios: FN Γout ΓN Γ↓ = = FS ΓS ΓN + Γ ↓
ΓS .
(4.38)
Equations (4.35) and (4.36) follow directly. The reader may find this result surprising. The grouping of quantities defined by Eq. (4.34) to be Γ↓ plays exactly the role of a tunneling rate in the two-stage analysis of the problem. Fermi’s Golden Rule, in general accurate only to second order in V , apparently provides an exact solution to this problem. The reason for this serendipitous success of Fermi’s Golden Rule is the simplicity of the two-level model. In the general problem of N s states, the Green function consists of a denominator, in which Vˆ appears to order Ns , and a numerator, in which Vˆ can be at most order Ns −1. As we saw in Sec. 3.3.2, the role of the denominator of G(E) is solely to determine the complex Rabi frequency of the problem in the two-level case. In problems with Ns > 2, it similarly only affects the problem in ways that can be absorbed into the poles of the Green function, and thus does not determine the relevance of Fermi’s Golden Rule. In this sense, the character of the time dynamics is governed by the numerator. In the two-level problem, therefore, the Golden Rule’s inclusion of perturbation theory to O V 2 is sufficient for an exact result for Γ ↓ , although a derivation of the branching ratios themselves requires use of the information in the denominator of G(E).
4.4.2
Measurement of the tunneling width
The relation (4.35) between Γ↓ and FN can be inverted to find [187] Γ↓ =
FN ΓN ΓS . ΓN − FN (ΓN + ΓS )
(4.39)
This is a central result, as it clearly demonstrates that Γ ↓ is a measurable quantity, in the sense that a typical SD-decay experiment determines all of the parameters necessary for its knowledge. Table 4.1 gives values of Γ ↓ , as extracted from the experimental data. Equation (4.39) demonstrates the measurable nature of Γ ↓ , but its form may trouble the reader. As an energy width, Γ↓ is inherently positive definite, and thus a negative
74 value is unphysical. We must thus infer the inequality [187] FN
Γ S
FN . FS
(4.41)
Experimental results for which these two inequalities break down indicate either a departure form the validity of the two-level approximation, or that one or more of the three experimental inputs is poorly known. In fact, Table 4.1 includes two SD states,
152 Dy(26)
and
192 Pb(10),
for which the
inequalities (4.40) and (4.41) are violated. As mentioned in Sec. 4.3.2, however, the standard error in ΓN is expected to be at least a factor of 2. In both cases, the only two currently known, this factor is more than sufficient to explain the negative result of Eq. (4.39). The conclusion that a third state plays an important role in the dynamics of these decays is therefore premature, for a major factor in the appearance of an anomalous sign is likely the use of Eq. (4.19) in place of the true, unknown density of states. In cases where the positivity of Eq. (4.39) is violated, then, it behooves us to remember that, like all experimentally-based results, the value quoted for Γ N is really the centroid of a probability distribution. For want of better information, we can assume a Gaussian shape of this distribution, and take the half-width to be the estimated error [188]: − 1 P0 (ΓN ) = 0 √ e ΓN 2π
„
ΓN −Γ0 √ 0N 2Γ N
«2
,
(4.42)
where the Γ0N is the quoted experimental value. The quoted experimental values of Γ S and FN could, of course, be treated similarly, but since their uncertainties are so much less than that of ΓN , we forego this process and treat them as known quantities. The probability distribution for Γ↓ follows from P0 (ΓN ): «2 „ 0 ↓ 2 − Γ√ N −ΓN Γ /Γ dΓ N N 2Γ0 N P0 (Γ↓ ) = P0 (ΓN ) ↓ = 0 √ . e dΓ ΓN 2π
(4.43)
Here, ΓN is given in terms of Γ↓ by Eq. (4.39).
A physical value for Γ↓ can be extracted by explicitly demanding the validity of the two-level model. Requiring that inequality (4.41) be satisfied imposes a similar restriction
75 on Γ↓ : Γ↓ > Γ S
FN , FS
(4.44)
due to the dual relationship between the two quantities. This result can be “forced” onto the probability density function (4.43) by truncating it at the minimum value, Γ↓ ≤ ΓS FFNS 0, „ « 2 , ΓN −Γ0 P(Γ↓ ) = 2 √ 0N 2Γ ↓ > Γ FN A ΓN e − N Γ S FS Γ↓
where A is the new normalization constant, determined as usual by Z ∞ dΓ↓ P(Γ↓ ) = 1.
(4.45)
(4.46)
−∞
A typical value for Γ↓ , as extracted from the experimental evidence, then, is the median of the distribution (4.45). That result is given, in the cases of the two anomalous decays, in Table 4.1. The preceding analysis is not intended for the ideal case, in which the three experimentally determined quantities, ΓS , ΓN , and FN , are well known. In such cases, violations of the inequalities (4.40) and (4.41), should they occur, would indicate that the original two-level approximation is invalid. With the current state of affairs, however, the method outlined above provides a statistical approach to continuing our analysis in cases, such as
152 Dy(26)
and
192 Pb(10),
where our current poor understanding of Γ N impedes
conclusions. It is to be hoped that this approach will eventually be outmoded by an improved understanding of the spectrum to which the ND level decays, or a more direct measurement of ΓN .
4.4.3
Limits of the tunneling width
One can view the two-level approximation as one of two possible limits of the true manylevel ND spectrum. In the general case, an approximate tunneling width into the ND well Γ↓ may still be calculated via Eq. (4.32), except that we should take ρ N (E) as the many-peaked density of states of the ND well, and V 2 is now a function of energy. For an average peak spacing in the ND well D N , we observe that the two-level model is equivalent to the limit V DN , for which we recover the result (4.34) for Γ ↓ . This reasoning is similar to that of Sec. 3.2.1.
