Sep 18, 2010 - a chain of coupled Majorana bound states generated at crossings between topological and ..... 2 A. Y. Kitaev, Physics-Uspekhi 44, 131 (2001).
Tunneling characteristic of a chain of Majorana bound states Karsten Flensberg
arXiv:1009.3533v1 [cond-mat.mes-hall] 18 Sep 2010
Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA Niels Bohr Institute, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark (Dated: September 21, 2010) We consider theoretically tunneling characteristic of a junction between a normal metal and a chain of coupled Majorana bound states generated at crossings between topological and nontopological superconducting sections, as a result of, for example, disorder in nanowires. While an isolated Majorana state supports a resonant Andreev process, yielding a zero bias differential conductance peak of height 2e2 /h, the situation with more coupled Majorana states is distinctively different with both zeros and 2e2 /h peaks in the differential conductance. We derive a general expression for the current between a normal metal and a network of coupled Majorana bound states and describe the differential conductance spectra for a generic set of situations, including regular, disordered, and infinite chains of bound states.
Topological materials are of large current interest, in part because of their potential for topological quantum computing and their interesting non-Abelian quasiparticles.1 One variant of this is topological superconductors where the low energy quasiparticles in addition are Majorana Fermions.2,3 Currently, there is an active search for materials that can host such particles, either in certain p-wave superconductors, or semiconductors with proximity induced superconductivity and strong spin-orbit coupling.4–7 Because it takes two Majorana Fermions to form a usual Dirac Fermion that can couple to other degrees of freedom, detection of the state of the Majorana fermion system requires non-local measurements or interferometry.8–10 In contrast, a local tunnel current, being independent of the parity of the topological superconductor, does not reveal information about the state of the Majorana fermions. Nevertheless, a tunneling probe could detect the presence of a Majorana bound state (MBS)7,11,12 and the detection of Majorana bound states is the first major challenge in this field. Tunneling contact to an isolated Majorana state give rise to a resonant Andreev process that gives a zero bias conductance peak of 2e2 /h.11 With two coupled Majorana states cross correlations of the current into each could also show their existence and non-local character.13,14 However, because of material difficulties, isolated Majorana states might be rare. Rather it is to be expected that density fluctuations will generate a random configuration of topological/non-topological boundaries, at which Majorana states will be located. For strong disorder the distances between these states are sufficiently short for the MBS to overlap and therefore it is important to understand how a network of coupled Majorana fermions maps onto the tunneling characteristic. This problem was recently considered in Ref. 15 in the weak coupling regime, using a renormalization group to reduce a chain to a sum of single Majorana pairs on a logarithmic energy scale. In this paper, a theory for tunneling between a metallic probe and a collection of coupled Majorana states in the strong coupling regime is developed, and experimentally relevant situations are addressed. This is done in
S N
TS
Γ
TS
t2,3
TS MBS
Density
Position
FIG. 1. (Color online) Illustration of a semiconductor with induced superconducting order parameter from an adjacent superconductor (S). Spatial variations in density or superconducting order parameter create crossings between topological (TS) and non-topological segments when the density crosses the critical density (vertical (red) lines). Majorana bound states (MBS) (circles) are located at each crossing point and the distances between them determine the coupling matrix elements tij of the resulting Majorana network. A tunneling contact (N) probes the network by tunneling into the end Majorana mode, with tunneling density of states Γ.
