Single Electron Tunneling at Large Conductance

Single Electron Tunneling at Large Conductance Inaugural{Dissertation zur Erlangung der Doktorwurde der Albert{Ludwigs{Universitat Freiburg i. Br.

vorgelegt von

Georg Goppert aus Schweighausen Januar 2000

Dekan Leiter der Arbeit Referent Koreferent

: : : :

Prof. Dr. K. Konigsmann Prof. Dr. H. Grabert Prof. Dr. H. Grabert Prof. J. St. Briggs

Tag der Verkundigung des Prufungsergebnisses: 01.03.2000

Contents 1 Introduction 1.1 1.2 1.3 1.4 1.5 1.6

Charging Eects in Metallic Nanostructures Single Electron Box . . . . . . . . . . . . . Single Electron Transistor . . . . . . . . . . Finite Tunneling Conductance . . . . . . . . Overview . . . . . . . . . . . . . . . . . . . Publications . . . . . . . . . . . . . . . . . .

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2.1 System and Model Hamiltonian . . . . . . . . . . . 2.2 Perturbation Expansion and Diagrammatic Rules . 2.2.1 Perturbation Expansion . . . . . . . . . . . 2.2.2 Diagrammatic Rules . . . . . . . . . . . . . 2.2.3 Re ected Graphs . . . . . . . . . . . . . . . 2.2.4 Insertions . . . . . . . . . . . . . . . . . . . 2.3 Zero Temperature Limit and Third Order . . . . . 2.3.1 Zero Temperature Limit . . . . . . . . . . . 2.3.2 First and Second Order Results . . . . . . . 2.3.3 Third Order . . . . . . . . . . . . . . . . . . 2.4 Average Charge Number . . . . . . . . . . . . . . . 2.5 Degeneracy Point . . . . . . . . . . . . . . . . . . . 2.6 Discussion of Results . . . . . . . . . . . . . . . . . 2.7 Quantum Monte Carlo . . . . . . . . . . . . . . . . 2.8 Finite Channel Numbers . . . . . . . . . . . . . . . 2.9 Appendix . . . . . . . . . . . . . . . . . . . . . . . 2.9.1 Third Order Integrands . . . . . . . . . . . 2.9.2 Third Order Result . . . . . . . . . . . . . .

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3 The Semiclassical Approach

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2 Perturbation Expansion

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3.1 Model and General Method . . . . . . . . . . . . . . . . . 3.2 Tunnel Junction with Environment . . . . . . . . . . . . . 3.2.1 Generating Functional . . . . . . . . . . . . . . . . 3.2.2 Conductance . . . . . . . . . . . . . . . . . . . . . 3.2.3 Semiclassical Limit . . . . . . . . . . . . . . . . . . 3.2.4 Results and Comparison with Experimental Data . 3.3 Array of Junctions with Environment . . . . . . . . . . . 3.3.1 Generating Functional and Conductance . . . . . . 3

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15 17 17 21 25 25 26 26 27 27 29 30 33 35 38 45 45 47

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51 53 53 57 58 61 64 64

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CONTENTS 3.3.2 Semiclassical Limit . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Discussion of Results . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Single Electron Transistor . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Generating Functional and Conductance . . . . . . . . . . . . . 3.4.2 Semiclassical Limit . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Discussion of Results and Comparison with Experimental Data

4 Quantum Monte Carlo Approach

4.1 General Method for Transport Coecients . . . . . . . . . . . . . . . . 4.2 Singular Value Decomposition . . . . . . . . . . . . . . . . . . . . . . . 4.3 Single Electron Transistor . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 System and General Considerations . . . . . . . . . . . . . . . 4.3.2 Quantum Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Singular Value Decomposition . . . . . . . . . . . . . . . . . . . 4.3.4 Discussion of Results and Comparison with Experimental Data

5 Conclusions Bibliography

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Chapter 1

Introduction 1.1 Charging Eects in Metallic Nanostructures In this work we study electron tunneling in nanofabricated metallic structures. Speci cally we consider systems containing metallic tunnel junctions that consist of two metallic electrodes separated by a thin oxide layer, schematically depicted in Fig. 1.1. Electrons in the leads are described by Fermi liquid theory and considered as free quasiparticles where the charge is screened and Coulomb energy is included implicitly in the dispersion relation. However, when an electron tunnels through the barrier it leaves the positively charged screening cloud behind and will be screened at the other side. Therefore, the electron tunneling is a charged particle and the Coulomb energy has to be taken into account. Since times for dressing and undressing of electrons are of the order of the inverse plasma frequency and tunneling times [1] in metallic tunnel junctions are much shorter than other relevant time scales [2] considered in this thesis, these processes can be assumed to be instantaneous. Provided the screening length in the leads is much smaller than the barrier thickness and sample size the electrostatic energy can be described in terms of a geometrical capacitance CT formed at the oxide layer [3]. The corresponding charging energy EC = e2 =2CT being the energy needed to charge the geometrical capacitance by one electron serves as relevant energy scale of the system. If an electron in the uncharged system tunnels from the left electrode through the barrier to the right electrode, the energy of the system is shifted by this amount. Therefore, at low temperatures T EC =kB , where kB is the Boltzmann factor, this transition is forbidden energetically and one expects a complete suppression of tunneling probabilities { the Coulomb blockade. At higher temperatures, however, the levels are statistically occupied and charging eects are smeared out [2]. Therefore, to observe strong Coulomb blockade phenomena we nd the condition EC 1 where = 1=kB T is the inverse temperature. To estimate the temperature range where such

metal

metal

Figure 1.1: Metallic tunnel junction. 5

6

CHAPTER 1. INTRODUCTION metallic lead electrode oxyde layer

metallic island

metallic gate electrode

Figure 1.2: Experimental realization of a single electron box. eects are observable, we consider tunnel junctions that are usually made by two{angle shadow evaporation [4]. Firstly, a mask is prepared by electron beam lithography and then a lower metal lm is vapor deposited at a certain angle corresponding to the left lead electrode in Fig. 1.2. After oxidizing the aluminum lm the second metal is vaporized from a dierent angle. This way aluminum lms overlap by a de nite small area of about 50nm 50nm and are separated by a thin oxide layer of about 10 A thickness, cf. Fig. 1.2. A rough estimation leads to a capacitance of 1fF and a corresponding temperature of 1K showing that in metallic nanostructures Coulomb blockade eects are observable at very low temperatures only. However, ongoing progress in lithographical fabrication techniques decreases the length scales continuously, and since the inverse capacitance and therefore the temperatures in the system scale with inverse length squared these eects become more and more observable at higher temperatures. Actually, in granular media charging eects appear even at room temperature but the contacts cannot be fabricated reproducibly. A necessary requirement imposed by these rough estimations is that charge on the electrodes has to be quantized. However, at nite phenomenological tunneling conductance GT an average current I = V GT , with V = e=C , ows back implying a nite dwell time for the additional electron in the electrode. The corresponding energy uncertainty has to be small compared to the charging energy leading to GT GK for charge to be quantized, where GK = e2 =h is the conductance quantum. Likewise, in experimental measurements an electrical circuit connected to a tunnel junction leads to a nite current ow and we nd the corresponding condition G GK , where G means the absolute value of the frequency dependent conductance of the outer circuit at the measurement frequency. Here G depends on the geometrical junction capacitance and the whole electromagnetic environment including the metallic leads. Provided the sample size is small enough that the diusion time of the electrons in the metals is much smaller than the dwell time of the electron in the lead, the electrodes can be considered as ideal conductors and one can concentrate on speci c electromagnetic environments to explore charging eects. The simplest environment one can think of is a capacitance that completely blocks charge transfer in dc measurements, or more accurately almost blocks the charge transfer in real low{frequency lock{ in measurements. This setup forms the so called single electron box (SEB) depicted schematically in Fig. 1.2 showing an isolated metallic lm where the excess charge is strictly quantized and

1.2. SINGLE ELECTRON BOX

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a)

EC/4

b)

n=1

n=0

En

CgUg/e

2 1

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0

1

2

CgUg/e

0

1

2

CgUg/e

G/Gcl 0.5

Figure 1.3: a) Electrostatic energy parabolas of the single electron box and the single electron transistor at vanishing driving voltage in dependence on the gate voltage Ug . In the weak tunneling regime and for zero temperature the average island charge number hni for the single electron box and the dierential conductance G of the single electron transistor are depicted in b) and c), respectively. therefore theoretically provides the simplest system showing Coulomb blockade phenomena.

1.2 Single Electron Box We consider a system consisting of a single tunnel junction where one metallic lm is coupled to a gate electrode purely capacitively, cf. Fig. 1.2. The excess charge on the metallic island in between the two leads is strictly quantized. Further there are two capacitances: the one characterizing the geometrical capacitance at the barrier CT and the other coupling to the external gate electrode Cg . Purely electrostatic considerations lead to the charging energy EC = e2 =2C that depends on the island capacitance C = CT + Cg only. Considering the entire electrostatic energy EC (n , Cg Ug =e)2 as a function of the applied gate voltage Ug , we get a family of parabolas labeled by the island excess charge number n. In Fig. 1.3 a) we show the energy parabolas for n = 0; 1; 2. The energies are degenerate at the values Cg Ug =e = n 21 for all integers n. At low temperatures and small tunneling conductance the system will follow the

8

CHAPTER 1. INTRODUCTION

CgUg/e

Figure 1.4: Experimental data by Lafarge et al. [5] for the average island charge number hni in the weak tunneling regime for two dierent temperatures in dependence on the dimensionless gate voltage ng = Cg Ug =e. minimum of the electrostatic energy. Therefore, at the degeneracy points where two parabolas intersect, the charge state will be changed by one when sweeping the gate voltage. This way we obtain the well known Coulomb staircase function for the average island charge number hni depicted in Fig. 1.3 b). The experimental data of the step function by Lafarge et al. [5] are shown in Fig. 1.4 for two dierent temperatures. One realizes the smeared charging eects with higher temperatures (solid line). These temperature eects are well described by statistical occupation of higher charge states. How to measure such a small excess charge? The average charge at the SEB island leads to a voltage drop V = hnie=C at the tunnel junction. This voltage has to be measured purely electrostatically, because any nite continuous current ow would in uence charging eects. In the experiment [5] they used the single electron transistor (SET) as a measuring device. This electrometer described in the next section is similar to the SEB and uses exactly the same strong dependence on a capacitively coupled gate voltage to detect the additional charge on the island.

1.3 Single Electron Transistor The SET consists of a series of two tunnel junctions described by the tunneling conductance G1 , G2 and capacitance C1 , C2 , respectively. The junctions may be biased by a transport voltage V and the island between the two barriers is coupled to a gate electrode purely capacitively, with capacitance Cg , cf. Fig. 1.5. In the experiment [5] the gate electrode correspond to the island of the SEB, thus, probing the voltage drop or equivalently the average excess charge number of the SEB. Since energetic considerations strongly depend on the driving voltage V , we consider

1.3. SINGLE ELECTRON TRANSISTOR

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gate

lead electrode

island

lead electrode

Figure 1.5: Single electron transistor. dierent regimes of the voltage separately. For vanishing transport voltage V = 0 the leads are shorted and the SET setup is equivalent the SEB: an island coupled to a lead (or equivalently to two shorted leads) via a tunneling barrier and to a gate electrode via the gate capacitance Cg . Here, the tunneling conductance is simply the parallel conductance GT = G1 + G2 and the island capacitance equals C = C1 + C2 + Cg . Therefore, energy considerations from the previous section hold here as well: At low temperatures and V = 0 tunneling is completely suppressed except at the degeneracy points. There, electrons may tunnel from one lead onto the island and leave it back or to the other lead without changing the charging energy intermediately. Hence, at the degeneracy points current ow from the left lead to the right one and vice versa is not forbidden energetically. While, at V = 0 the current to the left and to the right cancel and the net current is zero, current uctuations lead to a non{vanishing conductance that shows a periodic peak{like structure in dependence on the gate voltage. The peaks shown in Fig. 1.3 c) are located at the degeneracy points where current ow is allowed only. The height of the peaks equals Gcl =2, where Gcl = G1 G2 =(G1 + G2) is the classical conductance of the SET. This result is obtained by solving the corresponding master equation including only the two degenerate charge states. For nite driving voltage current ow in one direction will dominate at the degeneracy point leading to a nite net current. O the degeneracy point the current is suppressed as long as the quantity eV=2 does not exceed the electrostatic energy dierence between the charge states, i.e. the dierence of the parabolas in Fig. 1.3 a). This leads to the well known rhombic{ shaped stability diagram [2] where inside these rhombi all transitions are suppressed by Coulomb blockade and no current ows through the device. For higher driving voltages more charge states have to be included leading to an eectively higher temperature where charging eects are suppressed and nally the current will asymptotically reach Gcl V independent of Ug . In Fig. 1.6 we show experimental data by Lafarge et al. [5] for the current through the SET in dependence on the driving voltage V for two dierent gate voltages. Considering the dierential conductance G = @I=@V in dependence on the gate voltage, one realizes that the modulation is most pronounced at vanishing driving voltage V = 0. Therefore, in this work we focus on the linear dierential conductance of the SET, i.e. the smearing of the conductance peaks in Fig. 1.3 c) with temperature and nite tunneling conductance. Experimentally, these corrections are important since in recent years the reproducible fabrication of tunneling barriers with de nite tunneling conductance became feasible. When using these devices, e.g. as highly sensitive electrometers [5], in detectors [6], or for thermometry [7],

10


I(nA)

0.5 0

-0.5 -0.3

0

V(mV)

0.3

Figure 1.6: Experimental data by Lafarge et al. [5] for the current through a single electron transistor for two dierent gate voltages in dependence on the applied voltage V . a large current signal is desirable meaning large tunneling conductance. Therefore, we consider now the eect of nite tunneling conductance on Coulomb blockade phenomena.

1.4 Finite Tunneling Conductance In the weak tunneling regime systems are well described by Fermi Golden Rule type calculations [3, 8] in terms of the so called tunneling Hamiltonian as perturbation. This Hamiltonian describes the naive picture of annihilating a quasiparticle in one electrode and creating one in the other electrode. This description is equivalent to wave propagation through a barrier up to rst order in the transparency or third order in the tunneling matrix elements. During this process the charge of 1e is transferred charging the classical capacitance C . Here, the system is assumed to be small enough that time dependent propagation of the charge cloud need not to be considered. In this approach the charge acts as a collective mode completely decoupled from the quasiparticles in the unperturbed system. A charge shift operator in the tunneling Hamiltonian couples the quasiparticle excitations with the electromagnetic degrees of freedom of the environment including the capacitance CT . These excitations of quasiparticles appear only in pairs corresponding to contractions over two tunneling Hamiltonians and describe creation and annihilation of electron{hole pairs where the electron, or more adequately the quasiparticle, and the hole are created in dierent electrodes. When tracing out the fermionic degrees of freedom these mixed electron{hole pairs may be considered as single bosonic excitations leading to an eective dissipative environment for the collective electromagnetic mode. The trace over quasiparticles includes longitudinal and transversal quantum numbers. When using an averaged tunneling matrix element the sum over transversal quantum numbers can be performed explicitly leading to a prefactor N being the number of transversal \tunneling channels". Assuming the level spacing between the energies with respect to longitudinal quantum numbers to be small enough to justify a continuum description, only the density of states , 0 at the Fermi energy of the left and right electrode contribute. The perturbative series then proceeds in terms of

1.4. FINITE TUNNELING CONDUCTANCE

11

GT =GK being proportional to N 0 times the averaged transmission amplitude squared. This weak tunneling regime is well established theoretically [3, 8{11], whereby the predictions explain many experimental data [12{15].

So far only sequential tunneling processes have been considered that can be superimposed incoherently. Before focusing on higher order processes in the perturbative series, where excitations are strongly coupled by the electrostatic energy, we have to justify the use of HT for higher orders. Since in metallic junctions the contact area is typically much larger than the Fermi wavelength squared, the number of tunneling channels is very large, typically of the order N 104 and only leading order terms in the number of channels are considered. Therefore, neglecting errors of order 1=N we need to justify the tunneling Hamiltonian only in lowest nonvanishing order in the conductance per channel. Strong tunneling in our sense then means GT = N Gi GK and Gi GK , with Gi the tunneling conductance per channel. In this regime the tunneling Hamiltonian is justi ed to all orders in GT =GK . Higher orders in the perturbative series now lead to processes where an electron tunnels onto the island and during this excitation a dierent electron tunnels back or to the other lead electrode. Therefore, energy conservation is only violated during the virtual intermediate state. Since such excitations are coupled by the electrostatic energy they cannot be superimposed incoherently (with respect to the electromagnetic degrees of freedom). Here, we do not focus on higher order coherent processes of single electron{hole pairs because such events are restricted to exactly one channel and therefore are 1=N suppressed. So called cotunneling processes [16, 17], where two electron{hole pairs overlap in time are the leading corrections to Fermi's Golden Rule. Corresponding, higher order processes where more excitations are coupled are called resonant tunneling processes [18]. Due to these higher order events Coulomb blockade is not strictly obeyed at low temperatures and corrections are most pronounced in the region where rst order processes are forbidden energetically. On the other hand at higher temperatures where thermal uctuations dominate corrections of higher orders play a minor role. While the strong tunneling regime has been explored extensively by recent experiments [19{22], theoretical predictions remain limited. The theoretical work roughly splits into three groups. Firstly, higher order perturbative results [23{27] were successful in explaining some of the recent experimental data, yet, the analysis typically is restricted to conductances at most of order of the conductance quantum. Based on the diagrammatic expansion, partial resummation techniques were used to obtain nonperturbative results [18, 28{30], however, for a restricted set of charge states. The arbitrary cuto necessary in these latter theories limits their use for direct comparison with experimental ndings. Further progress can be made by using perturbative renormalization group techniques [31, 32]. Apart from these approaches based on diagrams generated by treating tunneling as a perturbation, a formally exact path integral expression [33] including all orders in the tunneling conductance may serve as a starting point for analytical predictions [34{39] in the semiclassical limit. While perturbation theory (PT) in the tunneling term usually starts from states with de nite electric charge, this latter approach employs the canonically conjugate phase variable and thus is well adapted in the semiclassical limit where the charge is smeared by thermal or quantum uctuations. Finally, numerical studies by quantum Monte Carlo (QMC) simulations may cover the entire parameter range relevant experimentally. As starting point of the calculation one may use either the perturbative series in charge basis [25] summing up all electron{hole pair con gurations, or alternatively the formal exact path integral expression in phase representation [40{43]. Whereas the former is preferred in the strong quantum regime the latter has advantages for large tunneling conductances.