76 In the opposite limit, V DN , the result for Γ↓ is lim
V /DN →∞
Γ↓ = 2π
hV 2 i , DN
(4.47)
where h· · · i represents an energy average. This is the limit in which an infinite number of ND levels participate significantly in the decay, often called the continuum limit. It is worth noting (see Table 4.1), that, experimentally, this is never an appropriate limit, as DN Γ N
(4.48)
always holds, indicating that the correct picture is of a discrete, well separated spectrum of levels in the ND well. Furthermore, many-level models consistently find [189] DN
p hV 2 i,
(4.49)
which is the central condition for the validity of a two- or few-level model.
4.5
Statistical Theory of Tunneling
While it was shown in the previous section that Γ ↓ is an experimentally fixed quantity, a method of determining V , and thus the shape of the potential barrier, is not a priori clear. Solving Eq. (4.34) for V requires knowledge of ∆, but the experimental spectrum of the ND rotational states is not generally known. Even if it were, only an estimate of ∆ would be possible, since the quantity is a theoretical construct, representing as it does the detuning between levels in the isolated wells, not the eigenenergies of the full Hamiltonian that are available from experiment. Our answer to this conundrum will be to construct an estimate based on theoretical considerations. We make use of a common tool of random-matrix theory, the Gaussian Orthogonal Ensemble (GOE). This statistical approach allows us to find a probability distribution for ∆, from which, in turn, follows a probability distribution for V . With the current types of SD-decay experiments, this is the most successful we can hope to be at determining the tunneling matrix element.
4.5.1
Gaussian orthogonal ensemble
Developed to deal with the problem of an unknown nuclear Hamiltonian [194–197], random matrix theory has in time found use in a host of other disciplines. Its wide utility
77 [45, 198] is due to an unorthodox, yet often successful, approach to a common problem: how can one best determine the eigenvalues of a Hamiltonian about which one knows little, aside from its space-time symmetries? The answer provided by random-matrix theory is simply to ignore any other information, and focus on the statistical properties of an ensemble of Hamiltonians with the appropriate symmetries. We posit a Hamiltonian H N D , the potential VN D (ε) of which is simply the isolated ND well of the SD-decay problem. We are not really sure what shape the well has, or even, perhaps, if it has a uniquely determined shape. All we need know is that HN D possesses both time-reversal symmetry and either rotational symmetry or integral angular momentum [45]. In other words, the spectrum corresponding to H N D ’s real eigenvalues is solely governed by level repulsion. The ensemble of such matrices is known as the Gaussian Orthogonal Ensemble. A probability density function for its energy levels is given by the Wigner “surmise” [199]: P(s) =
π −πs2 /4 se . 2
(4.50)
Here s is the spacing between two adjacent levels, in units of the average spacing D N . The average level spacing, and the space-time symmetries of the problem, are the only inputs into a generic random-matrix theory problem. We shall use Eq. (4.50) to build a statistical theory of the tunneling matrix element V .
4.5.2
Implications for tunneling
The first step in building a statistical theory of V is to consider the behavior of ∆ under the assumption of GOE-distributed states in the ND well [187]. First, consider only the two ND levels which lie just above and below the SD state in energy, one of which must be the nearest-energy ND level of interest to the two-level model. The two ND states are separated by a spacing sD N , whose probability distribution is given by the Wigner surmise (4.50). Since the ND and SD states are isolated from each other, ε S is uncorrelated with the energy of either state, and the probability distribution for the detuning of the SD level with either of the two ND levels must be rectangular: s 1 |∆| Ps (∆) = Θ − . sDN 2 DN The Heaviside step function chooses the nearer of the two ND states.
(4.51)
78
Figure 4.10: Probability distributions for the two ND levels on either side of the decaying SD level within the GOE, given by Eqs. (4.52) and (4.64). The average detuning of the nearest level is h|∆|i ≡ h|∆1 |i = DN /4, and the average detuning of the next-nearest level is h|∆2 |i = 3DN /4. From Ref. [187] The Total Probability Theorem now yields a distribution for ∆ [187]: Z ∞ √ |∆| π P (∆) = dsPs (∆) P (s) = π , erfc 2DN DN 0
(4.52)
where erfc (x) is the complementary error function of x. This distribution is illustrated in Fig. 4.10. The average detuning is Z h|∆|i =
∞
d∆|∆|P (∆) =
−∞
DN . 4
(4.53)
The typical deviation of |∆| from its average, for cases of real decays, is characterized by s r Z ∞ DN 2 1 1 d∆ ∆ − P(∆) = DN − ≈ .8353h|∆|i. (4.54) σ|∆| = 2 4 3π 16 0 The fact σ|∆| . h|∆|i reassures us that h|∆|i is a meaningful measure of the detunings typical to experiment. At this point, we note that the branching ratios (4.30) and (4.31) depend on only four parameters, ∆, V , ΓS , and ΓS . The data in Table 4.1 and the results (4.53) and (4.54) indicate that, among these four values, a separation of scales exists. In general, ΓS , ΓN V, ∆,
(4.55)
79 where we assume, as implied by the fact σ ∆ ∼ h∆i, that ∆ will not stray too far from h∆i. Thus, it is to be expected that only two parameters significantly affect the decay-out process, ΓS /ΓN and V /∆. Other combinations of the four parameters are expected to have little importance in practice. The pieces are now in place to construct a probability density function for V . In general, such a function is given by d∆ , P (V ) = 2P (∆) dV
(4.56)
where 2 d∆ dV is the Jacobian associated with the change of variables. Solving the definition
of Γ↓ (4.34) for |∆| yields
|∆| =
s
2Γ Γ↓
Γ↓ Γ V − . 2 2
(4.57)
The requirement that |∆| be real yields a lower bound on V : r 1 ↓ V ≥ Vmin = Γ Γ. 2
(4.58)
Taking the derivative of Eq. (4.57) gives v u d∆ u 2ΓΓ↓ t . ↓ dV = 1 − Γ Γ2
(4.59)
2V
Within the assumption of ND states obeying the GOE, then, V is drawn from the probability distribution [187] 0, V < Vmin √ , P(V ) = |∆| 2ΓV π ↓ π , V ≥ V erfc min D Γ |∆|D N
(4.60)
N
where |∆| is evaluated according to Eq. (4.57). The average value of V is thus s " !# Z ∞ 2 Γ↓ DN Γ hV i = . dV V P(V ) = +O 4 DN 2Γ −∞ The width of the distribution (4.60) is characterized by sZ r ∞ 2 1 2 σV = dV (V − hV i) P(V ) = DN − 3π 3/2 Vmin
. hV i.