the limit where the voltage eV , the tunneling broadenings Γ, and the hopping matrix elements between any two MBS all are much smaller than the superconducting energy gap ∆. The regime of large ∆ is well suited for characterization and detection of the MBS, because in absence of Majorana states the Andreev conductance is of the order16 (e2 /h)(Γ/∆)2 and thus much smaller than the resonant Andreev current carried by the Majorana states. Examples for different number and configurations of Majorana states is given and the case of a uniform infinite chain is solved exactly. Finally, disordered Majorana chains are addressed. Disorder is introduced as random nearest neighbor couplings and it is show to reduce to a finite chain, truncated by the first weak link (quantified below) in the chain. The borders of the topological superconductor segments give rise Majorana bound states. These states are zero energy solutions to the Boguliubov-de Gennes equa-
2 tions for the geometry in question. The general form of a MBS is XZ ∗ γi = dx fσ,i (x)Ψσ (x) + fσ,i (x)Ψ†σ (x) . (1) σ
The Majorana Fermion has the properties that γi = γi† and γi2 = 1. The superconductor Hamiltonian, describing the coupled Majorana state network, is iX HS = t γγ , 2 ij ij i j
(2)
The tunnel Hamiltonian between the normal metal and the superconductor is XZ dx t∗k (x)c†kσ Ψσ (x) + h.c. , HT = kσ
where ck,σ are lead-electron annihilation operators and Ψσ (x) the superconductor electron-field operator. As explained above, for (eV, Γ) ∆ the Majorana states contribution to the current dominates. Using Nambu representation, Ψ = (Ψ↑ , Ψ↓ , Ψ†↓ , Ψ†↑ ), the projection of the field operator Ψ onto the manifold of Majorana states is P ∗ ∗ (x)), which then Ψ(x) ≈ i γi (f↑,i (x), f↓,i (x), f↓,i (x), f↑,i leads to the effective tunnel Hamiltonian describing the coupling between the lead and the Majorana states X ∗ HT = (Vkσ,i c†kσ − Vkσ,i ckσ )γi , (3) kσi
R
where Vkσ,i = dx fσ,i (x)tk (x). The current operator is given by the rate of change of the number of electrons in the normal lead X 2e ∗ I = −eN˙ = −ie[HT , N ]/~ = Re Vkσ,i G< (0) , i,kσ ~ kσi (4) where the lesser Green’s function combining kσ and i is D E † < defined as Gi,kσ (t) = i ckσ γi (t) , which is written as G< i,kσ (t) =
Xh
(0)
Gij Vkσ,j Gkσ
i
4|t|. √ (4Γ) 2 2 2 |eV |+
(eV ) −(4t)
+(4Γ)
(17) This is an interesting expression with a line shape that strongly depends on the ratio t/Γ, which is shown in Fig. 3. Furthermore, for eV = 0 it reduces to dI 2e2 2Γ = , (18) dV V =0 h |t| + 2Γ which shows that for small tunnel broadening compared to the bandwidth of the chain the zero bias conductance is 2e2 /h, which was expected because it corresponds to tunneling into an effectively isolated Majorana state. As the last situation, which might also be the most experimentally relevant, we now discuss different realizations of long disordered chains, sampled for example by scanning the average density and creating a different set of crossings of the topological superconductor thresholds, as illustrated in Fig. 1. One could in this way study the average conductance of a network, given a distribution of nearest neighbor MBS couplings tij . The tunneling coupling between two neighboring MBS depends both on the distance between them and the deviation from the critical value for the topological/non-topological transition, with exponential dependence on both, as was shown by Shivamoggi et al.15 for a specific example. The distribution of tunneling couplings is thus a complicated convolution of amplitude fluctuations and the level-crossings
dIdV @e2hD
4 + + + + + 1.5 + + + ++ + + + + + 1. + + + + + + ++ + + + + + 0.5 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + 0. -5 0 5 eVG
+ + + + + ++ ++ ++ 1. ++ ++ ++ + ++ ++ +++ +++ +++ +++ ++ ++ ++ 0.5 + + + ++ ++ + + ++ + + + + + ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + 0. -5 0 5 eVG 1.5
FIG. 4. (Color online) Both panels show the conductance for a chain with 10 sites and one weak link with tunnel coupling tweak = 0.25Γ, while the other links have t = Γ. The weak links are the 1-2 and 5-6 connections for the left/right panel, respectively. The thick (blue) curve is the conductance for kT = 0.1Γ, the thin gray curve for kT = 0, while the crosses show the conductance for the chain truncate at the weak link also with kT = 0.1Γ. It is clearly seen that the truncated approximation works well, because the chain after the truncations leads to structure barely resolvable, because kB T is not much smaller than the width (≈ 0.125) of the additional resonances.