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1.5 Overview In this work we explore the strong tunneling regime in systems containing metallic tunnel junctions. We focus on all three theoretical approaches discussed above and therefore split this dissertation into three main parts which are organized as follows: In chapter 2 we consider the SEB as the simplest system featuring Coulomb blockade phenomena. For this system the perturbative series was already developed in Ref. [23]. Unfortunately, a huge number of diagrams limits the explicit calculation to lower orders and one is interested in the range of validity of these nite order results. To estimate this range we consider the zero temperature case where the PT is worst. In fact at T = 0 the PT even diverges at the degeneracy point showing that the range of validity depends on both the tunneling strength and the gate voltage. We calculate the third order PT contribution of the average island charge number analytically for all gate voltages. At the degeneracy point we explicitly resum the leading order logarithmic divergencies to get a nite result. Then, a quantum Monte Carlo algorithm in the charge basis has been developed to calculate the average island charge number and the charging energy numerically. The resulting data are compared with the analytical predictions to estimate the range of validity of the perturbative series. Further we compare the results with recent ndings of a renormalization group approach. Since all calculations assume the channel number N to be large enough for limiting considerations we additionally calculate the corrections of nite channel number in second order PT at zero temperature to justify the approximation used. In spite of restricting to the SEB, the structure of the perturbative series and the appearing integrals are similar in all systems containing metallic tunnel junctions so that the range of validity should hold also for other systems when considering diagrams with the same number of loops. In chapter 3 we study the linear conductance of single electron devices containing metallic tunnel junctions embedded in an arbitrary electromagnetic environment. We derive a formally exact path integral expression of the imaginary time correlation function for such systems. The correlators are calculated in the semiclassical limit and the conductance is obtained from the Kubo formula. Speci cally we employ this approach to three circuits: First, a single tunnel junction with electromagnetic environment is considered where the semiclassical regime includes large conductance and/or high temperatures. We speci cally focus on large tunneling conductance and moderate temperatures where the ndings are compared with experimental data of two groups. Then we employ the method to a linear array of N junctions where we calculate the conductance at high temperature explicitly depending on the array length and the electromagnetic environment. The predictions in the limit of weak tunneling are compared with recent results of master equation approach. To go beyond the leading order approximation we nally calculate the conductance of the SET in dependence of the gate voltage and compare our ndings with recent experimental data. In chapter 4 we calculate the linear conductance of the SET for arbitrary parameters focusing on the strong tunneling regime. Electron tunneling thereby is treated nonperturbatively by means of the path integral derived in chapter 3. A quantum Monte Carlo (QMC) simulation is used to calculate the correlation functions in imaginary time for arbitrary parameters. The conductance is obtained from an inverse Laplace transform of the correlation function. We employ singular value decomposition to invert the integral transform numerically. The result is improved by exploiting positivity of the spectral function and sum rules. The ndings for the linear conductance are compared in the moderate tunneling regime with second order perturbation theory and in the strong tunneling regime with semiclassical results from chapter 3. Further, we used our ndings to explain experimental data in the strong tunneling regime where

1.6. PUBLICATIONS

13

no analytical theory is available. Finally, we conclude and discuss some possible extensions in chapter 5.

1.6 Publications This thesis work has lead to the following publications: 1) G. Goppert, X. Wang, and H. Grabert, High{Temperature Anomaly of the Conductance of a Tunnel Junction, Phys. Rev. B 55, (Rapid Comm.) R10213 (1997). 2) G. Goppert, H. Grabert, N. V. Prokof'ev, and B. V. Svistunov, Eect of Tunneling Conductance on the Coulomb Staircase, Phys. Rev. Lett. 81, 2324 (1998). 3) G. Goppert and H. Grabert, High Temperature Conductance of the Single-Electron Transistor, Phys. Rev. B 58, (Rapid Comm.) R10155 (1998). 4) G. Goppert and H. Grabert, Path Integral Methods for the Single-Electron Transistor, in Paht Integrals from peV to TeV , edited by R. Casalbuoni R. Giachetti, V. Tognetti, R. Vaia, and P. Verrucchi (World Scienti c, Singapore, 1999), p. 395. 5) G. Goppert, H. Grabert, and C. Beck, Coulomb Charging Eects for Finite Channel Number, Europhys. Lett. 45, 249 (1999). 6) G. Goppert and H. Grabert, Frequency Dependent Conductance of a Tunnel Junction, C. R. Acad. Sci. 327, 885 (1999). 7) G. Goppert and H. Grabert, Semiclassical Methods for Single Electron Devices, in Quantum Physics at Mesoscopic Scale, edited by D. C. Glattli and M. Sanquer (Editions Frontiers, France, in press). 8) G. Goppert, B. Hupper, and H. Grabert, Conductance of the Single Electron Transistor in the Strong Tunneling Regime , Physica B, in press. 9) A. Komnik, G. Goppert, R. Egger, and H. Grabert, Transport and Coulomb Blockade in Carbon Nanotubes, Physica B, in press. 10) G. Goppert and H. Grabert, Single Electron Tunneling at Large Conductance: The Semiclassical Approach (submitted). 11) G. Goppert, B. Hupper, and H. Grabert, Conductance of the Single Electron Transistor for Arbitrary Tunneling Strength (submitted). 12) G. Goppert and H. Grabert, Charge Fluctuations in the Single Electron Box: Perturbation Expansion in the Tunneling Conductance II (in preparation). The results have been presented at the following conferences: 1) Coulomb Blockade bei starkem Tunneln and Semiklassische Methoden fur den Einzelelektronentransistor, DPG Fruhjahrstagung 23.-27.3.98 in Regensburg, Germany. 2) Path Integral Methods for the Single-Electron Transistor, International Conference on Path-Integrals From peV to TeV, 25.-29.8.98 in Florence, Italy.

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3) Semiclassical Methods for the Single-Electron Transistor, Nanoscience Seminar, 10.12.98 in Saclay, France. 4) Semiclassical Methods for Single Electron Devices, Rencontres de Moriond, \Quantum Physics at Mesoscopic Scale", 23.-30.1.99 in Les Arcs, France. 5) Semiklassische Theorie der frequenzabhangigen Leitfahigkeit eines Tunnelkontaktes, DPG Fruhjahrstagung 22.-26.3.99 in Munster, Germany. 6) Semiclassical Conductance of a Tunnel Junction, TMR, Phase Coherent Dynamics of Hybrid Nanostructures, 24.-29.5.99 in Bad Herrenalb, Germany. 7) Semiclassical Methods for Single Electron Devices, NATO ASI, Quantum Mesoscopic Phenomena and Mesoscopic Devices in Microelectronics, 13.-25.6.99 in Ankara, Turkey. 8) Conductance of an Array of Tunnel Junctions, Symposium on Micro- and Nanocryogenics, 1.-3.8.99 in Jyvaskyla, Finland. 9) Conductance of the Single Electron Transistor in the Strong Tunneling Regime and Transport and Coulomb Blockade in Carbon Nanotubes, 22nd International Conference on Low Temperature Physics, 4.-11.8.99 in Helsinki, Finland. 10) Conductance of the Single Electron Transistor in the Strong Tunneling Regime, TMR, Exotic States in Quantum Nanostructures, 16.-29.8.99 in Windsor, United Kingdom.

Chapter 2

Perturbation Expansion In this chapter we investigate quantum uctuations of the charge in the single electron box (SEB). Particularly, we determine the range of validity of the perturbation theory (PT). Since higher order perturbative corrections are most pronounced at low temperatures a minimal range of validity can be determined at zero temperature, where PT is worst and even diverges at the degeneracy points. Based on a former perturbative expansion [23] we calculate the average island charge number and the eective charging energy in third order perturbation theory in the dimensionless tunneling conductance = GT =GK . At the degeneracy point we resum the leading logarithmic divergencies to get a nite result. A quantum Monte Carlo (QMC) simulation based on the perturbative series summing up all intermediate electron{hole pairs in the charge basis is developed to calculate the average charge number and the charging energy numerically. The data are compared with the analytical predictions to discuss the range of validity. Further results from a recent renormalization group analysis [26] are used to fortify predictions. Whereas good agreement between the third order result and numerical data [25] justi es the perturbative approach in most of the parameter regime relevant experimentally, the resummation is shown to be insucient to describe strong tunneling eects quantitatively near the degeneracy points. Finally we focus on systems with nite channel number N . We therefore calculate the 1=N corrections in second order PT at zero temperature. Limiting considerations for large N are found to be justi ed [27] rather rapidly with increasing channel number.

2.1 System and Model Hamiltonian We consider a SEB consisting of a metallic grain that couples to a lead electrode via an oxide layer. The separation permits tunneling of single electrons with the corresponding phenomenological tunneling conductance GT . The geometrical capacitance between the grain an the lead reads CT . Furthermore, a gate electrode is capacitively coupled to the grain with gate capacitance Cg . This setup is shown schematically in Fig. 2.1, where the circuit is biased by a gate voltage Ug shifting the Coulomb energy continuously. According to Ref. [23] we describe the SEB by the Hamiltonian H = H0 + HT (2.1) where

H0 = HC + Hqp 15

(2.2)

16

CHAPTER 2. PERTURBATION EXPANSION

GT CT

Cg

n

Ug

Figure 2.1: Circuit diagram of a single electron box. represents the system in absence of tunneling. Here, X X Hqp = k ayk ak + q ayq aq describes free Fermions and

q

k

(2.3)

HC = EC (n , ng )2

(2.4) is the Coulomb energy for n excess charges on the island biased by the dimensionless external voltage ng = Cg Ug =e. The charging energy 2 EC = 2eC

(2.5)

depends solely on the island capacitance C = CT + Cg . The quasiparticle creation and annihilation operators for transversal and spin quantum number and longitudinal quantum number p = k; q for the island and the lead electrode, respectively are denoted by ayp and ap . Further, p are single quasiparticle energies for the corresponding quantum numbers. Spin and transversal quantum numbers are conserved during the tunneling process described by the tunneling Hamiltonian X y HT = tkq ak aq + H.c. ; (2.6) kq

with tkq the transition amplitude between states with quantum numbers k and q. The charge shift operator accounts for the Coulomb energy and is related to the charge number operator n by the relation y n = n + 1: (2.7) The excess charge number n can be expressed as a derivative of the system Hamiltonian with respect to ng . Similarly we nd for the average island charge number 1 @ ln Z ; (2.8) hni = ng + 2 E c @ng where Z = trfexp(, H )g (2.9)

2.2. PERTURBATION EXPANSION AND DIAGRAMMATIC RULES

17

is the partition function of the system. At low temperatures and in the limit of vanishing tunneling conductance the logarithm of the partition function reduces to the minimum of the electrostatic energy EC (n0 , ng )2 where n0 is the integer closest to ng . Hence, as a function of the applied voltage Ug the island charge number displays the well known Coulomb staircase hni = n0 observed experimentally [5]. Due to occupation of higher energy levels at nite temperatures the step function is smeared. Similarly, the Coulomb staircase is rounded by virtual occupation of higher levels caused by the nite tunneling conductance. Here, we restrict ourselves to zero temperature and discuss the in uence on higher order tunneling processes to charge uctuations. Because of the periodicity and symmetry of the partition function Z with respect to ng , it is sucient to consider 0 ng < 21 . The diagrammatic rules and some essential simpli cations used hold for arbitrary temperatures, therefore, we rst give the general results valid for nite temperature and then specify to the zero temperature case.

2.2 Perturbation Expansion and Diagrammatic Rules In this section we brie y illustrate the method in Ref. [23] and give the diagrammatic rules used in the remainder.

2.2.1 Perturbation Expansion

Starting from the partition function (2:9) we use that the ng dependence arises from the charging energy only and put , H Z := trtr ee, Hqp : (2.10) qp

Factorizing the exponential exp(, H ) into a part exp(, H0 ) in the absence of tunneling and an interaction part that we write in the series representation in the tunneling Hamiltonian HT we get

Z=

1 X m=0

(,1)m

Z 0

dm

Z m 0

dm,1 : : :

Z 2 0

n o tr e, H0 HT (m ) : : : HT (1 ) d1 ; trqp e, Hqp

(2.11)

where the tunneling Hamiltonians are in imaginary time interaction picture

HT ( ) = eH0 HT e,H0 :

(2.12)

Separating the trace in a Coulomb and a quasiparticle trace tr e, H : : : = trC e, HC trqp e, Hqp : : :

(2.13)

one is left with multi{point correlation functions of tunneling Hamiltonians in imaginary time interaction picture

Z= where

1 X

m=0

(,1)m

Z 0

dm

Z m 0

dm,1 : : :

Z 2 0

d1 trC e, HC hHT (m ) : : : HT (1 )i0 ;

n

trqp e, Hqp X hX i0 = tr e, Hqp qp

(2.14)

o (2.15)

18


is the thermal quasiparticle average. Due to Coulomb interaction these correlators do not decompose into a product of two{point correlators. However, the charge shift operators in interaction picture ( ) = exp(HC ) exp(,HC ) commute with the quasiparticle operators and therefore may be factored out of the quasiparticle trace leading to

Z =

1 X

m=0

(,1)m

1 X

n=,1

Z 0

dm

Z m 0

dm,1 : : :

0

d1

X

j =1

k1 q1 1 1

:::

X

tm 1 : : : m

km qm m m E akmm m (m )a,qmmm (m ) : : : ak11 1 (1 )a,q111 (1 ) 0

P , 2m (j ,j,1 )Enj D

e

Z 2

: (2.16)

Here, the Coulomb trace is explicitly represented by a sum over charge states labeled by n and

En = EC (n , ng )2

(2.17) is the corresponding Coulomb energy. The charge shift operators in interaction picture lead to exponentials exp[j (Enj+1 , Enj )] where the integers nj are de ned by

n1 = n;

nj =

jX ,1 k=1

k ;

(2.18)

with k = labeling excess charge number increasing or decreasing processes. Due to charge conservation the {sums in Eq. (2:16) are constrained by 2m X

j =1

j = 0:

(2.19)

Further, we have introduced the shorthand notation a+ = ay and a, = a, respectively, and t = tkq is a real averaged transmission amplitude. Now time dierence variables j , (j = 1; : : : ; 2m) between two subsequent operators are introduced and the cyclic invariance of the trace can be used to obtain

0 2m 1 Z1 Z1 P m j E n 1 1 X X X X , 2j=1 2m j @ A Z = t d d , e 1 2m j 2 m 0 0 m=0 j =1 1 ;:::;2m n=,1 0 1 0 1+ * j j 2m X X Y X X j akjj j @ l A a,qjjj @ l A : (2.20) k1 q1 1

k2m q2m 2m j =1

l=1

l=1

0

Since the free quasiparticle Hamiltonian Hqp is quadratic in the fermionic degrees of freedom the quasiparticle operators ayp ( ) and ap ( ) imply a Wick theorem and the average in (2:20) decompose into a sum over pair products of two{point correlators

D

E

1 (1 ,2 )p1 1 : 1 + e1 p1 1

ap111 (1 )ap22 2 (2 ) 0 = 1 ;,2 p1 ;p2 1 ;2 e

So far the result is valid for arbitrary numbers of tunneling channels X N = 1:

(2.21) (2.22)

In metallic junctions where the junction area is typically much larger than the Fermi{wavelength squared, N is very large. Experimentally the value is of the order N 104 that justi es limiting


a)

b)

19

c)

Figure 2.2: Representative circle diagrams of a) rst, b) second, and c) third order in the perturbative series. considerations. The 1=N corrections for the SEB are considered explicitly in Sec. 2.8 con rming the validity of the approximation. In leading order only the combination X Y (2 , 1 ) = t2 hak2 (2)a,k1 (1)i0 ha,q2 (2)aq1 (1)i0 k1 q1 k2 q2

Z 1 e,jj=D e,(2 ,1 ) = g d ,1 1 , e,

(2.23)

of two two{point correlators contributes. Here, we have replaced the sums over k and q by energy integrals and have already performed one of them. We introduced the notation g = t2N 0 = =42 where and 0 are densities of states at the Fermi level for the island and the lead electrode, respectively. The function Y ( ) is an electron{hole pair Green function where electron and hole are created in dierent electrodes and D is the electronic bandwidth. Representing the {function in Eq. (2:20) in terms of an energy integral over an auxiliary variable E Z ( ) = 21 dEe,iE (2.24) one can perform all imaginary time integrals j , (j = 1 : : : 2m) gaining energy denominators that are linear combinations of the auxiliary variable E as well as Enj , (j = 1 : : : 2m) and k , (k = 1 : : : m). The coecients of the linear combinations depend on the Wick decomposition and the {sums in (2:20). The remaining summations over pairs and j 's can be represented graphically by diagrams, whereby the partition function then reads Z1 X 1 dE ei E D (E ) : (2.25) Z = 2 ,1 diagrams In Fig. 2.2 representative circle diagrams of a) rst,Pb) second, and c) third order in PT are shown. Each diagram D(E ) includes the charge sum 1 n=,1 and any circle segment correspond to an energy denominator 1 (2.26) E + Pm + iE nj

k=1 jk k

where jk = 1 for a time interval j in between of two vertices connected by a tunnelon line, and jk = 0 otherwise. The straight tunnelon lines represent energy integrals

Z

,jk j=D g dk 1k,e e, k

(2.27)

20


(1)

(2)

Figure 2.3: Diagrams of rst order in the perturbative series. for k = 1; : : : ; m, stemming from electron{hole pair Green functions (2:23). Since with respect to the auxiliary variable E the whole integrand is a product of energy denominators, the integral can be performed explicitly by means of contour integration. One gains a sum over residua corresponding to poles at a certain circle segment. Since residua of higher order poles correspond to derivatives of the nondiverging parts with respect to E we get graphs with decorations. The decorations are placed on the nondiverging segments indicating a derivative of the corresponding energy denominator with respect to the auxiliary variable E evaluated at the pole position considered. The simple form of the energy denominators allows us to perform the derivatives explicitly. We gain a higher power of the energy denominator 1=E q+1 times (,i)q q! for a q{fold derivative of 1=E . This way the sum over residua can be represented by circle diagrams, where the divergent circle segments are discarded and the remaining segments are decorated with slashes corresponding to derivatives of the energy denominator evaluated at the pole position. Considering a circle diagram with a pole of order r one nds that when discarding the pole segments the graph decomposes into r peaces denoted by Tq with q = 1; : : : ; r. The partition function then may be arranged according to the order of poles

Z=

1 Xr X 1 X r=1 diag n=,1

e, En

rX ,1

r,s

2 r 3(s) 1 4Y T 5 q

s=0 (r , s , 1)! f q=1

(2.28)

where the sum over diagrams is restricted to those with pole of order r and the symbol [: : :](s) stands for the sum over all decorations (derivatives) between the brackets with s slashes. Further, the factor f is the number of identical subgroups in Tq , q = 1; : : : ; r. Introducing the quantity

2 r 3(s) X r 4 Y Tq 5 Ur(s) = q=1

diag

(2.29)

we may write the partition function in the form

9 8 1 1 = < pX p X Z= e, En :1 + p! p + s Up(+s)s ; : n=,1 p=1 s=0 1 X

(2.30)

Using analytical properties of Ur(s) one may show that [23]

!p 1 1 1 p X X ( s ) ( s ) p + s Up+s = s + 1 Us+1 :

s=0

s=0

(2.31)


(1)

(2)

(5)

(3)

(6)

(9)

(7)

(10)

(11)

21

(4)

(8)

(12)

Figure 2.4: Diagrams of second order. This way get an eective Coulomb representation of the partition function

Z= where

1 X

n=,1

e, (En +n) ;

2 r 3(r,1) 1 Xr 1 Y X 4 Tq 5 n = , r=1 diag

r

q=1

(2.32) (2.33)

is the energy correction to the Coulomb energy state n that may be sorted in powers of g n =

1 X

m=1

(nm) :

(2.34)

Here, (nm) is the contribution of order gm which is represented graphically by arc diagrams consisting of a vertical line and semicircles. The diagrammatic rules and some examples are given in the next subsection.