Γ↓ 32Γ
+
√ π 2 4 Γ
+O
(4.61)
p 3 Γ Γ↓ (4.62)
80 Equation (4.61) thus provides a typical value for V in the two-level model. Equation (4.60) is a central result, since it represents the maximal information we can have about V without prior knowledge of the shape of the potential. Earlier attempts to consider a statistical theory of SD decay [175, 176, 180, 184, 185] focused on average values of Γ↓ and FN . The fluctuations in FN are much larger than the mean, indicating that the average value has little meaning for comparison to experiment. Given the experimentally measured branching ratio FN , on the other hand, the approach outlined here allows the essential parameters ∆ and V to be determined, in the sense of “sharp” probability distributions whose typical values are comparable to their means.
4.6
Adding More Levels
Thus far we have treated only a two-level model of the SD decay process, and it is reasonable to ask what effect the inclusion of further ND levels has. To that end, we include discussion of the results of two investigations into similar models with more ND levels. Throughout this work, an emphasis has been the utility of adopting the minimally complex approach, while still describing realistic physics. In this light, we present investigations into three-level and infinite-level models as checks on the accuracy and limits of applicability for the two-level model.
4.6.1
Three-state model
As a first step toward determining the applicability of the two-level model, Cardamone, Stafford, and Barrett considered the addition of a third level, the energy level of the ND well next-nearest in energy to the SD level [187]. To analyze the situation, they made use of the GOE to determine likely placement of the second level, ∆ 2 ≡ εN 2 − εS . In this section, ∆ ≡ ∆1 ≡ εN 1 − εS , to make the notation uniform for clarity. Encouraged by the successes of a statistical approach previously, we again turn to the GOE to determine a probability distribution for ∆2 in analogy to P(∆1 ) given by Eq. (4.52). There are two cases of physical interest: the ND states may lie on the same side, or on opposite sides, of the SD level. Due to level repulsion, the first case is the less likely. In addition, level repulsion causes the the next-nearest ND level to be further from the SD level, on average, than it would be in the second case, that of bracketing ND levels,
81 and thus the effect of the third level on tunneling dynamics is appreciably suppressed in the same-side case relative to the bracketing case. While the Wigner surmise (4.50) does allow us to give each case its proper weight when constructing P(∆2 ), we would be including two different “regimes” of behavior in the same probability distribution. No real decay is represented by average values of a statistical distribution. Rather, we hope to create a distribution whose typical values reflect experiment. For this reason, it is undesirable to include both regimes in the same distribution; we opt instead to consider mainly the bracketing case, taking the role of “devil’s advocate” for the two-level model, and content ourselves with the knowledge that those rare decays for which the two nearest ND states fall on the same side of the SD level conform even better to the predictions of the two-level model than those cases studied here. Based on the assumption of bracketing ND levels, then, we can write down a conditional probability distribution for ∆ 2 : 1 Θ Ps (∆2 ) = sDN
|∆2 | s − DN 2
|∆2 | Θ s− DN
.
(4.63)
As in Eq. (4.51), the distribution is rectangular, since all of the ND levels’ placements are uncorrelated with each other. The two step functions specify that the level be the next-nearest neighbor. Proceeding, we arrive at the total probability distribution [187]: √ Z ∞ √ π π erf |∆2 | , P(∆2 ) = π|∆2 | − erf dsPs (∆2 )P(s) = 2DN 2 0
(4.64)
where erf(x) is the error function of x. This distribution is shown in Fig. 4.10. Its average detuning is h|∆2 |i =
Z
∞
−∞
d∆2 |∆2 |P(∆2 ) =
3DN , 4
(4.65)
as one might expect from Eq. (4.53).
Now that we have some idea where the next-nearest ND level lies, we move to the Green function treatment of the problem. The Hamiltonian of the three-level system can again be divided into three parts, as in Eq. (4.15). The first part, corresponding to the closed, nuclear system, is Hnuc =
εS c†S cS
+ εN 1 c†N 1 cN 1
+ εN 2 c†N 2 cN 2
+ V1
c†S cN 1
+
c†N 1 cS
+ V2
c†S cN 2
+
c†N 2 cS
.
(4.66)
82 Note that, since no observables depend on the phase differences of the states in the discrete system, we are free to choose both tunneling matrix elements V 1 and V2 to be real and positive, without loss of generality. This is a consequence of the fact that there is no direct tunneling between the two ND states. Also without loss of generality, we assume for notational convenience that the state N 1 is the ND state nearer in energy, and N 2 is the further. The other parts of the Hamiltonian, which describe the environmental electromagnetic field and its coupling to the discrete two-well system, are essentially the same as Eq. (4.17) and (4.18), with the modification that the sum over discrete states must include both N 1 and N 2.