statistics,19 in this case with two crossings. The resulting distribution of tunneling couplings is an interesting problem in itself. However, instead of pursuing this line, we focus at the generic behavior expected for a given configurations of the tunneling couplings in the chain. As we learned for the infinite chain, different behaviors occur depending on the ratio of Γ to the tunneling couplings tij . For the random chain the ratio of Γ to the spread of tunneling couplings turns out to be crucial. Clearly, if the spread in tunneling couplings is much smaller than Γ, the average conductance resembles that of the homogenous infinite chain, which we have verified by numerical simulation. In contrast, with large fluctuations in the tunneling couplings, the infinite chain will be effectively truncated
1 2 3
4
5 6
7 8 9
10 11
12
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into a finite chain, where one of the tunneling coupling happens to be much smaller than kB T . To see this, invert the matrix in Eq. (12) and pull out the dependence on the weak link and write is as 2D2,∞ , D1,∞ − 4t2n,n+1 D1,n Dn+1,∞ + iΓD2,∞ (19) where tweak = tn,n+1 is the weak link, and where Di,j is the determinant of the matrix (ω − 2t) for the isolated chain between sites i and j, but with tn,n+1 = 0. The weak link has two effects, 1) it gives small shifts of the existing resonances and 2) it creates new resonances. The new resonances, however, have widths that scale with the square of the weak coupling Γweak ∝ t2weak /htiΓ, where hti denotes typical couplings in the first part of the chain. Therefore the new resonances introduced by the chain after the weak link is not resolved if kB t & Γweak . An example of this is shown in Fig. 4, where the conductance of a truncated chain is compared with that of a full chain. In summary, the differential conductance for a junction between a normal metal and topological superconductor hosting a network of Majorana bound states has been studied. Different configurations of the interacting network of bound states give rise to distinct tunneling spectra. Long chains with fluctuating tunneling couplings is truncated into a finite chain once a coupling becomes smaller than a certain critical value, determined by temperature. GR (11, ω) =
ACKNOWLEDGMENTS
C.M. Marcus and M. Leijnse are gratefully acknowledged for stimulating discussions. The work was supported by The Danish Council for Independent Research | Natural Sciences.
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15
16
17
18
19
C. J. Bolech and E. Demler, Phys. Rev. Lett. 98, 237002 (2007). J. Nilsson, A. R. Akhmerov, and C. W. J. Beenakker, Phys. Rev. Lett. 101, 120403 (2008). V. Shivamoggi, G. Refael, and J. E. Moore, Phys. Rev. B 82, 041405 (2010). G. E. Blonder, M. Tinkham, and T. M. Klapwijk, Phys. Rev. B 25, 4515 (1982). by generalizing the k-sum in Eq. (9) to run over the different contacts as well and inserting a matrix that specifies the measurement contact in the trace inp Eq. (7). For |ω| < 4t this is g˜22 = (ω − i (4t)2 − ω 2 )/8t2 , whereasp for |ω| > 4t the result is g˜22 = (ω − sign(ω) ω 2 − (4t)2 )/8t2 . I. Blake and W. Lindsey, IEEE Transactions on Information Theory 19, 295 (1973).
5 Appendix A: Current formula for a Majorana state network coupled to normal-metal electrodes
The corresponding Keldysh contour Green’s functions are D E Gip (τ, τ 0 ) = −i TK (γi (τ )c†p (τ 0 )) ,
Gpi (τ, τ 0 ) = −i TK (cp (τ )γi (τ 0 )) .
The low energy model Hamiltonian projected onto the Majorana subspace is H = HN + HS + HT X HN = εp c†p cp ,
(A1a)
D E ¯ pi (τ, τ 0 ) = −i TK (c†p (τ )γi (τ 0 )) , G
kσ
iX tij γi γj , 2 ij X ∗ † (Vpi cp − Vpi cp )γi , HT =
(A1c)
(A4b)
Below we will also need
(A1b)
HS =
(A4a)
the Majorana Green’s function
(A1d)
Gij (τ, τ 0 ) = −i TK (γi (τ )γj (τ 0 )) ,
(A4c)
pi
where p = k, σ. In the non-equilibrium case the voltage on the normal metal is eV . The coupling matrix tij is real and obeys tij = −tji . The current operator is I = −eN˙ = −ie[HT , N ]/~ ie X ∗ † = Vpi cp γi − Vpi γi cp ~ pi eX ∗ < = Vpi Gip (0, 0) − Vpi G< pi (0, 0) , ~ pi
and the free lead electron Green’s functions D E G0p (τ, τ 0 ) = −i TK (cp (τ )c†p (τ 0 )) , 0 D E 0 † 0 ¯ (0) G (τ, τ ) = −i T c (τ )c (τ ) , K p p p
(A4d) (A4e)
0
where the unperturbed expectation value h·i0 is for the situation with HT = 0. The self energies (A2) Σeij (τ, τ 0 ) =
X
∗ Vpi Vpj G0p (τ, τ 0 ),
(A5a)
0 ¯ (0) Vpi∗ Vpj G p (τ, τ ),
(A5b)
p
and
Σhij (τ, τ 0 ) =
D E 0 † 0 G< ip (t, t ) = i cp (t )γi (t) ,
0 0 G< pi (t, t ) = i γi (t)cp (t ) .