2.2.2 Diagrammatic Rules

In this subsection we give the diagrammatic rules for the energy corrections (2:34). The term of order gm is given by (nm) and is composed of graphs containing a vertical line with m semicircles attached. Each semicircle corresponds to a tunneling event and represents an energy integral

g

Z1

,1

,jj=D

d 1e, e,

(2.35)

22


(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

Figure 2.5: Representants of distinct graphs of third order without insertions. stemming from the electron{hole pair Green function (2:23). Further each vertical line element contributes an energy denominator ,1=E where E is the excitation energy during the corresponding intermediate state, i.e. the Coulomb energy dierence

p = En+p , En

(2.36)

and the sum of all electron{hole pair excitation energies j present in the intermediate state corresponding to arcs that would be intersected by a horizontal line. There are two types of semicircles: in ected to the right or left, whereby the Coulomb state n is increased (decreased) by an arc to the right (left). For example in rst order in g there are just two processes depicted in Fig. 2.3. Whereas the graph (1) increases the excess charge number by one and therefore represents Z 1 e,jj=D 1 (1 ; 1) n = ,g d 1 , e, + ; (2.37) ,1

1

the graph (2) lowers the charge number and it's contribution (1n ;2) is obtained from the previous one by replacing 1 ! ,1 . In second order, cf. Fig. 2.4, there are two semicircles intersecting the vertical line into three parts. Each of them represents an energy denominator at the corresponding charging energy. The graphs (1) to (8) in Fig. 2.4 are given by all possibilities to attach two semicircles to a vertical line. Using the rules given above the graph (1) in Fig. 2.4 correspond to (2n ;1) = ,g2

Z1

,1

d1

Z1

,j1 j=D ,j2 j=D d2 11,e e, 1 12,e e, 2 ( +1 )2 + 1 + ,1 1 1 2 1 2

(2.38)

representing two tunneling processes with three intermediate energy denominators. Since an arc to the left hand side represents a tunneling process that lowers the charge number on the


(9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

23

Figure 2.6: Representants of distinct graphs of third order with insertions. island, the contribution of graph (2) diers from that of graph (1) by the replacement 2 ! 0. The rst eight graphs in Fig. 2.4 can be generated easily by the rules given previously. The graphs (9) to (12), however, dier by the prolongation of the interior arcs across the vertical line. These \insertions" represent separate graphs, i.e. each insertion represents a graph of lower order multiplied to the main graph. However, the vertical line of the main graph is intersected by the insertion and therefore the energy denominator is squared. Moreover, each insertion carries a factor (,1). Analogously, with two insertions the main graph has a cubic energy denominator, see for example the graph (16) in Fig. 2.6. Using these rules, the graph (11) in Fig. 2.4 leads to the energy correction Z 1 Z 1 1e,j1j=D 2e,j2j=D 1 (2n ;11) = g2 d1 d2 1 , e, 1 1 , e, 2 ( + )2 +1 (2.39) ,1 ,1 ,1 1 1 2 that factorizes into (,1) times the graph (2) in Fig. 2.3 with the energy denominator squared, multiplied with the graph (1) in Fig. 2.3. These insertions are shorthand notations for graphs with decorations that stem from a pole of higher order in the energy denominator multiplied with a lower order graph. Therefore the higher order denominators and the factors (,1) correspond to derivatives with respect to the auxiliary variable E . At zero temperature these graphs are related to terms in the Rayleigh{Schrodinger perturbative expansion stemming from normalization, cf. Sec. 2.8. In the high temperature limit, or equivalently in the limit EC ! 0, the Wick theorem implies that only \connected" graphs appear in the exponent. The \connected" graphs are those where all processes occur in one channel. In order gm , m > 1, these \connected" graphs, however, give only 1=N corrections that we have omitted. Hence, in the high temperature limit only the rst order survives and all higher order terms have to cancel. For the second order terms it turns out that each column in Fig. 2.4 cancels. For example, the contribution of the rst column, graphs

24


(1); (5), and (9), leads at vanishing charging energy to

;5;9) (2;1 n

EC 0

=

g2

0:

,j1 j=D 2 e,j2 j=D e 1 1 1 1 d1 d2 + , ,1 ,1 1 , e, 1 1 , e, 2 21 (1 + 2 ) 1 (1 + 2 )2 21 2

Z1

Z1

(2.40)

Higher order diagrams cancel likewise, and therefore in the limit of large channel numbers the perturbative expansion becomes exact at high temperatures where the charging energy EC is negligible compared to the thermal energy kB T . On the other hand, the 1=N corrections can survive even at high temperatures and may become relevant in that limit. The application of the diagrammatic rules to higher order graphs is obvious and we just present the contribution corresponding to graph (7) in Fig. 2.5 (3n ;7) = ,g3

Z1 ,1

d1

Z1 ,1

d2

Z1 ,1

,j1 j=D ,j2 j=D ,j3 j=D d3 11,e e, 1 12,e e, 2 13,e e, 3

1 (1 + 1 )(2 + 1 + 2 )(3 + 1 + 2 + 3 )(2 + 1 + 3 )(1 + 3 )

(2.41)

as an example of a third order term in the perturbative series. Here, three tunneling processes occur and we have to deal with ve energy denominators. In this graph the excess charge number is raised three times and then lowered stepwise. The energy denominators include the quasiparticle excitation energies of arcs that would be intersected by a horizontal line. The creation and annihilation times j , (j = 1; : : : ; 2m) of these quasiparticle excitations are already integrated out, however, the order of creation and annihilation of dierent excitations is crucial. On can think of the vertical line as an imaginary time ordering. Diagrams of third order PT without insertions are depicted in Fig. 2.5 and graphs with one or two insertions are shown in Fig. 2.6. Here, we have omitted dierent combinations of semicircles to the left and right but displayed only one representant. Hence, each representant in Figs. 2.5 and 2.6 stands for 8 dierent left{right combinations of the 3 semicircles. Further, the diagrams (7) and (8) in Fig. 2.5 and (11) and (12) in Fig. 2.6 are topologically distinct from graphs re ected at a horizontal line, but obviously, they lead to identical contributions and we have to count them twice. In summary, we have 80 topologically dierent graphs without insertions, 64 with one and 16 graphs with two insertions leading to 160 topologically dierent graphs of third order. All contributions in third order corresponding to representants in Figs. 2.5 and 2.6 are given in the Appendix 2.9.1. In general, there are always 2m left{right combinations for one representant of order m. Moreover, the number of representants exceeds the (2m , 1)!! possibilities to arrange m semicircles along the vertical line, simply as a consequence of the summation over insertions. After all, the number of graphs of order m exceeds (2m)!=m!, i.e. grows faster than the factorial. The rough estimation leads to 120 graphs in third and 1680 graphs in fourth order showing that a reasonable treatment is limited to third order. To obtain higher order or nonperturbative results, one has to investigate partial summations of graphs including the essential contributions. Unfortunately, in the in nite cuto limit D ! 1, each graph of the perturbative series represents a diverging integral and only the full sum in each order remains nite [23]. Hence, partial summations of higher orders need an arti cial cuto complicating a direct comparison with experimental ndings. Consequently, within PT, the systematical treatment of higher orders is the only tool to get results directly comparable with experiments. We proceed with two general simpli cations, valid for all orders.


25

= p

-p

Figure 2.7: Symmetry of two vertically re ected graphs.

2.2.3 Re ected Graphs

The analytical form of the integrals leads to general consequences for energy corrections. First we note that the particular form of the charging energy (2:4) implies

En(ng ) = En+1 (ng + 1) = E,n (,ng )

(2.42)

and the corresponding energy dierences in the denominators read

p = Enp(ng ) , En (ng ) = EC [p2 2p(n , ng )]

(2.43)

that depend on the dierence (n , ng ) only. Hence, the corrections to order m may be written in the form (nm) (ng ) = gm EC fm (n , ng ): (2.44) Since contributions of order m include always m integrals with 2m , 1 energy denominators, we gain by measuring all energies in units of EC a single factor EC , and in the limit of in nite bandwidth, D ! 1, the functions fm depend on EC only. Further, a re ection of a given graph on the vertical axis leads to the same contribution with the replacement Enp ! Enp, cf. Fig. 2.7. Equivalently, by virtue of Eq. (2:43), one can replace (n , ng ) by ,(n , ng ). Since the sum over diagrams includes all left{right permutations of arcs including pairs of vertically re ected graphs, one may write

fm (u) = gm (u) + gm (,u);

(2.45)

where the contribution of gm solely includes topologically dierent graphs where one arc is held xed.

2.2.4 Insertions

A further simpli cation can be achieved for graphs with insertions. It turns out that they factorize into a host graph, with energy denominator squared, and an insertion contribution. Since we have to sum over all possible left{right con gurations, the full lower order contribution may be inserted. In Fig. 2.8 a) we replace a rst order insertion and its vertically re ected companion by a circle representing a multiplication with the full rst order contribution (1) n . Likewise, insertions of order k and all possible left{right combinations lead to a multiplication with the full order k contribution (nk) , schematically depicted for k = 2 by the square in Fig. 2.8 b). Here, the arrow represents the sum over all possible left{right arrangements of semicircles

26


a) +

=

b) +

+

+

=

Figure 2.8: a) full rst and b) full second order inserted in graphs. belonging to the insertion. Therefore, all graphs belonging to the representants (13); (14); and (15) in Fig. 2.6 lead to Z 1 e,jj=D 1 (3n ;13,15) = ,g3 EC f2(u) d + (u ! ,u); (2.46) ,1 1 , e, (1 + )2 with u = n,ng . This is a multiplication of (2) n and the host graph (1) in Fig. 2.3 with the energy denominator squared, times (,1). Additionally, the contribution of the vertically re ected graph is added as the same term with the replacement u ! ,u. Likewise, more insertions can be included. For example, all graphs belonging to the representant (16) in Fig. 2.6 lead to Z 1 e,jj=D 1 (3 ; 16) 3 2 2 + (u ! ,u): (2.47) n = ,g EC f1 (u) d ,1 1 , e, (1 + )3

2

that is a multiplication of (1) and the host graph (1) in Fig. 2.3 with the energy denominator n cubed. This procedure holds for insertions of all order and is used in Sec. 2.5 where we resum the leading logarithmic divergencies in the two{state approximation. But rst, we consider the changes and simpli cations in the zero temperature limit and present results for the average charge number up to third order.

2.3 Zero Temperature Limit and Third Order In this section we perform the zero temperature limit and emphasize changes to the formulas. Then, we present the rst and second order results and exemplarily proceed with the calculation for one graph of third order.

2.3.1 Zero Temperature Limit

Generally, at zero temperature the partition function of a system with nondegenerate discrete energy levels reduces to the exponential of the ground state energy

Z=

+ X1

n=,1

e, (En +n ) ,! e, E :

(2.48)

2.3. ZERO TEMPERATURE LIMIT AND THIRD ORDER

27

Due to the restriction of the gate voltage to be in the range 0 ng < 12 , the ground state charge number is n = 0 and the ground state energy reads E = E0 + 0 . Therefore, the energy dierences in the denominators read

p = Ep , E0 = EC (p2 2png ) (2.49) and the functions fm and gm depend on ng only. The average island charge number reduces to @E : (2.50) hni = ng , 2E1 @n c g Bose factors in the integrands restrict the integration (2:35) to positive energies

Z 1 e,jj=D Z1 g d 1 , e, ,! g de,=D ,1 0

(2.51)

that facilitates the integration, because there are no poles from the energy denominator (2:26) that need to be taken care of, meaning there are no real excitations.

2.3.2 First and Second Order Results

The rst and second order calculations were already presented in Ref. [23]. The rst order with an exponential cuto D leads to

g1 (u) = ,(1 + 2u) ln(1 + 2u) , D=EC + ln(D=EC ) + ;

(2.52)

where = 0:577 : : : is Euler's constant. In the in nite cuto limit this term diverges, but the derivative with respect to ng remains nite. Considering only rst order in g, we may omit this divergence, however, in combination with higher order terms in the perturbative series, i.e. as lower order insertion, we have to use the full expression (2:52). Whereas the rst order contribution diverges, the full sum of graphs of each higher order m > 1 remains nite [23]. The second order contribution was also calculated leading to 5 1 , 2u 2 2 2 g2 (u) = 6 (1 + 2u + 8u ) , 2 , 4u + 2u ln2 4(1 , u) 1 , 2u 1 , 2u 1 2 2 + 4 + u + u ln 1 + 2u + (1 + 2u) ln 1 + 2u 1 , 2u 3 , 2u 2 ,4(1 , u) ln 4(1 , u) , (5 , 8u + 4u )Li2 4(1 , u) ; (2.53) where Li2 (u) is the dilogarithm function [44].

2.3.3 Third Order

Here, we motivate the calculation of the third order contribution exemplarily for the graph (7) in Fig. 2.5. The full analytic expression g3 (u) is presented in Appendix 2.9.2. Since the integral over the entire sum of 160 integrands remains nite, we may omit the cuto. However, to calculate the integral analytically we have to separate the whole expression into tractable parts. Each integral will be divergent and we introduce a sharp high energy cuto D. After integration we expand the expressions with respect to D=EC ! 1. There are divergent terms in each expression but the sum of divergencies of all graphs has to vanish. This

28


cancellation serves as a useful, nontrivial test of our calculation. We have to deal with eight dierent types of integrands without insertions depicted in Fig. 2.5 and six dierent types with insertions in Fig. 2.6 (the graphs (13) to (15) are merged to a single graph with the whole second order contribution inserted). We exemplarily proceed with graph (7) in Fig. 2.5 leading to an integral of type

h7 =

ZD 0

d1

ZD 0

d2

ZD 0

d3 ( + )( + + )( + 1 +2 3 + )( + + )( + ) : 1 1 2 1 2 3 1 2 3 4 1 3 5 3

(2.54) The full contribution of this representant consists of all possible left{right arrangements of the semicircles X (30 ;7) = ,g3 h7 (1 ; 2 ; 3 ; 4 ; 5 ); (2.55) 1 ;2 ;3 ;4 ;5

where the j {sum, with (j = 1; : : : ; 5), runs over possible combinations of allowed energy dierences that are: h7 (1 ; 2 ; 3 ; 2 ; 1 ), h7 (1 ; 0; 1 ; 2 ; 1 ), h7 (1 ; 0; ,1 ; 0; ,1 ), h7 (1 ; 2 ; 1 ; 0; ,1 ), and all terms with ng ! ,ng . The integrals may be performed by splitting denominators into partial fractions. Using 1 1 1 (2.56) (a + 2 )(b + 2 ) = (a , b)(b + 2 ) , (a , b)(a + 2 ) with a = 2 + 1 and b = 3 + 1 + 3 , we are able to perform the 2 integral leading to a logarithm function Z D ,2 d2 a + = a ln(a + D) , a ln(a) , D: (2.57) 0

2

The new denominator 1=(a,b) on the rhs of Eq. (2:56) has an arti cial pole at 3 = 2 ,3 . Since the whole expression is analytic in the integration region, the sum of all pole contributions has to cancel in the threefold integral. However, each pole contribution depends on the contour of integration and we have to specify the contour and use the same for all integrals. In the simplest case one has to choose a branch for the logarithm, i.e. insert a small imaginary part tending to zero, but in general this point is more involved. Since we have evaluated the integrals with the help of Mathematica which does not care about the contour and uses partial integration and variable transformations internally to solve integrals, one has to take care of this point explicitly. Next we consider the 1 integration. In the numerator now appears the logarithm function from the previous integration, ln(2 + 1 ), that diverges for 1 ! 0 when 2 0 and we temporarily introduce a lower integration limit. With the help of (2:56), where we replace 2 by 1 and the constants are now a = 1 and b = 4 + 3, we split the remaining fractions in (2:54) and perform the integral in terms of the dilogarithm function [44] Li2 (z ) = ,

Zz 0

0

dz0 ln(1z,0 z ) :

(2.58)

The third integration can then be performed using the trilogarithm function Li3 (z ), where the general polylogarithm functions are de ned by [44] Z z Lin,1(z0 ) Li (z ) = dz 0 : (2.59) n

0

z0

All terms emerging can be expressed by trilogarithms, dilogarithms, logarithms, and rational functions of ng . Here, the arguments of the transcendental functions are rational expressions

2.4. AVERAGE CHARGE NUMBER

29

of energy dierences. Since each integration increase the \order" of transcendental functions at most by one, the transcendental terms are of the form Lik lnl obeying k + l 3. Here lnl means a product of l logarithms of possibly dierent arguments. The \order" thereby characterizes the transcendental function for large arguments: e.g. Lik lnk is of order k. We nd in order m the perturbative series transcendental terms of the form (Lim )km (Li2 )k2 lnk1 where Pmof jk j =1 j m. In principle, the integrals in PT lead in all orders to analytically known functions but there are practical restrictions like the number of appearing integrals and rather lengthy expressions. Because of the length of the analytical result we present g3 (u) in Appendix 2.9.2. The result is rather unhandy and uses complicated functions that have to be calculated numerically. Therefore, one could think of a direct numerical evaluation of the three fold integrals, but there are some numerical problems. Firstly, only the full sum of the 160 integrals is nite and one has to use a huge integrand, cf. Appendix 2.9.1. Moreover, the integrand is not symmetric in the three integration variables, i.e. there are integration directions where the integrand leads to a diverging positive or negative contribution. For a numerical study one has to symmetrize the integrand that additionally enlarges it by a factor 6. Secondly, in spite of the smoothness and analyticity of the integrand and absence of poles in the integration region, the integrand contains oscillatory parts so that standard numerical routines, e.g. from the NAG{library fail. We used a statistical integration routine from the NAG{library to check the analytical predictions. The numerical evaluation of a single point with a few per cent accuracy took over one week to converge. Therefore, the numerical evaluation of the integral is hopeless in particular for delicate high precision studies regarding the limiting divergent behavior at the degeneracy point or calculations of the second order derivative needed to determine the charging energy. In the next section we give some analytical results for the average charge number.