Thus, the Green function of energy in the absence of inter-well tunneling is
now
1 ΓS E−ε S +i 2
G0 (E) =
0
0
0
1 Γ 1 E−εN 1 +i N 2
0
0
0
1 Γ 2 E−εN 2 +i N 2
while the self-energy for tunneling is given by 0 V 1 V2 Vˆ = V1 0 0 . V2 0 0
,
(4.67)
(4.68)
Once again including the effects of Vˆ exactly to all orders via Dyson’s Equation (2.22), we compute the full inverse Green function of energy, −V1 E − εS + i Γ2S −1 G (E) = −V1 E − εN 1 + i ΓN2 1 −V2 0
and hence the Green function itself: G(E)=
„
h“
E−εS +i
ΓS 2
”“
E−εN 1 +i
«„ « Γ 1 Γ 2 E−εN 1 +i N E−εN 2 +i N 2 2 „ « Γ 2 V1 E−εN 2 +i N 2 „ « Γ 1 V2 E−εN 1 +i N 2
ΓN 1 2
”“
E−εN 2 +i
ΓN 2 2
”
−V2 0 E − εN 2 + i ΓN2 2
,
(4.69)
“ ” “ ”i−1 Γ 2 Γ 1 +V12 E−εN 2 +i N +V22 E−εN 1 +i N × 2 2
„ « „ « Γ 2 Γ 1 V1 E−εN 2 +i N V2 E−εN 1 +i N 2 2 „ «„ « Γ Γ 2 V1 V2 E−εS +i 2S E−εN 2 +i N −V22 2 „ «„ « Γ Γ 1 V1 V2 E−εS +i 2S E−εN 1 +i N −V12 2
.
(4.70)
83
(a) ΓS = 10−3 ΓN (b) ΓS = ΓN
Figure 4.11: Numerical results comparing the two- and three-state total ND branching ratios. Superscripts indicate the number of levels included in the model. Since they decay via the same statistical process, the widths of the ND states in the three-level model are set equal, i.e. ΓN 1 = ΓN 2 = ΓN . In (a), ΓS /ΓN = 10−3 , while (b) shows the case ΓS = ΓN . In both, Γ/DN ≡ (ΓN + ΓS )/(2DN ) = 10−4 . The levels are taken at their mean detunings (4.53) and (4.65) within the GOE, under the assumption that they lie on opposite sides of the SD state, and the tunneling matrix elements are taken equal: V1 = V2 = V . From Ref. [187] The Parseval’s Theorem result, Eq. (4.29), once again yields the branching ratios. When a nucleus decays through the ND well, a measurable event, in the form of photon emission, occurs, and so decays through each of the ND states add classically. The full ND branching ratio in the three-level model is thus FN = F N 1 + F N 2 .
(4.71)
While the quantities GnS (E) are integrable analytically (e.g., by contour integration), it is more illuminating to simply plot the results. As noted in Sec. 4.5.2, only the combinations of parameters ΓS /ΓN and V /∆ are expected to be relevant to describing a particular decay. Figure 4.11 explores the parameter space numerically by plotting F N vs. 4V /h∆1 i, for cases of both large and small Γ S /ΓN . One conclusion of the numerical results is that the two-level model is, in general, well justified. For the more usual case that Γ S /ΓN is small, very little effect is found from
84 the addition of a third level. The conclusion for this regime is thus that adding levels beyond the nearest neighbors is largely superfluous to understanding the dynamics of SD decay-out [187]. The situation for ΓS /ΓN ∼ 1 is noticeably different: there is a distinct maximum in the three-level result. It is a consequence of Eq. (4.30) that as the SD width approaches the ND width, much larger tunneling matrix elements V are required to cause decay out. In analogy to Eq. (3.22), then, the separate Rabi oscillations involving N 1 and N 2 are faster. Interference between paths involving N 1 and N 2 is thus of greater significance, and the possibility for appreciable constructive interference exists. As V becomes very large, the two phases are essentially randomized with respect to each other, and interference effects become less important again. Obviously, this sort of physics is completely neglected in a two-level approach, and so some departure from that model’s results are to be expected. All things considered, however, the agreement between three- and two-level models is seen to remain quite good. A further important understanding that can be drawn from Fig. 4.11 is the physics behind the puzzling near-universality of SD decays in the A ≈ 190 region (Fig. 4.6). Table 4.1 includes a calculation of hV i/D N , and the two-level model shares in the common prediction that V increases with decreasing I. Comparing the values of hV i/D N with the graphs of Fig. 4.11, the universal character of many decays is shown to be a consequence of the steep character of FN vs. V in the relevant domain. As V increases, F N increases so quickly that the nuclei must decay over only a few states. After only one or two SD states of appreciable decay, at the usual rate of increasing V , the few-level models predict the branching ratios have become so high that essentially all strength is gone from the SD band.
4.6.2
Infinite-band approximation
Dzyublik and Utyuzh further investigated the applicability of the two-level approximation by examining the limit of many ND levels [186]. They considered an evenly spaced ND band, with each ND state coupling to the SD level with the same tunneling matrix p element, hV 2 i. Considering, as we have done, the tunneling as a perturbation, they noted that, since the ND levels are uncoupled, each contributes separately to the self-
85 energy due to tunneling: VˆSS =
X ν
|Vν |2 , E − εν + i ΓN2 ν
(4.72)
in analogy to Eq. (2.34). Here, Vν gives the coupling between the SD level and the ND state labeled by ν. After several reasonable approximations, the sum in Eq. (4.72) can be computed analytically [186]. An equally-spaced ND band is assumed, so that εν = νDN + ∆,
(4.73)
where ν runs over a set of consecutive integers. Further, the ground state of the ND well lies far below the SD level, so that the sum can be approximated by a sum on all integers. ΓN ν is set equal for each ND state. Finally, |V | 2 is approximated with its energy average,
hV 2 i. These approximations yield the result: VˆSS
∞ X
E − ∆ + i Γ2N hV 2 i πhV 2 i ≈ cot π = ΓN DN DN ν=−∞ E − νDN − ∆ + i 2
!
.
(4.74)
Dyson’s equation (2.23) includes Vˆ to all orders [186]: n o−1 GSS (E) = [G0 (E)]SS − VˆSS =
1 E − εS +
i Γ2S
−
Γ↓ 2
cot
E−∆+i π DN
ΓN 2
,
(4.75)
where G0 (E) is simply the Green function of a single width-broadened level given by Eq. (2.38). The branching ratios can be computed from Eq. (4.29). Results comparing FN in the infinite-band approximation and two-level approximation are shown in Fig. 4.12. As we have seen, the approximations of the infinite-band model tend to overestimate FN , while that of the two-level model tends to underestimate it. The two cases can thus be viewed as approximate upper and lower bounds on the true branching ratio. The proximity of the two curves, even in the unlikely, worst-case scenario shown in Fig. 4.12b, speaks to the determinative importance of the nearest-neighbor state in the decay-out process. Even when all of the states neglected by the two-level model are included, to all orders in perturbation theory, their contribution to the main experimentally observed quantity is slight.