(A3a) (A3b)
X p
will also appear.
¯ pi . From diagrammatics, direct expansion, or equation of motion, Now return to the mixed Green’s functions Gip , Gpi , and G we obtain Z X XZ Gip (τ, τ 0 )Vpi∗ = dτ 00 Gij (τ, τ 00 )Σeji (τ 00 , τ 0 ) = dτ 00 Tr G(τ, τ 00 )Σe (τ 00 , τ 0 ) , (A5ca) p0 i
X p0 i
X p0 i
ij
Vpi Gpi (τ, τ 0 ) =
XZ
dτ 00 Σeij (τ, τ 00 )Gji (τ 00 , τ 0 ) =
Z
dτ 00 Tr Σe (τ, τ 00 )G(τ 00 , τ 0 ) ,
(A5cb)
ij
¯ pi (τ, τ 0 ) = − Vpi∗ G
XZ
dτ 00 Σhij (τ, τ 00 )Gji (τ 00 , τ 0 ) = −
Z
h i dτ 00 Tr Σh (τ, τ 00 )G(τ 00 , τ 0 ) .
(A5cc)
ij
With this the current becomes Z e dω I= Tr (GΣe − Σe G)< ω . ~ 2π Next, let us find the Dyson equation for the Majorana Green’s function. Its equation of motion is
i∂τ Gij (τ, τ 0 ) = 2δij δ(τ, τ 0 ) + i TK ([H, γi ](τ )γj (τ 0 )) . (A4)
The factor of 2 because {γi , γj } = 2δij . The commutators: X X iX [H0 , γi ] = ti0 j 0 [γi0 γj 0 , γi ] = i (tji − tij ) γj = −2i tij γj . 2 0 0 j j i j i Xh ∗ † X ∗ † [HT , γi ] = (Vpj cp − Vpj cp )γj , γi = 2 (Vpi cp − Vpi cp ). pj
p
(A5) (A6)
6 Again the factors of 2 come from the unusual commutation relation. With this the equation of motion becomes X X ¯ pj (τ, τ 0 ) . i∂τ Gij (τ, τ 0 ) = 2δij δ(τ, τ 0 ) + 2i tij 0 Gj 0 j (τ, τ 0 ) + 2 Vpi Gpj (τ, τ 0 ) − Vpi∗ G j0
(A7)
p
By Eq. (A5c) the Dyson equation closes i∂τ Gij (τ, τ 0 ) = 2δij δ(τ, τ 0 ) + 2i
X
tij 0 Gj 0 j (τ, τ 0 ) + 2
j0
(i∂τ − 2it − 2Σ) G = 2,
(A9)
which has the solution (A10)
where the unperturbed Majorana Green’s function is (i∂τ − 2it) G0 = 2,
(A11)
and where the Majorana self energy is Σ = Σe + Σh .
(A12)
The same result can be derived using diagrammatics, in which case the factors of 2 then comes from the 2 ways 4 Majoranas can pair. The retarded components of the self energy are ΣeR ij (ω) =
X p
ΣhR ij (ω) =
∗ Vpi Vpj Γij (ω) =i + Λij (ω), ω−εp +iη 2
Vpi∗ Vpj
X p
=i
ω+εp +iη
Γji (−ω) − Λji (−ω), 2
dτ 00 Σeij0 (τ, τ 00 ) + Σhij0 (τ, τ 00 ) Gj0j (τ 00 , τ 0 ),
(A8)
j0
or in shorthand notation
G = G0 + G0 ΣG,
XZ
Now go back to current and use that (BC)< = B R C < + B < C A to get Z h i e dω A e< < eA eR I= Tr GR ω − Gω Σ ω + Gω Σ ω − Σ ω ~ 2π (A19) A In equilibrium G