2.4 Average Charge Number

The analytical result of the average island charge number hni at zero temperature and in rst order in g was calculated in [28, 45] (2.60) hni1 = g ln 11 ,+ 22nng g and can be readily obtained by using Eq. (2:50) with the energy correction (2:52). The function is well behaved except at the degeneracy point ng = 21 where it exhibits a logarithmic divergence hni1 ,g ln , with = 12 , ng . Therefore, the range of validity of the perturbative series at zero temperature is strictly restricted to ng < 21 . The contribution of second order in g presented in Ref. [24] reads !# ( " 2 + 2ng , 2n2g ) 1 , 2 n 4 g hni2 = ,g2 ng 3 + ln2 1 + 2n + (316(1 , 2ng )(1 + 2ng ) ln(1 , 2ng ) g " ! ! 1 , 2 n 3 , 2 n g g 2 +2(1 , ng ) ln 4(1 , n ) + 2Li2 4(1 , n ) g g # ) 8(1 , n ) g , (1 , 2n )(3 , 2n ) ln(4(1 , ng )) , (ng ! ,ng ) : (2.61) g g Here, (ng ! ,ng ) stands for the same sum of terms with ng replaced by ,ng showing explicitly the asymmetry of hni with respect to the applied voltage Ug . At the degeneracy point, a leading logarithmic divergency of hni2 ,2g2 ln2 appears. This indicates that near the degeneracy

30


1

Figure 2.9: Full ground state energy diagram in the non{crossing approximation of the two{state model. point g ln is the eective expansion parameter so that the larger g the smaller is the range of ng . The analytical result of third order is not given explicitly but can be easily calculated by dierentiation of the expression (3) 0 with respect to ng , cf. (2:50). Also the third order term shows a logarithmic divergency at the degeneracy point leading together with the lower order contributions to the asymptotic expansion

hni = ag2 + bg3 , (g + 6g2 + cg3 ) ln , (2g2 + 24g3 ) ln2 , 4g3 ln3 + O ; g4 ; (2.62) where the coecients a, b, and c read numerically a = ,9:7726:::, b = ,70:546:::, and c = 65:462::: . The leading order logarithmic terms in Eq. (2.62) read

hni ,g ln , 2g2 ln2 , 4g3 ln3 :

(2.63)

They are related to diagrams that contain only the charge states n = 0 and 1. Therefore, before comparing our ndings with numerical results, we consider a two{state approximation that include these degenerate charge states only.

2.5 Degeneracy Point

A further analysis of the perturbative series in the limit of ng ! 21 shows that the leading logarithmic divergencies (2:63) stem from diagrams including only charge states n = 0 and 1. This is a direct consequence of the degeneracy of these charge states at ng = 21 . A two{state approximation can be generated by restricting the perturbative series to graphs that permit the charge states n = 0; 1 only. This model was shown to be equivalent to an anisotropic multi channel Kondo Hamiltonian where EC corresponds to the magnetic eld and g to the exchange integral J0 [28]. Considering the leading order logarithmic divergencies we nd that crossed diagrams, like graph (6) in Fig. 2.5 with the middle semicircle to the left, do not contribute. Therefore, we restrict ourselves to noncrossing diagrams where it is possible to write the ground state energy as Z e0 = g D dF1 () (2.64) 0

with a sharp cuto D, graphically represented in Fig. 2.9. The generalized energy denominator F1 () corresponds to the bold line with index 1 and is determined by a Dyson equation graphically represented in Fig. 2.10. For convenience, we have rotated the graphs by 90 degrees and an upper (lower) semicircle increases (lowers) the charge number. In the Dyson equation depicted in Fig. 2.10 the thin line with index 0 correspond to the bare energy denominator ,1= of charge state 0 where is the energy variable. Analogously, the thin line with index 1 correspond to the bare energy denominator ,1=(1 + ) of charge state 1. In contrast, the bold line with index 0 (1)

2.5. DEGENERACY POINT 0

31

=

1

=

0

=

1

=

0

+

1

+

1

0

0

0 0

1

1 1

+

+

1

1

Figure 2.10: Dyson equation for the non{crossing approximation in the two{state model. represents the dressed n = 0 (n = 1) propagator F0 () (F1 ()) that can be generated gradually by inserting bare propagators and the interaction vertices. Hereby an interaction vertex of the propagator 0 (1) is represented by a circle with index 0 (1) and is composed of a semicircle and an insertion R de ned by the rules in Sec. 2.2. Each part of the interaction vertices contains an integral g 0D d0 0 but they dier in the explicit meaning of 0 : Whereas an insertion is just a multiplication where the propagator inside does not depend on the energy variable of the legs of the vertex, the semicircle propagator takes into account all excitation energies and therefore depends on + 0 . Due to the restriction of the perturbative series to charge states n = 0 and n = 1 the only possible contribution, except upward insertions, is an upward (downward) semicircle for the n = 0 (n = 1) energy denominator that increase (lowers) the charge state. This way the Dyson equation depicted in Fig. 2.10 generates all diagrams of the non{crossing approximation in the two{state model. Unfortunately, to be systematical one has to include an arbitrary cuto D that restricts electron{hole pair excitations to energies lower than the next Coulomb state. Therefore, in the in nite bandwidth limit the parameters used are eective renormalized quantities complicating comparison with experiments. One possible way to nd the renormalization of the parameters is a comparison of the limiting divergent behavior for ng ! 21 with the full third order result. This comparison leads to a power series of the renormalized parameters g and EC in terms of the bare conductance g. Iterating the Dyson equation we generate graphs including only charge states n = 0; 1. In Fig. 2.11 we depict the generated graphs of a) rst, b) second, and c) third order in g. However, not all of these graphs contribute to the leading behavior. We nd that only the graphs shown in Fig. 2.12 are responsible for the leading logarithmic divergencies in order gm . The diagram a) shows m , 1 downward arcs leading to an asymptotic behavior of e (0m;0) ,gm lnm :

(2.65)

32

CHAPTER 2. PERTURBATION EXPANSION a) b)

c)

Figure 2.11: Graphs of a) rst, b) second, and c) third order in the two{state model generated by the Dyson equation in Fig. 2.10. The graphs of the form b) in Fig. 2.12 show m , k , 1 arcs lowering the charge state and one multi{insertion represented as an insertion with a slash with label k which stands for the full order k result in the two{state approximation multiplied with the main graph, cf. Sec. 2.2.4. Assuming the order k result shows a leading behavior of (2.66) e (0k) , 21 (2g ln )k apparent for k = 1; 2 and 3 form Eq. (2:63), these graphs show the asymptotic behavior m,k k ,1 e (m;k) e (k) gm,k ln = ,gm 2 lnm : (2.67) 0

m,k

0

m,k

Interchange of the multi{insertion and arcs leads to topologically dierent contributions that have to be counted separately. This leads to a factor (m , k) canceling the denominator in Eq. (2:67). The sum over all k's starting with k = 0 represented by the graph a) in Fig. 2.12 and from k = 1 to k = m , 1 corresponding to the graph b) leads to an asymptotic behavior of the form mX ,1 e (0m) e (0m;k) = , 21 (2g ln )m (2.68) k=0

which proves the assumption (2:66) for all k by iteration. For the average island charge number (2:50) we obtain by summing over all orders in g : (2.69) hni = 1 ,,g2ln g ln This result was originally obtained by Matveev [28] with RG techniques. Whereas this result does not depend on the cuto D, it contains renormalized parameters g and depending on the

2.6. DISCUSSION OF RESULTS

33

a)

1

2

m-1

1

m-k-1

b) k

Figure 2.12: Diagrams of order gm contributing to the leading logarithmic divergencies. tunneling strength. The leading logarithmic divergencies up to third order coincide with those in (2:63), however, nonleading divergencies from (2:62) are missing. They can be generated by using renormalized parameters g and in (2:69). Since the energy dierence appears in the form EC and the renormalization is a pure multiplication we may write EC instead of EC where correspond to the charging energy renormalization. The renormalized quantities are found to read h i g = g 1 + 6g + ag2 + O(g3 ) (2.70) and h i = 1 + bg + cg2 + O(g3 ) (2.71) with a = ,26:3716 : : : , b = 9:7726 : : : , and c = 59:6626 : : : . In spite of generating the correct asymptotics, the rapidly increasing series coecient lead not to meaningful results in the vicinity of ng = 21 . Therefore, one has to resum also the nonleading divergencies in the two{state model to get useful renormalized parameters. This has not been considered so far but is the aim of future work.

2.6 Discussion of Results In this section we compare our analytical results with numerical data and estimate the range of validity. Therefore, we have developed a QMC approach explained in Sec. 2.7 that is able to sample the average island charge number for nite gate voltages and very low temperatures. Further, we con rm our data with recent renormalization group results [31]. The comparison of the rst, second, and third order PT with QMC data [25] and RG results [31] for = 5 and 10 is shown in Fig. 2.13. We give results here in terms of dimensionless tunneling conductance = GT =GK = 42 g. We nd good agreement for gate voltages near zero, but for ng ! 21 the

34


0.5

0.25

1. order 2. order 3. order resum RG QMC

0.0 0.0

=5

0.25

0.5

ng

0.5

0.25

0.0 0.0

1. order 2. order 3. order resum RG QMC

=10

0.25

0.5

ng

Figure 2.13: The average electron number hni as a function of the dimensionless voltage ng is shown in rst, second, and third order perturbation theory in , and compared with QMC data [25] and RG results [31]. The result (2:69) is also shown as a dotted line. analytic result diverges. The range of validity of PT shrinks with increasing gate voltage. We nd that third order PT in remains valid with errors below 4% up to ng 0:495 for dimensionless conductance = 2 (data not shown), up to ng 0:45 for = 5, and up to ng 0:4 for = 10. These estimations are con rmed by the RG results. Since for ng = 0:45 the charging energies for n = 0 and n = 1 dier only by 0:1EC , deviations from the third order result in can be observed only for temperatures well below EC =10kB even at ng = 21 . Further, Fig. 2.13 shows that the resummation of the leading logarithmic terms (2:69) does not suce to describe the behavior near ng = 21 . Subleading logarithms are important to obtain quantitatively meaningful results in the strong tunneling regime. In spite of the correct divergent behavior at ng = 21 , the inclusion of subleading logarithmic terms using renormalized parameters g and from (2:70) and (2:71) leads to results worse than with the bare parameters. This implies that one has to include consistently the nonleading logarithmic terms in the two{state approximation. Since for ng ! 0 the QMC and RG data perfectly coincide with the perturbative results, we used a more sensitive quantity to determine the range of validity of PT in this limit: For small external voltages, the average island charge grows linearly as

hQi = ehni = C Ug

(2.72)

where C is an eective capacitance of the box. In the absence of Coulomb blockade eects

2.7. QUANTUM MONTE CARLO

35

1

*

EC /EC

1. order 2. order 3. order RG QMC

0.5

0.0

0

10

20

Figure 2.14: Eective charging energy as a function of the dimensionless conduction = GT =GK . Perturbative ndings are compared with QMC data [25] and RG results [31].

C = C , while for strong Coulomb blockade, i.e., in the limit of vanishing tunneling conductance, C = 0. It is thus natural to characterize the strength of the Coulomb blockade eect by an eective charging energy EC de ned by [40] EC = 1 , C = 1 , @ hni : (2.73) EC C @ng ng =0 One may identify EC as an eective rounding of the energy parabolas in Fig. 1.3 a) at ng = 0 caused by the nite tunneling strength. The perturbative series gives

EC = 1 , 4g + Ag2 , Bg3 + O(g4 ); (2.74) EC where A = 5:066::: and B = 1:457::: are analytically known coecients whose explicit form can

be readily obtained by the analytical formulas. In Fig. 2.14 we compare our predictions for the eective charging energy with QMC data [25] and recent RG results [31]. We nd good agreement of the PT up to = 8 for second order PT, whereas the third order extends to = 16. The results shown are consistent with earlier MC data [40] (data not shown) calculated in phase representation whereas the new QMC results [25] are sampled in the charge representation following the prescription in the next section.

2.7 Quantum Monte Carlo To explore the range of validity of the higher order perturbative results we have carried out precise quantum Monte{Carlo simulations for the single electron box. In contrast to earlier attempts [29, 40{42], we do not work in the phase representation, but simulate con gurations

36


directly in the charge representation, keeping track of all intermediate electron{hole pairs. A general numeric scheme for evaluating a series of integrals directly in the continuum, i.e., without invoking arti cial discretization of the integration variables was explained in [46, 47]. It is possible then to develop an exact QMC algorithm summing the whole perturbative diagrammatic series [48] without systematic errors. Here, the integration variables are imaginary times of charge transfer events, and we have employed this \diagrammatic" QMC for summing all graphs of the partition function (2:16) of the SEB. The eciency of the new algorithm has allowed us to simulate the SEB at extremely low temperatures ( EC as large as 104 ) necessary to compare with analytic zero temperature results in the vicinity of the degeneracy point. While in the phase representation, at low temperatures ng 6= 0 results in a sign{problem, in the charge representation all contributions are positive de nite even at nite external voltage. This has enabled us to obtain for the rst time QMC data for the entire staircase function. More explicitly, we start with the partition function in imaginary time perturbation expansion in the tunneling Hamiltonian [23]

Z =

1 X m=0

(,1)m

1 X

n=,1

Z 0

d m

Z m 0

d m,1 : : :

0

d 1

X

j =1

k1 q1 1 1

:::

X

tm 1 : : : m

km qm m m E akmm m ( m )a,qmmm ( m ) : : : ak11 1 ( 1 )a,q111 ( 1 ) 0

P , 2m ( j , j,1 )Enj D

e

Z 2

(2.75)

where the Coulomb trace is performed explicitly leading to a charge sum labeled by n. Further the Coulomb P energy dierences are En = EC (n , ng )2 and the indices nj are de ned by n1 = n and nj = n + jk,=11 k . The average denotes the free quasiparticle average and all de nitions from Sec. 2.2 hold here as well. The average over quasiparticle creation and annihilation operators obey Wick's theorem and we may decompose the average into a sum over products of two{point correlation functions. In the limit of large channel numbers only speci c combinations of the Wick decomposition (2:23) contribute and we get

Z=

1 Z X

m=0 0

d 2m

Z 2m 0

d 2m,1 : : :

Z 2 0

d 1

1 X X0 X pairs 1 :::2m n=,1

e,

P2m ( j , j,1)En Y m j j =1

j =1

Y ( j1 , j2 ) ; (2.76)

where the {sum is restricted by the speci c pair con guration leading to the constraint j1 ,j2 for each pair (j1 ; j2 ) with ji 2 1; : : : ; 2m and j1 6= j2 . The sum over charge states n, over the j , (j = 1 : : : 2m), and over the dierent pair con gurations may be represented by arc diagrams. These diagrams consist of a horizontal line corresponding to the imaginary time interval [0; ] and m semicircles, up or down, characterizing the electron{hole pairs increasing or lowering the intermediate charge states. In Fig. 2.15 we depicted a graph of order m = 4, where the labels denote the corresponding charge state. We may write these dierent graphs as a sum over con gurations labeled by m

Z=

1 XZ X

m=0 m 0

d 2m

Z 2m 0

d 2m,1 : : :

Here the weight function

F (m ; ng ; 1 ; : : : ; 2m ) = e,

Z 2 0

d 1 F (m ; ng ; 1 ; : : : ; 2m ):

P2m ( j , j,1)En Y m j j =1

j =1

Y ( j1 , j2 )

(2.77)

(2.78)

2.7. QUANTUM MONTE CARLO

1

2

3

37

2

1

0

1

0

2

1

β

Figure 2.15: A graph of order m = 4 contributing to the partition function. The solid line correspond to the imaginary time interval [0; ] and the labels denote the corresponding charge states of the propagators. depends strongly on the speci c choice of the graph m . Due to the positivity of the weight function F one can employ importance sampling methods for the calculation of the multiple integrals. However, there is an important dierence to standard sampling techniques: here, we have to deal with a varying number of integrals. Therefore, the QMC scheme decomposes into two dierent classes of updates: (I) those which do not change the type of the diagram, i.e. updates that shift the integration variables of the function F , obeying time ordering, and (II) those which do change the structure of the diagram. The updates of class I are rather straightforward, being identical to those of simulating a continuous distribution. Here, we use a standard Metropolis algorithm [49, 50]: We choose randomly an electron{hole pair j with j 2 f1; : : : ; mg and shift the time variables ji ! ji + d ji for i = 1; 2 with d ji equally in [,a; a], obeying the time ordering 1 < : : : < 2m . The update will be accepted with probability

(

Pupdate = F1 ( nal)=F (initial)

if F ( nal)=F (initial) < 1 else

(2.79)

where F ( nal) is the weight function after and F (initial) before the trial step. Here a is chosen to be small enough to ensure reasonable acceptance rates. The updates of class II are more involved and we utilize the scheme described in [48] where the polaron problem is studied by diagrammatic QMC: We consider updates A that create an additional electron{hole pair and transform the function F (m ; ng ; 1 ; : : : ; 2m ) into F (m+1 ; ng ; 1 ; : : : ; 2m+1 ; : : : ; 2m+2 ; : : : ; 2m ), and its counterpart B performing the inverse transformation, i.e. annihilate an excitation. The update procedure involves two steps. First it proposes a change, selecting a new type of diagram m+1 , in particular, the new diagram is characterized by creation and annihilation times 2m+1 and 2m+2 , charge state n and the direction of the arc (up or down), the rest of the diagram being unchanged. Here, 2m+1 is chosen uniquely distributed in [0; ]. 2m+2 may be distributed by W ( 2m+2 ; 2m+1 ) in the remaining interval [ 2m+1 ; ] obeying time ordering. Here, we chose W const:, but in general one may use a distribution for all other labels ( 2m+1 ; n; ) similar to the single excitation case W F (1 ; ng ; 2m+1 ; 2m+2 ) that may be calculated exactly [48]. The update is accepted with probability Pcreate , or rejected. For balanced updates we employ the Metropolis{like prescription [47]

Pcreate =

(

R=W 1

if R < W else

(2.80)

38


where W equals the distribution used for the labels and (2.81) R = ppB F (m+1 ; ng ; F1(;: : :;;n 2;m +1; :; :: :: ;: ; 2m) +2 ; : : : ; 2m ) A m g 1 2m is the ratio of weight functions F (final) and F (initial). Further, pA and pB correspond to probabilities for updates A and B, respectively. According to Ref. [47] the relative weight for the creation process is given by pB =pA = Nlab =(Nlab + Nadd ) where Nlab is the number of labels before the trial step and Nadd the number of additional labels due to the process. This leads in our case to pB =pA = m=(m + 1) for the creation process A. Equivalently, the annihilation process B is accepted with probability

Pannihilate =

(

W=R 1

if R > W : else

(2.82)

Here, the ratio of probabilities is given by pB =pA = (Nlab , Nadd )=Nlab = (m , 1)=m. Using this scheme one can sample the partition function or equivalently the average excess charge number hni by summing over electron{hole pair con gurations. Successive measurements are usually correlated and we therefore wait a suciently long time between the measurements to sample uncorrelated data and the errors are obtained by standard deviations of the mean. Since for nite gate voltages and low temperatures there is no sign{problem, it is possible to obtain data for really low temperatures that can be compared with zero temperature theories. Here we use temperatures down to EC 104 necessary for gate voltages near the Coulomb step where nite temperature eects are intensi ed by the small energy dierence between the Coulomb states n = 0 and 1. For very large tunneling conductances and low temperatures the number of electron{hole pairs increase rapidly. Not only bookkeeping of the huge number of arcs becomes dicult but also problems concerning memory of the computer and equilibration time limit the calculations. For example: for = 10 and EC = 104 we count typically m 2 105 electron{hole pairs showing that the QMC scheme is restricted to moderately large tunneling conductances. In Sec. 2.6 the numerical data were compared with third order PT and RG results.