86
1.0
1.0
FN
FN
0.5
0.5
0.2
0
0.4
√
(a)
hV 2 i/∆
0
0.2
0.4
√
(b)
hV 2 i/∆
Figure 4.12: Comparison of results of the infinite-level model (solid curves) of Ref. [186] and the two-level model (dotted curves). (a) shows the result when ∆ is taken at its typical value in the GOE, h∆i = D4N , while (b) shows the result for ∆ = D2N , its maximum value consistent with the model of Ref. [186]. In both graphs, adapted from Ref. [186], DN = 100eV, ΓS = 0.1meV, and ΓN = 10mev.
4.7
Summary
As this chapter has shown, the Green function formalism can be employed to solve the double-well model of decay-out of SD bands in several different approximations. The most important of these is the two-level model, in which only one level in each well participates meaningfully in the dynamics. This model yields a very clear picture of the decay process, as well as several important results. It explains why the decay profile of A ≈ 190 nuclei are all so similar, and why the decay-out happens so quickly. Furthermore, the tunneling width Γ↓ takes on a rigorous, experimentally measurable meaning in the two-level model. The rate Γ↓ /~ is the net rate for tunneling through the barrier between SD and ND wells. Because of its simplicity, a statistical theory of tunneling between the two levels can be extracted from the two-level model. Once a statistical assumption for the probability distribution of level spacings in the ND well is made, for example via the Wigner surmise of random matrix theory, a probability distribution for the tunneling matrix element V can be calculated. This represents the maximal knowledge extractable for the current style of SD-decay experiments, and thus a great success. Knowledge of V can be used to
87 determine the shape of the barrier between SD and ND wells, which in turn may speak volumes about nuclear structure. The Green function formalism, considered as a controlled expansion in the number of levels, further predicts the accuracy of the two-level model. Including a third level is a numerically and analytically simple way to do this; the result is that the two-level approximation is excellent for small Γ S /ΓN , and remains quite good over the full range of that ratio which occurs in nature. When Γ S /ΓN approaches the maximum of its known range, as is the case in the A ≈ 150 region of SD nuclei, the stronger couplings V 1 and V2 can lead to important interference effects, which cause some modification to F N . The solution of an infinite-ND-band model via the Green function approach has also been presented. With various reasonable assumptions, this model is also solvable exactly. It, in essence, predicts an approximate upper bound on experimental results, whereas the two-level model gives an approximate lower bound, which is expected to be especially accurate in the 190 mass region, where interference effects are almost absent. That the two models hardly differ in their calculated branching ratios is a great victory for the extraordinarily transparent and simple two-level model. This and the previous chapter have shown how powerful the Green function approach can be. The use of Dyson’s equation to take perturbations into account exactly is especially useful when appropriate approximations are available to simplify a problem to its bare essentials. The next chapter will take an important step beyond what we have studied so far, to expand the equilibrium theory described in Chapter 2 to cases where timereversal symmetry is violated. Such a theory is essential to describing non-equilibrium processes, such as charge transport at finite voltage.
88
CHAPTER 5 MOLECULAR ELECTRONICS As a final example of the utility, and versatility, of the Dyson’s equation approach to mesoscopic systems, we shall study the physics of small, single-molecular devices. Within the last decade, experimentalists have grown increasingly adept at making electrical contacts to single molecules, and at fashioning such devices with specific properties. It has not, of course, escaped the notice of the community, or the world at large, that such devices may well play an important role in the technology of the near future. Moreover, the advent of such techniques has opened the gates to a host of systems of basic scientific interest. Molecular electronic systems are an elegant physical example of the archetype at the heart of this work: the coherent quantum system which interfaces with the classical environment. While the molecules themselves are, generally speaking, good quantum mechanical systems, the leads introduce decoherence and dephasing into the system. As we shall see, not only are these effects often non-negligible, but they can give rise to interesting, and useful, physical phenomena in their own right. For this reason, the method explored in previous chapters is ideal to apply to molecular electronic systems. Treating exactly, as it does, the decoherent effects of external systems on the coherent dynamics of electrons in the molecule, Dyson’s equation obviates the need for a priori assumption of a system’s character. Thus, if interesting physics is to be found from coherent or incoherent behavior, or from a combination of these effects, the approach can reveal it.
5.1
Fabrication of Single-Molecular Systems
The conductance theory of molecular electronic systems is, in many ways, simply an idealization of that of semiconductor nanostructures [4, 200]. Atomic site orbitals, of known number and character, take the place of energy levels in each dot, and the parameters of the molecular system are generally well known through chemistry and chemical physics. From a fabrication point of view, however, the problems are entirely different. A single molecule must be suspended between at least two electrically active surfaces, which serve as the macroscopic leads (see Fig. 5.1). Although simply described in words, this system
89
Figure 5.1: (color) Artist’s conception of the molecular device proposed in this chapter, the Quantum Interference Effect Transistor. The small molecular component is in contact with three macroscopic leads. The colored spheres represent individual carbon (green), hydrogen (purple), and sulfur (yellow) atoms, while the three gold structures are metallic contacts. Image by Helen Giesel. is obviously a major challenge to create in practice. Throughout the last decade, however, it has been done, and with increasing control and precision.
5.1.1
Scanning-tunneling microscopic techniques
The first successful investigations of the conductance properties of single molecules were made with a scanning-tunneling microscope (STM). The device, a central tool in today’s experimental condensed matter world, consists of two electrically active surfaces: a monocrystalline substrate, above which a metallic, atomic-scale tip can be positioned with a precision of fractions of an angstrom in three dimensions [201]. The relative bias of the two surfaces is controllable, so that when they are brought near to each other, a tunnel junction is formed.
90 Although the device is famous for its remarkable topographic images, it is also ideal for measurements of molecular electronic properties. The tip is positioned over a molecule deposited on the lower surface, and biased relative to the molecule and substrate. A twoterminal current measurement can thus be made across an individual molecule, and the differential conductance found. Monatomic molecules were the first investigated using this method [202–205]. While STMs are remarkably capable of positioning single atoms, however, difficulty in expanding the technique arose from their inability to manipulate larger molecules. In particular, their was no clear way to make a single molecule “stand up”, so that the STM would measure conductance through the whole molecule. For this reason, the first successful STM measurements of conductance through larger molecules were on the highly symmetric and stable C60 system [206, 207]. A further development came with the use of the self-assembled monolayer (SAM), a mechanically stable single layer of molecular conductors which automatically construct themselves in a parallel arrangement [208, 209]. Many earlier measurements had been made of the conductance of entire monolayers, but use of an STM tip as the second terminal allowed construction of circuits containing only a few, parallel molecular elements. The first such experiments, shown schematically in Fig. 5.2a, made use of a metallic nanograin contact between tip and molecules [210, 211]. As facility with the technique grew, it became possible to contact single molecules of the SAM directly with an atomically sharp tip [212]. Originally, this was achieved through use of a secondary, shorter SAM, which was used to “prop up” the conducter, as in Fig. 5.2b.
5.1.2
Mechanically controllable break junction
A second method of making conductance measurements on single molecules, making use of the experimental technique called the mechanically controllable break junction (MCBJ) and shown in Fig. 5.3, has also seen much success. The MCBJ is simply a well controlled method of fracturing a thin (∼ .1mm) metallic wire [213–215]. It produces two clean, atomically sharp contacts, whose spacing can be adjusted on the atomic scale. If a SAM is allowed to grow in the system before breaking, a single-molecular junction can be fabricated [216–218].
91
(a)
(b)
Figure 5.2: Schematic diagrams of methods by which an STM has been used to make conductance measurements on small molecules. (a), from Ref. [211], shows a gold nanograin being used to contact a small number of molecules in a SAM. (b), from Ref. [212], shows the method of using an auxiliary SAM to support a single molecule for STM conductance measurement.
(b)
(a)
Figure 5.3: MCBJ technique. (a) Scanning electron micrograph of a lithographically constructed gold MCBJ, before breaking. The scale bar is ∼ 1µm. From [219]. (b) Schematic of MCBJ device used in a single-molecule conductance experiment. The piezo “e” is used to bend the bar “a” and fracture the notched gold wire “c”. The SAM is formed from the solution “f”. From [216].
92
5.1.3
Other techniques
A few other techniques have seen success in producing single-molecular junctions, but they are of less general use than the STM and MCBJ methods. Conventional lithographic contacts have been used with much success to measure the electrical properties of carbon nanotubes [220–222]. This approach has the advantage that multiple-lead measurements are straightforward, and four-terminal experiments have been successful [221]. Further, an atomic force microscope can be used to simultaneously take topographic and electrical data [220]. Nevertheless, simple lithography is obviously not appropriate for generic molecules, which are much smaller than nanotubes. As of yet, no general technique exists for connecting more than two leads to a small molecule. Recent success has been reported, however, in combining the STM technique with a nearby single-atom probe, whose state was shown to affect the molecule suspended in the STM tunnel junction [223]. Although this technique currently only provides an electrostatic interaction, it is a promising avenue of exploration toward eventual multiplelead configurations. Another is to combine the STM and MCBJ techniques.
5.2
Modeling Molecular Electronics
The success of the Dyson’s equation approach at modeling the systems of the previous chapters motivates its application to the issues of molecular electronics. Like the others, these systems possess both a discrete, fully quantum-mechanical component, as well as the classical leads. One important change in our model at this stage, however, will be in the addition of a higher-order term to the discrete system’s Hamiltonian, representing the Coulomb force between electrons. In this work, a self-consistent mean-field picture will be used to treat this physics. Another difference from previous chapters is the essential non-equilibrium nature of a molecular electronic system modeled at nonzero voltage. Since time-reversal symmetry, an assumption of the derivations of Sec. 2.1, is not valid in such a system, the retarded Green function and self-energy do not provide a full picture of the system’s time dynamics. Later in this section, we shall turn to the Keldysh non-equilibrium Green function to move beyond this limitation.
93
5.2.1
Hamiltonian
We write the Hamiltonian of this system, in the usual way, as the sum of three terms: H = Hmol + Hleads + Htun .
(5.1)
The first is the extended Hubbard model molecular Hamiltonian [224] Hmol =
X nσ
εn d†nσ dnσ −
X
hnmiσ
XU nm tnm d†nσ dnσ + H.c. + Qn Qm , 2 nm
(5.2)
where dnσ annihilates an electron on atomic site n with spin σ, ε n are the atomic site energies, and tnm are the tunneling matrix elements. The final term of Eq. (5.2) contains intersite and same-site Coulomb interactions, as well as the electrostatic effects of the leads. The interaction energies are modeled according to the Ohno parameterization [225, 226] 11.13eV Unm = q 2 , 1 + 0.6117 Rnm /˚ A
(5.3)
where Rnm is the distance between sites n and m. Qn =
X σ
d†nσ dnσ −
X Cnα Vα α
e
−1
(5.4)
is the effective charge operator for atomic site n, where the second term represents the polarization charge on site n due to capacitive coupling with lead α. Here V α is the voltage on lead α, and Cnα is the capacitance between site n and α, chosen to correspond with the interaction energies of Eq. (5.3). That is [18], C = U −1 /e2 ,
(5.5)
where C and U are the full capacitance and interaction matrices, respectively, each of which includes the leads as well as the atomic sites. This amounts to an approximation that the presence of a macroscopic lead does not alter the internal electrostatics of the molecule or other leads too strongly, which is consistent with the general point of view taken in this work that correlations between the continua and discrete systems are negligible. The final degrees of freedom in the lead site-capacitances C nα are determined by the locations of the leads [227].