2.8 Finite Channel Numbers In this section we consider quantum uctuations of the charge in the single electron box by means of Rayleigh{Schrodinger perturbation theory in the tunneling Hamiltonian. In particular we discuss the dependence of charging eects on the number N of tunneling channels. Since the junction area is typically much larger than 2F , where F is the Fermi wavelength, the number of tunneling channels N is often very large for metallic junctions, typically about 104 . Terms of order 1=N are thus dropped in most approaches. In contrast to lithographically fabricated metallic junctions, in break junctions [51] only a small number of channels may be available and terms proportional to 1=N cannot be neglected. Also for tunnel barriers [52] in a two-dimensional electron gas there are typically only 2 spin degenerate transport channels contributing to the tunneling current. Thus, it is interesting to investigate how Coulomb blockade eects are modi ed by such 1=N corrections. Here we focus on systems where the dimensionless conductance per channel, g0 = GT =42 N GK , remains small, that is to cases where the channels contributing to charge transfer are weakly transmitting. Further we restrict ourselves to the zero temperature case. This covers only partly the range of experimental interest but the results indicate how relevant 1=N corrections are.

2.8. FINITE CHANNEL NUMBERS

a)

CT G T N

n

39

b)

Cg

e Ug

Figure 2.16: a) Circuit diagram of a metallic grain (area within dashed line) coupled by a tunnel junction to the left electrode and capacitively to the right electrode. b) Electron{hole pair excitation in channel of the tunnel junction. We use the Hamiltonian and de nitions given in Sec. 2.1 that are valid for arbitrary channel numbers. At zero temperature the system is described by the ground state energy E and in view of the free Hamiltonian (2:2) the expectation value of the island charge may be written @E : (2.83) hni = ng , 2E1 @n C g To calculate E we make use of Rayleigh{Schrodinger perturbation theory. The perturbation (2:6) contains products of one creation and one annihilation operator, thus only even powers in t contribute to E , and it may be written E = E (0) + E (2) + E (4) + O(t6 ): (2.84) The zeroth order term E (0) without tunneling is given by the minimum of the electrostatic energy and reads EC (n0 , ng )2 where n0 is the integer closest to ng . Hence, the average island charge in zeroth order perturbation theory is given by the well known Coulomb staircase. E depends on ng only via the electrostatic energy. It is thus an antisymmetric and quasiperiodic function of ng which allows us to con ne ourselves to 0 ng < 21 . For the second order term we get formally E (2) = h0jHT QHT j0i (2.85) with the auxiliary operator ih0j : Q = 1E,,j0H (2.86) 0 0 In the same way the fourth order term reads E (4) = h0jHT QHT QHT QHT j0i , h0jHT Q2HT j0ih0jHT QHT j0i (2.87) where terms with the energy denominator squared arise from the normalization of the ground state wave function. Inserting the tunneling Hamiltonian (2:6) into the second order contribution (2:85) one gets X 2 " (,q )(k ) (q )(,k ) # (2) E = , t + , + , + (2.88) kq

1

k

q

,1

k

q

40


a) b)

c)

d)

Figure 2.17: Graphical representation of Rayleigh{Schrodinger perturbation theory. a) Graphs of second order. b), c), and d) Graphs of fourth order where graphs with inverted arrows are omitted. with the Coulomb energy dierences p = EC (p2 , 2png ). Both summands correspond to the virtual creation of an electron{hole pair with electron and hole sitting on dierent electrodes, cf. Fig. 2.16b). This can be represented in terms of Goldstone graphs depicted in Fig. 2.17. The contributions of second order correspond to the diagrams in Fig. 2.17a) where the upper arc with an arrow to the right (left) electrode represents the creation of an electron{hole pair with the electron sitting on the right (left) and the hole sitting on the other electrode of the junction. The lower arc destroys this pair and we may omit the arrow since the process is uniquely determined by the upper one. If we assume that the one particle energy is separable into a longitudinal and channel part, the summand in Eq. (2:88) is independent of the channel apart from the factor t2 . The sum over the transversal P and spin quantum numbers leads to an overall factor N multiplied by the average t2 = t2 =N of the tunneling matrix elements. We assume the bandwidth D to be large compared to EC and take the limit D=EC ! 1 at the end of the calculation. To keep the nite bandwidth in intermediate formulas, we introduce an exponential cuto and replace the sum over the longitudinal quantum numbers by

X k

F (k ) ,!

Z1

,1

de,jj=D F ()

(2.89)

where is the density of states at the Fermi level. Analogously, we introduce an integral with 0 and D for the q sum. Formula (2:88) then takes the form 1 Z1 1 (2) 2 0 ,j j =D (2.90) E = ,t N d e 1 + + ,1 + : 0


41

1 1. order 2. order, without 1/N 2. order, with 1/N

*

EC /EC

N=5

0.6

0.2

0

4

8

GT /GK

Figure 2.18: Eective charging energy for channel number N = 5 as a function of the dimensionless tunneling conductance GT =GK in rst and second order perturbation theory with and without 1=N corrections. We now de ne the dimensionless tunneling conductance per channel by g0 = t2 0 . Whereas the integral (2:90) is divergent for D ! 1, the average island charge to rst order in g0 remains nite leading to

!

hni = g0 N ln 11 ,+ 22nng + O(g2 ) g

(2.91)

in accordance with the previous result (2:60) Along these lines the fourth order term (2:87) is given in terms of double sums over kq. These contributions can also be represented graphically as twofold virtual electron{hole pair creation and annihilation. We distinguish three groups of graphs depicted in Figs. 2.17b), c) and d). The rst set of the graphs represents virtual electron hole pair creation and annihilation in two distinct channels and 0 which are only coupled by the energy denominator. Fig. 2.17c) represents graphs arising from the normalization of the wave function where the dash signs that the energy denominator is squared, cf. Eq. (2:87). These graphs with dashes correspond to graphs with insertions in the imaginary time perturbation expansion in Sec. 2.2 where the energy denominator is squared as well. If there were no Coulomb interaction, i.e. EC = 0, the graphs b) and c) would cancel in accordance with the linked cluster theorem for uncorrelated fermions. Here we have to take these terms into account since the Coulomb energy correlates the electrons in the two electrodes. The channels of the graphs in b) and c) are not restricted and we get by summation over the transversal quantum numbers a factor N 2 . On the other hand, the graphs in Fig 2.17d) describe processes within one channel because the electron created recombines not with the hole created at the same time but with a dierent one which has to have the same channel number. Thus we get by summation over the channels only a factor N . Therefore, we

42

CHAPTER 2. PERTURBATION EXPANSION 0.5

1. order 2. order, without 1/N 2. order, with 1/N

GT/GK=5 N=5

0.25

0.0 0.0

0.25

0.5

ng Figure 2.19: The average island charge number for dimensionless tunneling conductance GT =GK = 5 and channel number N = 5 as a function of the dimensionless gate voltage ng . write the average island charge in the form

!

hni = g0 N ln 11 ,+ 22nng + (g0 N )2 [c(ng ) , c(,ng )] + g02 N [d(ng ) , d(,ng )] + O(g03 ): (2.92) g Previous theories for N 1 introduce the dimensionless conductance g = g0 N of the junction and the terms proportional to g02 N = g=N are dropped, cf. Sec. 2.2. The graphs in Fig. 2.17 b) and c) do not include 1=N {corrections and they lead to the known second order result for the island charge [24]

"

#

2u 16(1 + 2u , 2u2 ) 2 c(u) = ,u 43 + ln2 11 , + 2u , (3 , 2u)(1 + 2u) ln(1 , 2u) 3 , 2u 1 , 2u 8(1 , u) ln[4(1 , u)] ; (2.93) + 2Li , ,2(1 , u) ln2 4(1 2 4(1 , u) , u) (1 , 2u)(3 , 2u)

corresponding to Eq. (2:61). The graphs in Fig. 2.17 d) can also be integrated out analytically leading to a new contribution of order 1=N . We nd

4 , 4u 26 8 3 d(u) = 3 ln 1 , 2u , 3 ln3 (1 , 2u) + 15 ln3 (3 , 2u) 1 , 2u 2 +[15 ln(2) + 16 ln(4 , 4u)] ln 3 , 2u + [2 ln(1 + 2u) , 7 ln(3 , 2u)] ln2 (1 , 2u) "

# 4 , 4u 102 4 3 , 2u 1 , 2 u 2 2 + 1 , 2u , 3 + 38 ln (3 , 2u) ln 3 , 2u + 3 ln 4 , 4u 2u 1 , 2u 3 , 2u 8(1 , u) +4 ln 13 , 3 Li + 3 Li + 2 Li 2 3 , 2u 2 4 , 4u 2 (3 , 2u)2 1,,2u2u 3 , 2u 1 , 2u +6 Li3 3 , 2u , 8 Li3 4 , 4u , 8 Li3 4 , 4u (2.94) 10 ln2


43

k

Figure 2.20: Goldstone graph of order g0k which gives the leading asymptotic contribution for ng ! 21 . where Li2 (z ) is the dilogarithm and Li3 (z ) the trilogarithm function [44]. Within second order perturbation theory the result (2.92) is valid for arbitrary channel numbers including N = 1. We now compare our result to earlier ndings. One of the mostly frequently discussed quantities is the eective charging energy [38] EC characterizing the eective strength of the Coulomb blockade eect. This quantity is de ned by

2 EC =EC = 1 @ E2

(2.95) 2 @ng ng =0 or equivalently by Eq. (2:73). For small tunneling conductance, g0 ! 0, the eective charging energy approaches the bare charging energy EC whereas for strong electron tunneling EC vanishes. Our analytic expression leads to

EC =EC = 1 , 4g + 5:066 : : : g2 , 7:167 : : : g2 =N

(2.96)

where the constants are analytically known and can be readily obtained from the expressions for the charging energy and/or the average charge number. In Fig. 2.18 we show the normalized eective charging energy in rst and second order perturbation theory without 1=N corrections compared to the complete second order result as a function of the dimensionless conductance GT =GK . We see that the 1=N corrections become more signi cant for larger tunneling conductance. In Fig. 2.19 the average island charge hni in rst and second order in g0 with and without the 1=N corrections is depicted for the case N = 5 and GT =GK = 5. We see that the 1=N corrections become signi cant especially for larger external voltages. From the comparison with quantum Monte Carlo data in Fig. 2.13 one estimates that results of second order perturbation theory are reliable for ng up to 0:3. In the vicinity of the step, i.e., for ng ! 21 , nite order perturbation theory diverges and one has to sum diagrams of all orders, cf. Sec. 2.5. From our Eq. (2:94) one sees that the 1=N {corrections become relevant near ng = 21 even for large N since the qualitative behavior is changed. For ng ! 12 , the leading N terms of second order in

44


Eq. (2:63) show the logarithmic divergence 2(g0 N )2 ln2 where = 21 , ng , whereas the 1=N corrections lead to , 43 g02 N ln3 which eventually dominates the asymptotic behavior. For the special case N = 1, this divergence is in accordance with earlier ndings by Matveev [28]. In general, for the nite N corrections of order g0k have the leading asymptotic behavior

hni g0k N ln2k,1 + O(gk+1 )

(2.97)

arising from the diagram depicted in Fig. 2.20. These terms dominate the asymptotics for ng ! 21 , even for large N . In summary, we found that for small voltages, ng 0, the 1=N corrections are signi cant up to N 6 and become increasingly important as ng = 12 is approached. While near the step nite order perturbative results as derived here are not sucient, the expression obtained for the eective charging energy is valid within an experimentally relevant range of parameters.

2.9. APPENDIX

45

2.9 Appendix

2.9.1 Third Order Integrands

In this appendix we show all integrals of third order represented by the graphs in Fig. 2.5 and 2.6. The general form of the contributions in third order is a three fold integral weighted by bosonic spectral densities and includes ve energy denominators

Z1

Z1

Z1

,j1 j=D ,j2 j=D ,j3 j=D d3 11,e e, 1 12,e e, 2 13,e e, 3 Ii :

(2.98)

1 1 2 1 2 3 1 3 with the possible combinations (1 ; 2 ; 3 ) ! (1 ; 2 ; 2 ), (1 ; 0; 0), and 2 (1 ; 2 ; 0).

The fourth

(3n ;i) = ,g3

,1

d1

,1

d2

,1

Here, Ii for i = 1; : : : ; 16 represents the dierent types of energy denominators that are given below. The contributions of all graphs corresponding to a representant are generated by replacing the corresponding i by energy dierences from Eq. (2:43). Since a full re ection of the graphs lead to the replacement (n , ng ) ! ,(n , ng ) we give just half of the possible energy dierences for the j s and the remaining expressions are obtained by replacing the arguments. The integrand corresponding to the graph (1) in Fig. 2.5 is given by (2.99) I1 = ( + )2 ( + + 1 )2 ( + + + ) ; 1 1 2 1 2 3 1 2 3 the possible energy dierences are (1 ; 2 ; 3 ) ! (1 ; 2 ; 3 ), (1 ; 2 ; 1 ), (1 ; 0; 1 ), and (1 ; 0; ,1 ). The second graph leads to I2 = ( + )2 ( + + )( 1+ + + )( + + ) (2.100) 1 1 2 1 2 3 1 2 3 4 1 3 with (1 ; 2 ; 3 ; 4 ) ! (1 ; 2 ; 3 ; 2 ), (1 ; 0; ,1 ; 0), and 2 (1 ; 2 ; 1 ; 0). The third term is given by (2.101) I3 = ( + )3 ( + 1+ )( + + ) graph is represented by

(2.102) I4 = ( + )( + + )( +1 )( + + )( + ) : 1 1 2 1 2 3 2 4 2 3 5 3 The possible replacements are (1 ; 2 ; 3 ; 4 ; 5 ) ! (1 ; 2 ; 1 ; 2 ; 1 ), (1 ; 0; ,1 ; 0; 1 ), and 2 (1 ; 0; ,1 ; ,2 ; ,1 ). The fth term reads

I5 = ( + )( + + )2 (1 + + + )( + ) (2.103) 1 1 2 1 2 3 1 2 3 4 2 with (1 ; 2 ; 3 ; 4 ) ! (1 ; 2 ; 3 ; 1 ), (1 ; 2 ; 1 ; 1 ), and 2 (1 ; 0; 1 ; ,1 ). The sixth graph is given by

1 I6 = ( + )( + + )( + + (2.104) 1 1 2 1 2 3 1 2 + 3 )(4 + 2 + 3 )(5 + 3 ) with the possible combinations (1 ; 2 ; 3 ; 4 ; 5 ) ! (1 ; 2 ; 3 ; 2 ; 1 ), (1 ; 0; 1 ; 0; 1 ), and 2 (1 ; 2 ; 1 ; 0; ,1 ). The seventh term is counted twice because it's horizontally re ected version is a topologically dierent graph leading to the identical integrand that reads I7 = ( + )( + + )( + +1 + )( + + )( + ) : 1 1 2 1 2 3 1 2 3 4 1 3 5 3

(2.105)

46


Here, the possible energy dierences are (1 ; 2 ; 3 ; 4 ; 5 ) ! (1 ; 2 ; 3 ; 2 ; 1 ), (1 ; 0; 1 ; 2 ; 1 ), (1 ; 0; ,1 ; 0; ,1 ), and (1 ; 2 ; 1 ; 0; ,1 ). The eighth term yields the integrand (2.106) I8 = ( + )2 ( + + 1)( + + )( + ) 1 1 2 1 2 3 1 3 4 3 and is counted twice because of the same argument as for graph seven. The dierent replacements are (1 ; 2 ; 3 ; 4 ) ! (1 ; 2 ; 2 ; 1 ), (1 ; 0; 2 ; 1 ), (1 ; 0; 0; ,1 ), and (1 ; 2 ; 0; ,1 ). The following six integrands contain one insertion and therefore gain a minus sign. The ninth integrand reads I9 = ( + )2 ( +, 1 + )2( + ) (2.107) 1 1 2 1 2 3 3 and the possible combinations are (1 ; 2 ; 3 ) ! (1 ; 2 ; 1 ), (1 ; 2 ; ,1 ), (1 ; 0; 1 ), and (1 ; 0; ,1 ). The tenth graph is given by (2.108) I10 = ( + )( + +, 1)2 ( + )( + ) 1 1 2 1 2 3 3 4 2 where the replacements read (1 ; 2 ; 3 ; 4 ) ! (1 ; 2 ; 1 ; 1 ), (1 ; 2 ; ,1 ; 1 ), (1 ; 0; 1 ; ,1 ), and 2 (1 ; 0; ,1 ; ,1 ). The eleventh term is counted twice and reads (2.109) I11 = ( + )2 ( + )(,1 + + )( + ) 1 1 2 2 3 1 3 4 3 with the possible combinations (1 ; 2 ; 3 ; 4 ) ! (1 ; 1 ; 2 ; 1 ), (1 ; ,1 ; 2 ; 1 ), (1 ; 1 ; 0; ,1 ), and (1 ; ,1 ; 0; ,1 ). The twelveth integrand I12 = ( + )3 ( +,1 )( + + ) (2.110) 1 1 2 2 3 1 3 is counted twice as well, with the allowed energy dierences (1 ; 2 ; 3 ) ! (1 ; 1 ; 2 ), (1 ; 1 ; 0), (1 ; ,1 ; 2 ), and (1 ; ,1 ; 0). The thirteenth integrand reads I13 = ( + )2 ( + )(,1 + + )( + ) : (2.111) 1 1 2 2 3 2 3 4 3 The possible combinations are given by (1 ; 2 ; 3 ; 4 ) ! (1 ; 1 ; 2 ; 1 ), (1 ; ,1 ; ,2 ; ,1 ), and 2 (1 ; 1 ; 0; ,1 ). The fourteenth graph leads to the integrand (2.112) I14 = ( + )2 ( +, 1)2 ( + + ) 1 1 2 2 3 2 3 with the replacements (1 ; 2 ; 3 ) ! (1 ; 1 ; 2 ), (1 ; 1 ; 0), (1 ; ,1 ; 0), and (1 ; ,1 ; ,2 ). The following two graphs contain two insertions and read I15 = ( + )2 ( +1 )2 ( + ) (2.113) 1 1 2 2 3 3 representing the fteenth graph with the replacements (1 ; 2 ; 3 ) ! (1 ; 1 ; 1 ), (1 ; 1 ; ,1 ), (1 ; ,1 ; 1 ), and (1 ; ,1 ; ,1 ). The sixteenth graph nally gives I16 = ( + )3 ( 1+ )( + ) (2.114) 1 1 2 2 3 3 where the possible combinations read (1 ; 2 ; 3 ) ! (1 ; 1 ; 1 ), (1 ; 1 ; ,1 ), (1 ; ,1 ; 1 ), and (1 ; ,1 ; ,1 ). Inserting the given combinations of energy dierences and adding all terms with (n , ng ) ! ,(n , ng ) the entire 160 integrands represented by the graphs in Figs. 2.5 and 2.6 are obtained.

2.9. APPENDIX

47

2.9.2 Third Order Result

In this appendix we give the explicit analytic result of the third order contribution to the ground state energy E . According to the transcendental functions appearing, we split the result into six dierent terms g3 (u) = P 3(u) + P 2(u) + L3(u) + L2(u) + L1(u) + R(u) : (2.115) Here P 3 denotes terms containing trilogarithms Li3 and P 2 dilogarithms Li2 . Terms consisting of logarithms and no other transcendental functions are split into three types: In L3 expressions containing ln3 and in L2 terms with ln2 are listed. Simple logarithms appear in L1 and the remaining rational functions of ng and constants are gathered in R. Terms containing the trilogarithm function are given by

P 3(u) = (2.116) ! 207 223 u 1 79 211 u 2 1 2 3 2 3 4 + 2 + 105 u + 34 u Li3 2 + 2 + 2 + 102 u + 34 u Li3 2 " 1 1 2 ! 3 ! 3 ! 3 !# 3 , 1 2 2 2 +1 Li3 2 , 2 Li3 2 2 + Li3 2 3 + Li3 2 , 2 Li3 2 2 + Li3 2 3 2 2 1 1 1 1 1 127 55 u 9 u2 ! " 2 3 ! 63 31 u , 4 + 2 + 9 u2 + 2 u3 Li3 , 8 + 4 + 2 + u3 Li3 1 3 3 2 1 !# ! " ! , 2 , 2+ ! 3 2 (,1 + 2 ) 3 7 15 u 9 u 1 1 , 1 2 3 + Li3 3 + 8 + 4 + 2 + u , Li3 3 2 Li3 2 3 2 2 1 (1 ) 1 !# 1 3 5 3 u 15 , 1 1 2 3 2 + Li3 3 , 2 Li3 , 2 , 2 , 6 u , 2 u Li3 3 + 4 , 6 u , 3 u 3 1 ! !2 " !# 2 2 2! 3 3 119 47 u 9 u 2 3 1 2 Li3 3 , (18 + 8 u) Li3 + 8 + 4 + 2 + u Li3 3 + Li3 3 2 1 1 ! 3 ,2 2 ! 2 3 15 u , 9 u , u Li 1 1 : + 25 , 3 3 2 16 8 4 2

(2 ) The expressions containing dilogarithms read

P 2(u) = ,1 1 ,,1 33 1 (2.117) 4 1 + 4 u2 ln Li2 2 , Li2 , 2 + 59 u + 62 u2 + 20 u3 ln 2

! 1 2 2

1

1 2 127 55 u 9 u2 1 ! Li2 2 2 + 33 + 118 u + 124 u2 + 40 u3 ln 2 Li2 2 , 8 + 4 + 2 + u3 (1 )! 1 1 ! ! ! 2 2 2 3! " 3 119 47 u 9 u 3 1 2 1 1 2 3 ln 2 Li2 3 + 8 + 4 + 2 + u ln 3 Li2 3 , 49 + 72 u + 36 u2 2 1 1 3 1 ! 2 49 2 37 39 u 3 u 145 425 u 239 u2

+ 4 + 6 u + 3 u2 ln(2 ) + 8 + 4 + 2 + u3 ln(,1 ) + 8 + 4 + 2 ! ! ! # 2 2 145 425 u 239 u 63 u 9 u 135 3 3 2 3 3 +39 u ln(1 ) , 8 + 4 + 2 + 39 u ln(1 ) , 8 + 4 + 2 + u ln(2 )

48


0 1 0 ,2 1 2 7 + 15 u + @ ,2(1) A Li2 @ ,1 +3 22 3 A + ln 16 8 4 2 (1 ) ,1 + 2 3 ! ! ! ,2 + 2 ! 3 2 7 15 u 9 u 1 , 1 2 3 Li Li2 2 , 8 + 4 + 2 + u ln ,2 13 ,1 + 2 2 2 , 6 u , 3 u2 ln(2) + ,1 23 [,1 ln(,1) + 18 25 , 30 u , 36 u2 , 8 u3 ln 3 , 8 15 4 2 1 ,22 1 , 3 , 3 , 3 3 2 +2 ln(1 )] Li2 3 , 2 ,1 1 , (11 + 6 u) ln 3 , 2 ,1 ,2 + ,1 ln 3 2 1 ! ! 2 2 105 17 u 9 u 7 15 u 9 u 3 3 , 8 + 4 , 2 , u ln(1 ) + 8 + 4 + 2 + u ln(,,12 + 2 ) ! ! ! 2 2 3 247 95 u 9 u 135 63 u 9 u 1 3 2 3 3 + 8 + 4 + 2 + u ln(1 ) , 8 + 4 + 2 + u ln(2 ) Li2 3 2 1 ! ! 1 3 2 2 3 95 39 u 9 u 1 2 , 2 3 +1 ln 3 3 Li2 2 3 , 8 + 4 + 2 + u ln 3 + 12 ,2 ,1 + ,1 ln 3 2 1 2 135 63 u 9 u2 3 ! 55 u + 9 u2 + 2 u3 ln( 2 ) 1 + , 2 ,2 1 + ,2 , 8 + 4 + 2 + u ln(1 ) + 127 1 4 2 ! 2 3! h 3 ! i 119 47 u 9 u 3 3 2 3 2 , 8 + 4 + 2 + u ln(2 ) Li2 , 2 1 ln(1 ) , 2 1 ln(2 ) Li2 2 2 :

2! 1 Li2 + 2 2 ! , +1 ln 31 2

9 u2

! u3

3

1

Terms involving ln3 are given by

L3(u) =

(2.118)

1 129 + 98 u , 164 u2 + 56 u3 ln( ) ln( )2 , 1 131 , 282 u + 276 u2 , 88 u3 ln( )3 1 2 2 16 8 1 295 , 798 u + 804 u2 , 264 u3 ln( )2 ln( 2 ) + 1 559 , 1742 u + 1796 u2 , 584 u3 + 16 1 1 16 49 2 2 3 2 ln(2 ) ln(1 ) , 2 , 65 u + 62 u , 20 u ln(1) ln(12 )2 , 33 , 118 u + 124 u2 , 40 u3 ln(2 ) ln(12 )2 , 18 (1 )2 (,,12 + 2) ln(1 )2 ln(13 ) + 481 1087 , 3054 u + 3012 u2 , 968 u3 ln(12 )3 , 161 89 , 142 u + 60 u2 , 8 u3 ln(2)2 ln(23 ) + 18 119 , 94 u + 36 u2 , 8 u3 h i ln(12) ln(13)2 + ln(1 ) ln(2 ) ln(23 ) + ln(1 ) ln(13 ) ln(23 ) , 2 ln(1) ln(12 ) ln(23 ) 183 71 u 2 3 + 4 , 2 + 9 u , 2 u ln(1 ) ln(12 ) ln(13 ) , 81 105 , 34 u , 36 u2 + 8 u3 ln(1 ) ln(13 )2 , 18 (23 )2 ,1 ln(2) ln(12) ln(23 ) , 121 (1)2 (,,12 + 2) ln(13)3 + 81 127 , 110 u + 36 u2 , 8 u3 h i ln(2)2 ln(13 ) + ln(2 ) ln(13 )2 , 2 ln(1 ) ln(2 ) ln(13 ) , ln(13 )2 ln(23 ) + ln(2 13 ) ln(23 )2 1 433 , 530 u + 252 u2 , 56 u3 + 18 135 , 126 u + 36 u2 , 8 u3 ln(12 ) ln(13 ) ln(23 ) , 16 ln(12 ) ln(23 )2 , 241 55 + 78 u , 60 u2 + 8 u3 ln(23 )3 + 81 293 , 266 u + 108 u2 , 24 u3

2.9. APPENDIX

49

!

39 u + 3 u2 , u3 ln(1 ln(,1 ) + 37 , 8 4 2 ln(1 ) ln(2 ) ln(,1 ) , 13 10 + 48 u + 24 u2 + 32 u3 ln(,1 )3 + 18 (1)2 ,,23 ln(1)2 ln(,2) , 161 67 + 30 u , 12 u2 + 8 u3 ln(1 ) ln(,2 )2 , 81 (1 )2 (,,12 + 2) ln(1) ln(13 ) ln(,,12 + 2) , 18 373 , 314 u + 108 u2 , 24 u3 ln(12)2 ln(13 ) + 81 (1)2 (,,12 + 2) ln(12) ln(13 ) ln(,,12 + 2) , 18 31 + 146 u , 188 u2 + 56 u3 ln(1 ) ln(2 ) ln(12 ) + 161 35 , 66 u , 12 u2 + 8 u3 ln(1 ) h i ln(,,23 )2 + 18 1 (,,23 )2 ln(1) ln(,2) ln(,,12 )+ 161 (1)2(,,12 + 2) ln(1)2 , ln(12)2 ln(,,12 + 2) ,81 1 (,,23)2 ln(1 ) ln(,,12 ) ln(,,23 ) , 81 (1 )2 ,,23 ln(1) ln(,2) ln(,,23 ) , 161 (1)2 ,,23 ln(1 )2 ln(,,23) :

ln(12 )2

1 ln(23 ) , 16

69 , 18 u + 140 u2 + 264 u3

)2

Terms containing only two logarithm functions multiplied are given by

L2(u) =

(2.119)

,1 h49 + 12 ln 2 , u (38 , 16 ln 2) , 4 u2 (45 , 8 ln 2) + 8 u3 (15 + 8 ln 2) , 64 u4 ln 2i 12 ,1 ln(1)2 + [49 + 2 ln 2 , 2 ln 3 + 4 u (,18 , ln 2 + ln 3) + 36 u2 ] ln(1 ) ln(2 ) , 221 [127 + 6 ln 2 , 6 ln 3 , 4 u (123 + 7 ln 2 , 7 ln 3) + 8 u2 (92 + 5 ln 2 , 5 ln 3) 1 1 h i ,16 u3 (33 ln 2 , ln 3) + 144 u4 ]ln(2)2 + 22 ,19 + 35 u , 20 u2 + 4 u3 ln(2) ln(12) 1 h i 2 , 2 14 + 6 ln 2 + 18 ln 3 , u (23 + 16 ln 2 + 18 ln 3) + 4 u2 (4 + 2 ln 2 + ln 3) , 4 u3 1

ln(1 ) ln(12 ) , 1 ln 2 ln(13 )2 + 161 [176 + 32 ln 2 + 263 ln 3 , 2 u (96 + 32 ln 2 + 79 ln 3) +4 u2 (16 + 9 ln 3) , 8 u3 ln 3] ln(12 )2 + 18 [16 ln 2 + 231 ln 3 , 2 u (16 ln 2 + 79 ln 3) +36 u2 ln 3 , 8 u3 ln 3 ] ln(1 ) ln(13 ) , 81 [119 , 94 u + 36 u2 , 8 u3 ] ln 3 ln(1 ) ln(23 ) , 81 [373 , 314 u + 108 u2 , 24 u3 ] ln 3 ln(12) ln(13) + 81 [213 , 218 u + 108 u2 , 24 u3 ] ln 3 ln(12 ) ln(23 ) , 18 (23)2 ,1 ln 3 ln(2 ) ln(23 ) + 18 (135 , 126 u + 36 u2 , 8 u3 ) ln 3 ln(13 ) ln(23 ) , 161 (179 , 310 u + 180 u2 , 40 u3 ) ln 3 ln(23 )2 , 12 [13 , 8 ln 2 , 39 u ln 2 + u2 (36 , 32 ln 2) ,4 u3 ln 2] ln(1 ) ln(,1 ) + 8 1 [48 + 25 ln 3 + 4 u (12 + 5 ln 3) + 32 u2 (2 , 3 ln 3) ,1 2! 1 2 1 3 4 , 2 +16 u (4 , 5 ln 3) , 16 u ln 3] ln(1 ) ln(,2 ) , 8 (1 ) (,1 + 2 ) ln 3 ln 3 ln(,,12 + 2 )

, 81 1 (,,23)2 ln 3 ln(1 ) ln(,,23 ) : Single logarithms are listed in

1

50


"L1(u) =

(2.120)

# 3 2 2 2 2 6 2 + 295 , 798 u + 804 u , 264 u 48 , 1 ln 2 , 19 , 16 u + 4 u ln 3 2 2 ln(2 ) 1 3 2 , 79 , 402 u + 356 u2 , 376 u3 48 + 7 , 8 u + 4 u2 ln 3 2 1 2 + 2 (3 , u) ln(3)2 1 2 +1 (15 + 2 ln 2 ln 3) ln(1 ) , 3 12 + 861 , 1354 u + 1100 u2 , 344 u3 96 , 2 1 ln(2)2 2 1 2 2 3 2 3 + 48 197 + 22 u + 12 u , 24 u + 6 47 , 62 u + 36 u , 8 u ln(3) ln(23 ) ! 2 2 119 47 u 9 u 2 2 2 3 , 2 , 1 ln(2) , (1 ) (,1 + 2) 24 + 8 , 4 + 2 , u ln(3) ln(13 ) ,2 11 , 12 u + 4 u2 ln 3 , 161 263 , 158 u + 36 u2 , 8 u3 ln(3)2 ln(12 ) 1 ( )2 ( ,2 + ) h2 , 6 ln(3)2 i ln( ,2 + ) : + 96 1 ,1 2 ,1 2 2 3

The remaining rational functions and constants are

R(u) = (2.121) " # 2 2 , + 6 2 ln 2 , 48 ln(2)3 , 9 , 265 , ln(2)2 ln 3 + (11 + 37 ln 2) ln(3)2 , 953 ln(3)3 2

2

96

1

1

48

,38 ln 3 Li2 3 , 18 ln 3 Li2 4 + 18 Li3 4 + 38 Li3 3 + 18 Li3 34 , 1845 (3) 16 # " 2 2 3 3 651 4 2 3 2 +u 3 , 8 ln 3 + 4 ln(3) + 4 ln(3) , 4 (3) :

Here, we used the abbreviation

3

nm = m , n for energy dierences where we have to set EC = 1 throughout this appendix.

(2.122)

Chapter 3

The Semiclassical Approach In this chapter we study the linear conductance of single electron devices showing Coulomb blockade phenomena. We exemplarily derive a formally exact path integral representation describing electron tunneling nonperturbatively. The electromagnetic environment of the devices is treated in terms of the Caldeira{Leggett model. We obtain the linear conductance from the Kubo formula leading to a formally exact expression that is evaluated in the semiclassical limit. Speci cally we consider three models: First, the in uence of an electromagnetic environment of arbitrary impedance on a single tunnel junction is studied focusing on the limits of large tunneling conductance and high to moderately low temperatures. The predictions are compared with recent experimental data. Second, the conductance of an array of N tunnel junctions is determined in the high temperature limit in dependence on the length N of the array and the environmental impedance. The predictions are compared in the weak tunneling regime with recent numerical data based on master equation approach. Finally, we consider a single electron transistor. Here, we go beyond the leading order corrections and determine the conductance in dependence on the gate voltage. The ndings are compared with recent experimental data in the strong tunneling regime.

3.1 Model and General Method In this section we introduce the Hamiltonian for a single tunnel junction and model the electromagnetic environment in terms of a set of LC circuits. A metal - oxide layer - metal tunnel junction consists of two metallic leads separated by a thin oxide layer [2, 4]. Provided the screening length in the metal is small compared to typical electrode and oxide barrier dimensions, one may introduce a geometrical capacitance C . The energy shift for an electron tunneling from one lead to the other is determined by the charging energy EC = e2 =2C . The corresponding Coulomb Hamiltonian reads 2 (3.1) HC (Q) = 2QC ;

where Q is the charge operator on the capacitance. The leads are described in second quantization by X X Hqp = k ayk ak + q ayq aq ; (3.2) q

k

where the p are quasiparticle energies, and ayp and ap are creation and annihilation operators for states on the two electrodes, respectively. The indices p = k, q are longitudinal wave numbers and is the channel index including transversal and spin quantum numbers. Provided the 51

52

CHAPTER 3. THE SEMICLASSICAL APPROACH

tunneling amplitudes are small, we may describe barrier transmission by a tunneling Hamiltonian [2, 53] X y HT(') = tkq ak aq + H:c: ; (3.3) kq tkq

preserving the channel index . Here is the tunneling amplitude and the charge shift y operator obeying Q = Q + e. De ning a conjugate phase ' by [Q; '] = ie, we may write = exp(,i'): (3.4) The total Hamiltonian of a tunnel junction then reads HJ (Q; ') = HC (Q) + Hqp + HT('); (3.5) where the dependence on the charge and conjugate phase operators is made explicit to emphasize the similarity between the charging energy and a kinetic energy and between the tunneling Hamiltonian and an eective potential energy. The electromagnetic environment can be described by a Caldeira-Leggett model [54] as a linear combination of LC circuits # h 2 N " Q2 X 1 2 n + (' , n ) ; (3.6) Hem(') = 2 C n 2Ln e n=1 coupled to the phase operator ' of the device. The parameters of the LC-circuits are related to the environmental admittance by

Y (!) =

p

N X L [(! + !n) + (! , !n)] ;

n=1 n

(3.7)

where the !n = 1= LnCn are the eigenfrequencies of the oscillators. A single electron tunneling device consists of tunnel junctions, capacitances, and admittances. The Hamiltonian of this system can be constituted from the elements discussed above. Then, the bosonic degrees of freedom of the admittance and the fermionic degrees of freedom of the electrodes may be traced out. In the next section we exemplarily derive the Hamiltonian and the eective action for a tunnel junction embedded in an environment of arbitrary impedance. Speci cally, in this article we determine the linear conductance G = @I=@V jV =0 of single electron devices. Here I is the measured current and V the applied voltage. Within linear response theory one may use the Kubo formula for the linear conductance Z h 1 lim d ein hI (1) ( )I (2) (0)i ; (3.8) G(!) = ih ! i ! n !+i 0 where I (1) is the measured current and I (2) a current operator determined by the coupling to the applied voltage V , see below. The n = 2n=h are Matsubara frequencies. Correlation functions can be written as variational derivatives 2 2 Z [1 ; 2 ] h (1) (2) 0 hI ( )I ( )i = Z [0; 0] ( ) ( 0) (3.9) 1 2 i 0

of a generating functional [34]

9 8 Z < 1 h X (i) = Z [1; 2 ] = tr T exp:, h d H , I i( ) ; 0

i=1;2

(3.10)

3.2. TUNNEL JUNCTION WITH ENVIRONMENT

53

C , GT

Y( )

V

I

Figure 3.1: Circuit diagram of a tunnel junction in series with an admittance. depending on auxiliary elds i . Here, the Hamiltonian H describes the system at vanishing external voltage, V = 0, and T is the Matsubara time ordering operator. In subsequent sections we apply this generating functional approach to derive an explicit expression for the linear conductance in the semiclassical regime.