94 With lead-site interactions treated at the level of capacitances, the electronic situation of the leads is completely determined by the externally controlled voltages V α , along with their Fermi energies and temperatures. Each lead possesses a continuum of states, so that their Hamiltonian is Hleads =
XX
k c†kσ ckσ ,
(5.6)
α k∈α σ
where k is the energy of the single-particle level k in lead α, and the c kσ are the annihilation operators for the states in the leads. Tunneling between molecule and leads is provided by the final term of the Hamiltonian, Htun =
XX
hαai k∈α σ
Vak d†aσ ckσ + H.c. .
(5.7)
Vak are the tunneling matrix elements for moving from a level k within lead α to the nearby site a. Coupling of the leads to the molecule via inert molecular chains, as may be desirable for fabrication purposes, can be included in the effective V nk , as can the effect of any substituents used to bond the leads to the molecule [227]. In equilibrium, this system is directly analogous to those of the previous chapters. If no lead couples to more than one site, the self-energy due to the leads is X i Γα (E)δna , Σnm (E) = − δnm 2
(5.8)
haαi
by analogy to Eq. (2.36). Here the notation haαi refers to sites a that tunnel with lead α. The tunneling rate is given by Eq. (2.37): Γα (E) = 2π
X
k∈α σ
|Vnk |2 δ E − ε0k .
(5.9)
As usual, we shall take the broad-band limit in the leads, and approximate Γ α (E) with a constant parameter characterizing the lead-site coupling, which shifts the poles of the Green function into the complex plane. It is important to note that through this process the density of states (2.41) becomes a continuous, width-broadened function. Due to the open nature of the system, electrons can occupy all energies [227].
95
5.2.2
Hartree-Fock approximation
Since we wish to calculate the response of molecular electronic systems to finite bias, self-consistency requires a treatment of electron-electron interactions, represented by the third term of Eq. (5.2). Assuming that many-body correlations do not play a significant role, a qualitatively accurate picture of the physics is attainable through a mean-field approach. To treat Coulomb interactions, we choose the well known Hartree-Fock approximation [227], E E D D d†nσ dnσ d†mρ dmρ ∼ = d†mρ dmρ d†nσ dnσ − d†nρ dnσ d†nσ dmρ δσρ .
(5.10)
This result can be viewed as a consequence of Wick’s theorem (2.14), although not rigorous. Nevertheless, the large body of previous work indicates that, in general, this approximation is justified at the qualitative level important to our investigation. Application of Eq. (5.10) to Hmol yields its Hartree-Fock approximation [227], HF Hmol
=
X nσ
εn −
X mα
! X Cnα Vα Unm d†nσ dnσ + tnm d†nσ dmσ + H.c. e hnmiσ D E D E X Unm d†mρ dmρ d†nσ dnσ − d†mρ dnσ d†nσ dmρ δσρ . (5.11) + nσ mρ
In this approximation, the retarded molecular Green function is Gmol (E) =
5.2.3
1 E−
HF Hmol
+ i0+
.
(5.12)
Non-equilibrium Green function theory
Realistic modeling of a molecular device requires a consideration of the molecule’s current response to finite voltages. Such a system clearly does not posses the time-reversal symmetry exploited in Sec. 2.1.1, and so the conclusions of that analysis must be revisited. In Sec. 2.1, we considered the effects of adding an additional piece to the Hamiltonian by adiabatically turning it on at t = −∞, time-evolving forward to t = ∞, and adiabatically switching it off again. If the Hamiltonian possesses time-reversal symmetry, the ending state is the same as the starting one, and so this is a prescription for taking an expectation value. In the more general case, there is no guarantee that the two states are
96 the same, so it is inappropriate to use them both in the calculation of expectation values. The analysis of Sec. 2.1.1 breaks down from the point of Eq. (2.9). The breakthrough of Keldysh [228] was to construct a rigorous mapping of the nonequilibrium problem to an equilibrium one of larger Hilbert space, and construct the Green function in the new space. The question of whether the initial and final state are the same, after all, is merely a question of boundary values, so we should expect the general approach of Green function theory to remain valid. The problem is to evaluate the matrix elements of Eq. (2.8) without appealing to time-reversal symmetry. Clearly, a simple way to do this is to move forward from time −∞ to t, and then turn around and go back to −∞ again. In fact, we could even go on past t on the first leg of the journey, and turn around some time later, as long as we are careful to consider the possibility that the operator O(t) might be evaluated on either the outgoing or return trip. If we wish to be able to calculate hO(t)i for all experimentally accessible times, then, it would behoove us to construct a theory in which we wait until t → ∞ to turn around. This is the root of the time-loop contour which lies at the heart of Keldysh theory. On this contour, shown in Fig. 5.4, the complete discussion of Sec. 2.1 holds, where the parameter s, defined by dt, first half of contour , ds = −dt, second half of contour
(5.13)
replaces t, and a contour-ordering operator T s replaces T. In particular, Equation (2.13) becomes ∞ X i n − ~ n=0 ∞ X i n − S = ~ n=0
Ts [OI (s)S] =
1 n! 1 n!
Z
∞
−∞ Z ∞ −∞
ds1 · · · ds1 · · ·
Z
∞
−∞ Z ∞ −∞
dsn Ts [OI (s)HI (s1 ) · · · HI (sn )]
.
dsn Ts [HI (s1 ) · · · HI (sn )] (5.14)
Although we focused on the retarded quantities, the derivation of Dyson’s equation in Sec. 2.1.2 applies equally well to the time-ordered and anti-time-ordered Green functions [9]: D h iE Gtnm (t1 , t2 ) = −i T an (t2 )a†m (t1 ) ,
D h iE Gtnm (t1 , t2 ) = −i T an (t2 )a†m (t1 ) ,
(5.15)
97
t s Figure 5.4: The Keldysh contour, used to map a non-time-reversal-symmetric problem onto a symmetric one in a larger space. Adiabatic switching on and off still occurs at t = ±∞. The contour consists of two branches, one which time evolves forward from t = −∞ to t = ∞, and one which evolves backward from ∞ to −∞. When the two branches are taken in sequence, contour-reversal symmetry holds. The Green function on this contour, which obeys Dyson’s equation, consists of parts which contour evolve within each branch, as well as parts which take the system between the two. so long as the appropriate self-energies are used. Here T is an ordering operator, like T and N, that places later operators to the right. Thus, in the non-equilibrium case, the contour-ordered Green function D h iE Gsnm (s1 , s2 ) = −i Ts an (s2 )a†m (s1 )
(5.16)
obeys Dyson’s equation. Experiments, of course, are not done on time-loop contours, and we must make contact with real notions of time and energy. We note that, where solution of an equilibrium problem required only one Green function, it is clear from Eq. (5.16) that three independent Green functions are required in the non-equilibrium case: G s contains pieces which propagate forward in time along the first branch of the contour, pieces which go backward in time along the second branch, and pieces which move the system from the first branch to the second branch. It is often convenient to choose a 2 × 2 matrix representation, in which one of the elements is not independent: Green functions of different representations are then connected by similarity transforms [229]. The representation most useful to us is due to Craig [230]: Gt −G< , G= G> −Gt
(5.17)
where, in the time domain,
D E † G< (t , t ) ≡ i a (t )a (t ) , 1 2 1 n 2 nm m
D E † G> (t , t ) ≡ −i a (t )a (t ) . 1 2 n 2 1 nm m
(5.18)
98 We label the elements of the self-energy correspondingly t < Σ −Σ . Σ= Σ> −Σt
(5.19)
In the event that we can consider only stationary, non-transient behavior in the system, it remains meaningful to work in the energy domain. Dyson’s equation then reads 0 ˜ ˜ G(E) = G0 (E) + G0 (E)Σ(E)G(E) = G0 (E) + G(E)Σ(E)G (E),
(5.20)
which represents four coupled equations. Of the different elements, we shall be most interested in G < , since we note that it is, in fact, precisely the two-time correlation function. Returning to the definitions (2.17), (5.15), and (5.18) of Green functions allows us to solve Eq. (5.20) and write the answer in terms of retarded functions of energy [231]: h i G< (E) = [1 + G(E)Σ(E)] G0< (E) 1 + Σ† (E)G† (E) + G(E)Σ< (E)G† (E).
(5.21)
This result is known as the Keldysh equation. Typically, the self-energy is used to treat the source of non-equilibrium behavior in the system, for example leads at a finite voltage. When this is the case, the first term of Eq. (5.21) consists of entirely equilibrium quantities. As such, it can be fixed by comparison with a related equilibrium problem.
5.2.4
Equal-time correlation functions
Equation (5.11) gives an approximate molecular Hamiltonian in terms of the equal-time D E correlation functions d†nσ dmρ . Through the Keldysh approach, we now compute an inverse expression for the correlation functions in terms of the Hamiltonian. This selfconsistent loop can then be evaluated numerically. As noted in the previous section, correlation functions arise naturally in the Keldysh formalism: G< nσ,mρ (t1 , t2 )
≡i
D
d†mσ (t1 )dnρ (t2 )
E
=
Z
∞
−∞
dE < G (E)e−iω(t2 −t1 )/~ 2π nσ,mρ
(5.22)
G< (E) is given by the Keldysh result (5.21). −iΣ < gives the rate for electrons to enter the molecule from each lead [4, 9]: Σ< ab (E) = iδab
X
hαai
Γα (E)fα (E)
(5.23)
99 where the Fermi distribution function for electrons in lead α is fα (E) =
1 1+
e(E−µα )/kB T
,
(5.24)
with Boltzmann’s constant kB , lead temperature T , and electrochemical potential µ α in lead α. The first term of Eq. (5.21) is fixed by the equilibrium conditions of the system. In terms of the correlation functions, then, it corresponds to a simple gate voltage applied to the entire molecule. It can be thus be absorbed into the atomic site energies in the molecule. The remaining term of the Keldysh result, inserted into Eq. (5.22), yields [227] D E X Z ∞ dE † (5.25) Gnσ,aσ0 (E)G∗aσ0 ,mρ (E)Γα (E)fα (E). dmσ dnρ = −∞ 2π hαai
Although the retarded Green function G(E) alone no longer completely determines the time-dynamics of the system, Dyson’s equation remains true for it. Thus, it is given by Eq. (2.23): G−1 (E) = G−1 mol (E) − Σ(E),
(5.26)
which completes the self-consistent loop.
5.2.5
Landauer-B¨ uttiker formalism
The observables of primary interest in molecular electronic systems are the currents in each lead. A formalism to treat this topic was originally developed as part of scattering matrix theory by Landauer [232, 233] and B¨ uttiker [234] for nanoscale systems, and later shown to be consistent with the Keldysh approach [235]. This formalism has seen much success in the molecular electronics literature, as well [200]. To derive the multi-terminal current formula from the Keldysh formalism, we begin with the definition of current in lead α, ie dNα = − h[H, Nα ]i , Iα (t) ≡ −e dt ~
(5.27)
where Nα is the number of electrons in lead α. Our task is therefore to evaluate this expectation value. Htun is the only part of the Hamiltonian which does not commute with Nα . Using the fermion anticommutator relations, the commutator is evaluated: D E 2e X X ∗ Iα (t) = Re iVak c†kσ (t)daσ (t) . (5.28) ~ hαai k∈α σ
100 D E i c†kσ (t)daσ (t) is an equal-time “