3.2 Tunnel Junction with Environment

3.2.1 Generating Functional

We consider a tunnel junction characterized by the geometrical capacitance C and the tunneling conductance GT embedded in an electromagnetic environment. Via network transformations it is always possible to transform the environmental degrees of freedom into an admittance Y (!) in series with the junction biased by a voltage source V , cf. Fig. 3.1. In this subsection we obtain the eective action characterizing the generating functional introduced above. Readers familiar with path integral techniques for single electron devices may directly proceed to the next subsection. The Hamiltonian is given by H = HJ (QJ ; 'J ) + Hem('em ). Here the phases 'J and 'em are related to the voltages VJ and Vem across the tunnel junction and the admittance, respectively, by '_ J = he VJ and '_ em = he Vem . Further, one has to take care of constraints for the variables imposed by the circuit. Using Kirchho's law for the voltages, we nd that the sum of the phases in the circuit loop in Fig. 3.1 has to be constant, i.e. 'J + 'em + = const:, where we have described the voltage source in terms of an additional phase [8] (t) = he

Zt

,1

dt0 V (t0 ):

(3.11)

Similar relations hold for each loop of more complicated circuits. For an adequate handling of these constraints we start from the Lagrangian description, L = T , U . In general, the kinetic energy T is given by the sum of Coulomb energy terms and the eective potentials are the tunneling and environmental Hamiltonians. The constraints are naturally implemented by expressing the variables through generalized coordinates. De ning generalized momenta in the standard way, one can derive the Hamiltonian via a Legendre transformation. To de ne conjugate momenta non-ambiguously, we use the discrete Caldeira-Leggett model and perform the continuum limit only afterwards. Shunt capacitors need to be treated separately and will

54


be discussed in Sec. V. Since (t) is controlled externally, the phase 'em may be eliminated in favor of 'J ' and we may write

HJE(Q; ') = HJ (Q; ') + Hem(' + );

(3.12)

where Q = he @ L=@ '_ is the momentum canonically conjugate to '. In the second term, we have absorbed the minus sign in front of ' + into the arbitrary de nition of the sign of the phase of the environment. The current may be de ned as the time derivative of the charge where

Q_ = hi [HJE ; Q] = IT + Iem;

(3.13)

X y tkq ak aq , H:c: IT = hi [HT('); Q] = , ieh

(3.14)

is the current through the tunnel junction and

kq

N X Iem(') = hi [Hem('); Q] = he L1 (' , n ) n=1 n

(3.15)

the current through the admittance at vanishing external voltage. To determine the linear conductance (3:8), we choose the measured current I (1) to be the current Iem, and I (2) follows from the coupling to the phase in linear approximation: HJE(Q; '; ) = HJE (Q; ')+ he I (2) . Via a unitary transformation U = exp(,i1 Q=e), it is always possible to write the external voltage partly as a shift of the phase variable '. Using U y'U = ' , 1 and the general relation H 0 = U yHU + ih U y @t@ U , we get 0 (Q; ') = HJ (Q + 1 V C; ' , 1 ) + Hem(' + 2 ); HJE

(3.16)

where 1 is an arbitrary shift and 2 = 1 , 1 . Here we choose 1 = 0 so that the voltage couples solely to the environmental degrees of freedom and then get Hem(' + ) = Hem(') + he Iem . Hence, in this case I (2) coincides with the measured current I (2) = I (1) = Iem. To derive the path integral representation of the generating functional (3:10), we de ne

He em(')= Hem(') , ( )Iem (')= Hem [' , he ( )] + ind:

(3.17)

where ind. denotes a '-independent term that may be omitted. Further, we separate the exponential in Eq. (3:10) into a free part A0 (h ) for vanishing tunneling and a tunneling part AT (h ) according to 1 R h , T e, h 0 d He0 +HT = A0 (h )AT (h ); (3.18) where 1 R A0 ( ) = T e, h 0 d 0 He0 ( 0 ) (3.19) describes the system in presence of the unperturbed Hamiltonian He 0 = He em + HC + Hqp . Using the series expansion of AT ( ) in powers of HT and separating the trace in Eq. (3:10) into partial traces over the charge degrees of freedom of the device, the quasiparticle components, and the environmental degrees of freedom, we obtain an expression of the generating functional as a sum of averages of the unperturbed system, cf. [23]. Due to the Coulomb interaction Hamiltonian HC in the unperturbed Hamiltonian He 0, contributions of a given order in HT cannot simply be


55

evaluated with the help of Wick's theorem, however, the partial traces over the quasiparticle components are averages weighted with the free fermionic density matrix exp(, Hqp ) and, accordingly, products of quasi-particle creation and annihilation operators in the interaction picturePdecompose into products of two-pair correlators. In the limit of large channel number N = 1 1, only speci c combinations of contractions contribute that may be written in terms of two-time correlators of the tunneling Hamiltonian G(; 0 ) = 12 hHT( )HT( 0 )iqp h 2 X X X 1 ( ),2 ( 0)hak11 1 ( )a,k222 ( 0 )iqp ha,q111 ( )aq22 2 ( 0 )iqp = t2 h k1 q1 1 k2 q2 2 1 ;2 = 1 2 2 X ( 0 , )(k ,q ) = t2 ( ), ( 0 ) (1 + ee k )(1 + e, q ) ; h kq

(3.20)

with a real averaged tunneling matrix element t = tkq . Here h: : :iqp denotes the thermal average over the quasiparticles with Hamiltonian Hqp . The time dependence in the interaction picture reads HT ( ) = exp h Hqp HT exp , h Hqp (3.21)

and

( ) = A0 (, ) A0 ( ): (3.22) Further, we have introduced the notation a+ = ay, a, = a, and = exp(i'). Performing the continuum limit for the longitudinal quantum numbers k and q, we nd h i G(; 0 ) = h1 GGT ( , 0) y ( )( 0 ) + y( 0 )( ) (3.23) K

where GT =GK = 42 t2 N 0 is the classical dimensionless tunneling conductance with the densities of states and 0 at the Fermi level in the left and right electrode, respectively. In our approach the limit of strong tunneling is de ned by N 1, t2 0 1 such that 42 t2 N 0 1. Since for lithographically fabricated metallic tunnel junctions typically N > 104 , GT =GK can become very large, although each single channel is weakly transmitting only. The quasiparticle excitations generated by HT are described by an electron-hole pair Green function [23]

Z 1 e,jj=D (k ,q ) X d 1 , e,h e, ( ) = 42 N1 h0 (1 + e,e k )(1 + e q ) = 4h2 ,1 kq

(3.24)

where the electron and hole propagate on dierent electrodes. D is the electronic bandwidth which may be set to in nity at the end of the calculation since D EC ; kB T . Due to analytic properties of thermal Green functions, we may write

with Fourier coecients

1 X 1 ( ) = h e (n )e,in n=,1

(3.25)

e(n ) = , 4h jnje,jn j=D :

(3.26)

56


Here and in the remainder of the article the absolute value is de ned by

(

jzj = ,zz

Re(z ) > 0 (3.27) Re(z ) < 0 ; which leads to a unique analytical continuation [55] of the Fourier coecients (3:26). Along these lines the partial traces over the quasiparticle components may be evaluated in terms of the tunneling kernel ( ). To proceed we need to consider next the partial trace over the charge degreesRof freedom. It is convenient, to change to the phase representation and insert identity operators d' j' ih' j at each imaginary time slice n = hN n; n = 0 : : : ; N with N ! 1. The charge shift operators in the interaction picture then become ( ) = exp(i' ). Dividing the generating functional (3:10) by the quasiparticle partition function trqp exp(, Hqp ) which has no eect on the correlator (3:9), we get

ZJE [ ] =

Z

N Z Y

D['] n=1 1 Z h X

m=0 0

d2m

1 D[n] exp , h S0['; n ; ] Z 2m Z 2 X Y m 0

d2m,1 : : :

0

d1

pairs k=1

G(k1 ; k2 );

(3.28)

where S0 ['; n ; ] = SC ['] + Sem['; n ; ] contains the environmental and the Coulomb actions speci ed below. Since the integrandR is invariant under exchange of an arbitrary pair of variables we may extend the integrations to 0h di (i = 1; : : : ; 2m) and compensate the larger integration region by a factor 1=(2m)!. Further the sum over pairs leads to a factor (2m , 1)!!. Interchanging integrals and product we get

#m X 1 1 1 " 1 Z h Z h 0 0 d d G(; ) ZJE [] = D['] D[n] exp , h S0['; n ; ] 0 m=0 m! 2 0 n=1 Z N Z Y = D['] D[n] exp , h1 SJE['; n ; ] ; (3.29) Z

N Z Y

n=1

where the eective Euclidean action splits into three parts

SJE ['; n ; ] = SC ['] + ST ['] + Sem['; n ; ] :

(3.30)

Z h h 2C SC ['] = d 2e2 '_ 2

(3.31)

'( ) , '( 0 ) Z h Z h G T 0 0 2 ST [']=2 G d d ( , ) sin 2 0 K 0

(3.32)

Here describes Coulomb charging and

0

quasi-particle tunneling across the junction. The environmental action is given by

2 N Z h h 2C 2 X e h n 2 _ Sem['; n ; ] = d 2e2 n + 2e2 L ' , h , n : n n=1 0

(3.33)


57

The remaining trace over environmental degrees of freedom in Eq. (3:29) can be evaluated exactly [56] leading to a quadratic nonlocal action

2 Z h Z h e e 1 0 0 0 0 d d k( , ) '( ) , h ( ) , '( ) + h ( ) ; SY ['; ] = 2 0 0 where the kernel k( ) can be written as a Fourier series (3:25) with coecients b ke(n ) = , 4h Y (Gjn j) jnj: K

(3.34)

(3.35)

Here Yb (s) is the Laplace transform of the environmental response function Y (t), cf. Ref. [56]. Due to causality, for Re(s) > 0, one may write Yb (s) = Y (is) where Y (!) is the frequency dependent admittance (3:7) of the environment. This way the generating functional reads Z ZJE[ ] = D['] exp , h1 SJE ['; ] ; (3.36) with the eective action SJE ['; ] = SC ['] + ST ['] + SY ['; ] : (3.37) The explicit form of the generating functional serves as a starting point to calculate the correlator in the next subsection.

3.2.2 Conductance

We now perform the functional derivatives in Eq. (3:9) explicitly and get for the correlator [57]

! 1 e2 Z 1 hIem( )Iem(0)i = Z D['] exp , h SJE['; 0] 2 h k( ) + Iem['; ]Iem['; 0] ; (3.38) JE

where ZJE = ZJE [0] denotes the partition function. The current functional Iem ['; ] arising as variational derivative of the eective action (3:37) reads

Z h 2 e Iem['; ] = h d 0 k( , 0 )'( 0 ) : 0 The conductance (3:8) now splits into two pieces. (2) GJE (!) = G(1) JE (! ) + GJE (! )

where

2

1 2e e G(1) JE (! ) = ih! h k (,i! + ) = Y (! ) corresponds to the rst term in Eq. (3:38), and with

1 G(2) JE (! ) = ih! FJE (,i! + )

1 Z 1 D['] exp , h SJE['; 0] F ['; n ] FJE (n) = Z JE

(3.39) (3.40) (3.41) (3.42) (3.43)

58


to the second term in Eq. (3:38). The explicit form of the auxiliary functional F ['; n ] in terms of the Fourier components 'e(m ) reads + 2 X1 e 4 e e k(m )'e(m ): F ['; n ] = h k(n)'e(n) m=,1

(3.44)

So far no approximations have been made and Eqs. (3:40) , (3:44) give a formally exact representation of the linear conductance. To proceed we evaluate the path integral (3:43) in the semiclassical limit.

3.2.3 Semiclassical Limit

The classical trajectory of the phase ' is de ned by SJE ['; 0]=' = 0, and we obtain from Eq. (3:37) ' = '0 = const. Since the action is invariant under a global phase shift, we may put '0 = 0. Writing the action in terms of Fourier coecients of the phase

'e(n) = h1

Z h 0

dein '( )

and expanding in powers of 'e(n ), we get

2k [']; SJE

(3.46)

JE(n )je'(n)j2

(3.47)

0 ['] + SJE ['] = SJE

where

0 ['] = h SJE

1 X n=1

1 X

(3.45)

k=2

is the second order variational action with the eigenvalues

2 h i JE (n ) = he2 jn j Gb 0 (n ) + Yb (jn j) :

Here

(3.48)

Gb 0 (n) = jn jC + GT

(3.49) describes the tunnel junction as a capacitance in parallel with an Ohmic resistor characterized by the classical tunneling conductance. Further 2k ['] SJE

!

k ,1 2 k GT (,1)k+1 2X = G (,1)l h K (2k)! l=1 l

X0

n1 ;;n2k,1

0 l 1 0 2k,1 1 X X e @, np A 'e(n1 ) 'e(n2k,1 )'e @, np A p=1

p=1

(3.50)

is the variational action of order 2k. The summation over the ni is over all integers with ni = 0 omitted. Neglecting sixth and higher order terms, we get from Eq. (3:43) 1 e ( ) , e ( ) , e ( ) # 2 ke(n )2 " X 4 e G 2 T n+m n m : (3.51) FJE (n) = h ( ) 1 + G ( ) JE (m ) JE n K JE n m=,1 m6=0


59

C

a)

C , GT *

G( ) b)

Ceff C , GT *

G ( =0) Figure 3.2: Eective circuit diagrams for a tunnel junction in the semiclassical limit a) for arbitrary frequency and b) in the low frequency limit. The convergence of this expansion depends crucially on the eigenvalues (3:48). To estimate the range of validity of the truncated series, we write the smallest eigenvalue in more appropriate units as 22 + GT + Yb (1 ) : JE (1 ) = E (3.52) GK C This eigenvalue has to be large compared to 1, and we see that the expansion is useful for large conductances GT + Yb (1 ) GK and/or high temperatures EC 22 . Hence, we eectively expand in powers of ! G E K C : (3.53) = Min ; GT + Yb (1 ) 22 Performing the limit in ! ! + i, the analytically continued eigenvalue (3:48) reads 2 (,i!) = ,i! h [G (!) + Y (!)] ; (3.54) JE

where

e2

0

G0 (!) = GT , i!C

(3.55) is the analytic continuation of the Laplace transform of Eq. (3:49). The relative minus sign of the capacitive term is due to the usual de nition of the Fourier transform in quantum mechanics, the electro-technical convention is obtained by replacing ! ! ,!. For small frequencies the analytically continued eigenvalue (3:54) is no longer large compared to 1 and we are faced with a problem of order reduction. In the limit in ! ! + i each 1=JE (n ) in Eq. (3:51) becomes of order 1 while the 1=JE (m ) factors for m 6= n are not analytically continued and remain of order . The correction term of order 2 in Eq. (3:51), that is the term proportional to GT =GK , becomes of order after analytic continuation. Hence we loose one factor of . Generally, one nds that the higher order variational actions (3:50) include at most one 1=JE (n ) factor and consequently are reduced at most by one order in . Thus, products of the form 2k1 [']S 2k2 ['] : : : S 2kl ['] SJE (3.56) JE JE

60


*

Re[G ( )/GT]

1

0.8

g 1 10 40 100

EC = 1

*

Im[G ( )/GT]

0.65 0.2 g 1 10 40 100

EC = 1 0

-0.2 -10

0

10

Figure 3.3: Real and imaginary parts of G (!)=GT in the ohmic damping case for EC = 1 and various values of the dimensionless conductance g in dependence on the dimensionless frequency

= h !=2EC . of quartic or higher-order variational actions, as they arise from an expansion in powers of 'e(k ), give quantum corrections of order k1 +k2+:::+kl and after analytical continuation of order k1 +k2 +:::+kl ,l and of higher orders. This proves that the terms of the expansion of FJE given explicitly in Eq. (3:51) suce to calculate the leading order quantum corrections. After performing the analytical continuation we get

Y (!)2 1 + GT G(2) ( ! ) = , U ( ! ) JE G (!) + Y (!) G (! ) + Y (! ) 0

0

(3.57)

with the quantum correction factor

1 X 1 1 2 U (!) = i! m ( , i!) , ( ) : JE m JE m m=1

(3.58)

Hence, for the total conductance we may write

GJE (!) = GGe(!(!) )+YY(!(!) ) e

(3.59)

with an eective linear conductance of the junction

Ge (!) = GT [1 , U (!)] , i!C:

(3.60)

This describes a linear element G(!) = GT [1 ,U (!)], depending on the whole circuit, in parallel with the geometrical junction capacitance C , cf. Fig. 3.2a. The general form (3:59) is valid only


61

1

GJE(0)/Gcl

GT=GK

EC = 1/4

0.95

0.9

0

1

2

GK/Y

Figure 3.4: The ratio of the total dc conductance GJE (0) and the classical conductance Gcl = GT Y=(GT + Y ) shown vs. the dimensionless environmental resistance GK =Y for GT = GK and EC = 1=4. The solid line is the result (3:59) for ! = 0 and ohmic damping, and the dotted line corresponds to the approximation (3:66) for moderate to small Y . to rst order in . A systematic treatment of higher order contributions does not allow for a description of the tunnel junction in terms of an eective linear element. However, a partial resummation of higher order terms according to a self-consistent harmonic approximation [20, 58] leads again to the form (3:59).

3.2.4 Results and Comparison with Experimental Data

For further discussion and comparison with experimental data we restrict ourselves to ohmic dissipation Y (!) = Y . The eective linear element (3:60) then reads G (!) = 1 , (1 + u + !e ) , (1 + !e ) + (1 + u + !e ) , (1 + u) EC ; (3.61)

GT

!e

u

2

where is the logarithmic derivative of the gamma function and ! C !e = 2hi (3.62) u = g E 2 2 ; are auxiliary quantities. We also have introduced the dimensionless parallel conductance g = (GT + Y )=GK : (3.63) The quantum corrections depend only on this combination of conductances. The real and imaginary parts of G (!)=GT are depicted in Fig. 3.3 for EC = 1 and various values of g. The quantum corrections are most pronounced at zero frequency and disappear nonalgebraically for large ! and/or u, due to the logarithmic behavior of the psi-function for large arguments. For the dc conductance we get from (3:61) G (! = 0) = 1 , + (1 + u) + 0 (1 + u) EC : (3.64)

GT

u

2

62


*

G / GT

1

0.8

g = 23.8 g = 4.2

0.6

0

4

8

EC

*

G / GT

1

0.9

g = 34.2 g = 4.52

0.8

0

1

2

EC

Figure 3.5: The renormalized conductance (3:64) versus the dimensionless temperature compared with experimental data by Joyez et al. [20] for dimensionless parallel conductance g = 4:2 and 23:8 (upper plot) and with experimental data by Farhangfar et al. [21] for g = 4:52 and 34:2 (lower plot). In particular, in the limit of a very low resistance environment, the total conductance (3:59) approaches the classical limit nonanalytically, cf. Fig. 3.4, leading to the asymptotic expansion [57] GK GK j ln( EC )jGK GJE (! = 0) = GT 1 + 2 Y ln Y + O : (3.65) Y On the other hand for moderate to large environmental resistance, we may expand Eq. (3:64) with respect to u leading to a total conductance

h i GJE (! = 0) = GGT+YY 1 , G Y+ Y E3 C + O ( EC )2 ; u EC : T

(3.66)

This approximation correspond to the dotted line in Fig. 3.4 and remains analytic in the limit of large environmental conductance where it obviously fails. In Fig. 3.5 we compare our prediction (3:64) with recent experimental data by Joyez et al. [20] for dimensionless conductance g = 4:2 and 23:8 (upper plot) and by Farhangfar et al. [21] for g = 4:52 and g = 34:2 (lower plot). Fig. 3.5 shows that in the limit of large conductance we are able to explain the whole range of temperatures explored experimentally, whereas for moderate conductance only the high temperature part is covered by the semiclassical theory. Here, the parameters g and EC have not been adjusted to improve the t but coincide with those given in the experimental papers.


63

0.8

EC 20 40 80 160

*

Re[G ( )/GT]

0.86

g = 60 0.74

*

Im[G ( )/GT]

/g

EC 20 40 80 160

0

g = 60 - /g -0.1

0

0.1

Figure 3.6: Real and imaginary parts of G (!)=GT in the Ohmic damping case for dimensionless conductance g = 60 and inverse temperatures EC = 20; 40; 80, and 160 in dependence on the dimensionless frequency = !h =2EC . We conclude this section with some remarks on the frequency dependence of the conductance that has not been studied experimentally, so far. For small frequencies we may expand the result (3:61) and write G (!) = G (! = 0) , i!C + O(!2 ); (3.67) where C leads to a renormalization of the junction capacitance C . The renormalized capacitance Ce = C + C reads

Ce =1 + GT C GK

" 2 3

# , 2 0 (1 + u) , 00 (1 + u) ( EC )2 : u 44

(3.68)

The correction shows a quadratic dependence on EC and therefore is suppressed at high temperatures. It also vanishes linearly for large conductance g due to the analytical properties of the psi function. The semiclassical treatment covers only the region of weak Coulomb blockade. Whereas for small tunneling conductance low temperatures imply strong Coulomb blockade, these eects are suppressed for highly conducting tunnel junctions and the semiclassical theory is restored. A closer examination of Eq. (3:61) shows that for g 2 ln( EC ) the quantum corrections are always small. For xed g 1, our predictions are therefore valid for a very large range of temperatures covering in fact the entire range of parameters presently attainable experimentally for metallic junctions with strong tunneling [20{22]. Fig. 3.6 depicts the real and imaginary parts of G (!)=GT for g = 60 and various temperatures. With decreasing temperature the real part shows for ! = 0 a logarithmic decrease, G =GT = 1 , 2 ln( EC )=g, as long as kB T EC exp(,g=2). Thus, for large conductance the semiclassical treatment is an eective high temperature expansion valid for kB T large compared with the renormalized charging

64


Ceff / C

1.3 Y/GK 1 5 10 20

1.15

1

0

GT/GK = 20

1

2

EC

Figure 3.7: Renormalized capacitance Ce =C in the Ohmic damping case for tunneling conductance GT =GK = 20 and various environmental conductances Y=GK = 1; 5; 10, and 20 in dependence on the dimensionless inverse temperature EC . energy EC EC exp(,g=2) [35, 38]. Moreover, the analytical form of the quantum corrections indicates that Coulomb blockade survives for arbitrary large conductance but becomes strong only for temperatures below EC =kB . In the limit T ! 0, g ! 1 such that kB T EC exp(,g=2), the imaginary part of G (!) becomes a step function of width 2=g, cf. Fig. 3.6, leading to a divergent renormalized capacitance of the form Ce =C = EC GT =6(GT + Y ). The linear dependence of Ce on starts already at very high temperatures, cf. Fig. 3.7, and only saturates for EC of order exp(g=2). The large renormalized capacitance is a strong tunneling eect due to multiple electron tunneling and is found likewise for non-Ohmic environmental impedances. In Fig. 3.7 we show the renormalized capacitance Ce =C for GT =GK = 20 and various values of Y=GK in dependence on the dimensionless inverse temperature EC . Note that the linear behavior of the capacitance starts already near EC = 1. The renormalized capacitance describes the frequency dependence of the conductance for small frequencies !Ce GT =g, cf. Fig. 3.6. Rewriting this inequality we get ! 6kB T=h 1011 T Hz, where T is the temperature measured in Kelvin. Thus for all accessible temperatures the frequency range of strong 1=f noise can be avoided, and the eect predicted should be clearly observable experimentally.

3.3 Array of Junctions with Environment 3.3.1 Generating Functional and Conductance As a rst extension of the method, we now consider linear arrays of N tunnel junctions embedded in an electromagnetic environment. The junctions are characterized by classical tunneling conductances Gj and geometrical capacitances Cj in parallel. Like in the previous section, the environment can be transformed into an admittance Y (!) in series with an array of junctions biased by a voltage source V , cf. Fig 3.8. We start with a Lagrangian description depending on phase variables 'j of each junction j = 1 : : : N and an environmental phase 'em with the P constraint Nj=1 'j + 'em + = const:, where describes the applied voltage and is given by

3.3. ARRAY OF JUNCTIONS WITH ENVIRONMENT

Y( )

65

CN , GN

C 1 , G1

V

I

Figure 3.8: Circuit diagram of an array of N tunnel junctions in series with an admittance Y (!). Eq. (3:11). Using the 'j , j = 1 : : : N as generalized variables we nd for the total Hamiltonian

HAE (fQj g; f'j g)=

N X j =1

0N 1 X HJ(Qj ; 'j ) + Hem@ 'j + A ; j =1

(3.69)

with the junction and environmental Hamiltonians de ned by Eqs. (3:1) , (3:6). We follow the analysis in the previous section and rst derive a formally exact expression for the linear conductance. As measured current I (1) we choose again the current P owing throughPthe environmental impedance given by Eq. (3:15) with ' replaced by Nj=1 'j and Q by Nj=1 Qj , respectively. The second current operator I (2) is determined by the linear coupling to and we get I (1) = I (2) = Iem. Following the lines of reasoning in the previous sections, the generating functional is found to read 1 Z (3.70) ZAE[] = D[f'j g] exp , h SAE[f'j g; ] ; with the eective Euclidean action

SAE [f'j g; ] = SY

XN

X N

i=j 'j ; +

i=j

Sj ['j ] :

(3.71)

Here SY was introduced in Eq. (3:34) and Sj ['j ] = SjC ['j ] + SjT ['j ] describes the j 'th junction where the Coulomb action SjC and the tunneling action SjT are given by Eqs. (3:31) and (3:32), with the replacements GT ! Gj and C ! Cj . Performing the functional derivatives explicitly, the current-current correlator is found to be of the form (3:38) with the replacement ZJE ! ZAE = ZAE[0] and the appropriate actionPSAE [f'j g; 0]. Further, the current functionals Iem['; ] now depend on the sum of phases, ' ! Nj=1 'j , and the functional integral is de ned over all con gurations of the phases 'j . As in Eq. (3:41) the rst term can be handled exactly, and we nd GAE(!P) = Y (!) + G(2) (!), where G(2) AE AE(!) given by Eq. (3:42) and Eq. (3:43) with N F ['; n ] ! F [ j =1 'j ; n]. So far no approximations have been made and the result follows from a straightforward extension of our ndings for a single junction. The qualitative dierence lies in the topological structure of the phase con guration space and becomes clear when one evaluates the path integral. Again, we restrict ourselves to the semiclassical limit.

66


3.3.2 Semiclassical Limit

Determining the classical path one has to take into account the topological structure of the con guration space. The phases 'j , j = 1; : : : ; N are canonically conjugate to the charges Qj on the junction capacitances. Now the array has N , 1 metallic islands in between the junctions carrying the charges qj = Qj , Qj +1 for j = 1; : : : ; N , 1. These island charges are quantized in units of the elementary charge e, and the phases j canonically conjugate to the qj are compact, i.e., the con guration space of the phases is a (N , 1)-dimensional torus. Accordingly, the path integral is over all con gurations of the phases j with j (h ) = j (0) + 2kj where the winding numbers kj are integers. On the other hand, the environmental phase is extended and conjugate to a certain linear combination Q of the Qj . Since the environment transfers charges continuously, Q is not quantized and the path integral over the environmental phase is over all con gurations with 'em (0) = 'em (h ). Rather than making the canonical transform to the charges qj (j = 1; : : : ; N , 1), Q and the conjugate phases explicitly, one nds that equivalently we may integrate over all con gurations of thePphases 'j with 'j (h ) = 'j (0) + 2kj where the integer winding numbers obey the constraint Nj=1 kj = 0. At high temperatures the classical paths are straight line ips '(jkj ) ( ) = '0j + kj running from '0j to '0j + 2kj . The action is invariant under global shifts of the 'j and we may set '0j = 0 for all j = 1; : : : ; N . All paths with winding number kj 6= 0 are exponentially suppressed by the classical action contribution Sjcl 2 kj2 = ECj + jkj jGj =2GK . Thus, to obtain the leading order quantum corrections, we may restrict ourselves to winding numbers kj = 0 for j = 1 : : : N . Finite winding numbers are considered in the next section where we focus on the single electron transistor and go beyond the leading order quantum correction. The action may be expanded in powers of the Fourier coecients 'ei (n ) yielding a result of the form (3:46) where the second order variational action reads

XN

0 [f' g] = S 0 SAE j Y

j =1 'j

X N +

j =1

Sj0 ['j ]:

(3.72)

The environmental contribution is given by

SY0 ['] = h

1 X

n=1

Y (n)je'(n )j2 ;

(3.73)

with the eigenvalues

2 Y (n ) = he2 jnjYb (jn j): The second order variational tunneling action for junction j reads

Sj0 ['j ] = h with the eigenvalues

1 X

n=1

j (n ) j'ej (n )j2

2 j (n ) = he2 jnjGb 0j (n );

(3.74) (3.75) (3.76)

where Gb 0j (n ) is given by Eq. (3:49) adapted to a junction with capacitance Cj in parallel with an Ohmic resistor 1=Gj . The higher order variational actions are given by straightforward 2k ['], k = 2; 3; : : :, extensions of Eq. (3:50). Expanding in powers of the higher order terms SAE


67

we are left with expectation values of products of the phase variables 'ej (n ). It is now useful to de ne a Gaussian average

hX i0 = Z10

AE

Z Y 1 Y N

0 [f' g] X d'ej (n )d'ej (n ) exp , h1 SAE j n=1 j =1

(3.77)

0 de ned by the requirement h1i0 = 1. The dierence with the Gaussian partition function ZAE 0 and the full partition function ZAE is of order ( EC )2 and may be neglected between ZAE j 0 [f'j g]=h], the averages of products of here. Due to the Gaussian form of the measure exp[,SAE Fourier coecients 'ej (n ) decompose into sums over products of two point expectations. For two dierent phase variables l 6= l0 we obtain

he'l (n)'el0 (m )i0 = ,n;,m with

(n ) =

NY +1 j 6=l;l0

NX +1 NY +1 i=1 j 6=i

,

j (n) (n );

j (n):

(3.78) (3.79)

Here and in the remainder we de ne N +1 (n ) = Y (n ) and note that summation and multiplication indices run from 1 if not otherwise speci ed. For phase variables of the same junction we nd he'l (n)'el (m )i0 = (l)1( ) n;,m; (3.80) n where 2 (l) (n) = he2 jn jGb 0l (n ) (3.81) plays the role of an eective eigenvalue for phase uctuations in junction l with all other phases 'j , j 6= l already traced out. Here,

3,1 2 N X Gb 0l(n ) = Gb 0l (n ) + 4 b 1 + b 0 1 5 Y (jn j) j6=l Gj (n)

(3.82)

may be considered as the Laplace transform of an eective response function describing the circuit seen from junction l, i.e., a series of N , 1 junctions and an environmental impedance in parallel to junction l where the junctions are described eectively by linear elements. Including the fourth order variational derivative of the action, we get as a generalization of Eq. (3:51) N Y N 2 ke(n )2 X 4 e j (n ) FAE(n ) = h ( ) n l=1 j 6=l " N 1 e ( ) , e ( ) , e ( ) # Y X 2 G n+m n m : l 1 + G ( ) j (n) ( l ) ( ) K n j 6=l m m=,1

(3.83)

m6=0

Now, the convergence of the expansion depends on the eective eigenvalues (3:81). To estimate the range of validity, we consider the smallest eigenvalues which at high temperatures are given by (l) (1 ) 22 = ECl . Again the analytic continuation gives rise to a reduction of the order of the quantum corrections in the expansion parameter. For contributions with vanishing winding

68

CHAPTER 3. THE SEMICLASSICAL APPROACH 1.0 0.9 0.8 0.7

N=20

G/GK 30 20 10 5 1 0.1

*

G /G

Y/GK=20

0

0.5

1

EC

Figure 3.9: Renormalized conductance G =G of an array of N = 20 tunnel junctions in dependence on EC for Y=GK = 20 and various tunneling conductances G=GK . number the arguments given in the previous section apply likewise to the present problem. On the other hand, for lower temperatures one has to take into account winding numbers kj 6= 0 and nds that some of the eigenvalues tend to zero. The marginally stable uctuation modes lead to a breakdown of the simple semiclassical treatment. The appropriate extension of the semiclassical approximation was discussed elsewhere [38]. The topological structure of the phase space leading here to a breakdown of the simple semiclassical approximation at low temperatures even for large conductance is the main dierence between the single tunnel junction with environment and circuits containing many junctions. As a result one nds that the truncated expression (3:83) is valid up to rst order in = Max ECj : j = 1; : : : ; N . After the analytical continuation n ! ,i! + we can write the total conductance in the compact form

2 3,1 N X 1 1 5 ; GAE (!) = 4 Y (!) + j j =1 Ge (!)

(3.84)

describing N + 1 linear elements in series: Gje (!) with j = 1; : : : ; N and the admittance Y (!). Here, the Gje (!) are of the form (3:60) where the auxiliary functions Uj are given by Eq. (3:58) with JE replaced by (j ) introduced in Eq. (3:81). This is a straightforward extension of the result in the previous section valid to linear order in for arbitrary conductances Gj and admittances Y (!).

3.3.3 Discussion of Results

For a more explicit discussion of the results we consider now N identical junctions Gj = G and Cj = C . The eigenvalues (3:81) then read " # 2 Yb (jn j)Gb 0j (n) h 0 ( j ) b (3.85) (n) = (n ) = e2 jnj Gj (n ) + (N , 1)Yb (jn j) + Gb 0j (n ) and coincide for all junctions. For the total conductance of the array (3:84) we obtain (!) : G(!) = G G(e!)(!+)YNY (3.86) (! ) e


69

1.0 N 1 2 5 10

EC=1

0.9

*

G /G

G/GK = N

0.8 0.7

0

0.5

1

GK/Y

Figure 3.10: Renormalized conductance G =G of an array of N = 1; 2; 5, and 10 equivalent tunnel junctions leading to the same classical series conductance for EC = 1 as a function of the inverse environmental conductance GK =Y . where

Ge (!) = G[1 , U (!)] , i!C

(3.87) is the eective linear conductance of one junction. In this order each junction can be described by a linear element G (!) = G[1 ,U (!)], depending on the circuit, in parallel with the geometrical capacitance C as depicted in Fig. 3.2a. To proceed we consider an Ohmic environment Y (!) = Y and nd for the eective linear element G(!) = 1 , (N , 1)[ (1 + uT + !~ ) , (1 + !~ )] + (1 + uN + !~ ) , (1 + !~ )

G

where

uT uN C + (N , 1)[ (1 + uT + !~ ) , (1 + u!~T )] + (1 + uN + !~ ) , (1 + uN ) E 2 N ; (3.88)

C ; u = G EC ; !~ = h ! (3.89) uN = G +G NY E T G 22 2 2i K 2 K are auxiliary quantities and EC = e2 =2C is the charging energy for one junction. For N = 1 we recover the results of Sec. III, of course. For a large array with N 1, terms in Eq. (3:88) containing uN drop out, and the quantum suppression becomes independent of Y . Furthermore the high-temperature anomaly, cf. Fig. 3.4, is now a 1=N eect and the limiting result for N ! 1

is analytic. For small frequencies the eective element behaves like an Ohmic resistor 1=G (! = 0) with a renormalized capacitance in parallel. The dc conductance is given by G (! = 0) = 1 , (N , 1) + (1 + uT ) + 0 (1 + u )

G

T

uT

EC

+ uN ) + 0 (1 + u ) + + (1 N N2 : uN For N = 2 and Y=GK ! 1 this reduces to G = 1 , + (1 + uT ) + 0 (1 + u ) EC =2

G

uT

T

2

(3.90) (3.91)

70

CHAPTER 3. THE SEMICLASSICAL APPROACH 1.5

EC = 0.0442

*

1-G / G [%]

N = 20

1.4

Master eq. semiclassic

0

0.25

0.5

GK/Y Figure 3.11: Zero bias dip 1 , G =GT in per cent for an array of length N = 20 and EC = 0:0442 as a function of the inverse environmental conductance GK =Y in the perturbative limit compared with a numerical master equation approach by Farhangfar et al. [61]. coinciding with earlier ndings for the symmetrical SET [37, 59]. On the other hand, in the limit of large N and moderate G