4.3 Model
4.5.2 Single-trajectory Andreev level with scattering
We investigate a simple model for the anticrossing and calculate the Andreev bound-state energies for a 1D-model with impurity scattering modeled by a scattering matrix. Although this model takes only backward scattering into the same trajectory into account and neglects the complex interference effects of three-dimensional impurity scattering which are covered by our original Green’s function calculations, the results provide an understanding of the phase-dependent Andreev level density of states. The bound-state energies are obtained from the scattering matrices in the normal region [Naz09]. We consider the geometry shown in Fig. 4.14. The normal scattering matrix encompasses the backscattering at the impurity as well as the dynamical phases along the trajectory to the superconductor and is given by
SNe(E,x) =✓
re2ixE/ETh teiE/ETh teiE/ETh ≠re2i(1≠x)E/ETh
◆ ,
where xœ[0,1] accounts for the position of the impurity along the path and t2 =T = 1≠r2 is the transmission probability. The normal region scattering matrix for holes is given by SNh(E) =SNeú(≠E). The scattering matrices for electron-hole conversion at the interface to the superconductors are given by SAeæh(E,Ï) = exp[≠iarccos(E/∆)≠iÏ/2‡3] and SAhæe(E,Ï) = SAhe(E,≠Ï), respectively. Note that the ‡-space is not Nambu space. An electron arriving
4.5 1D scattering model 63
(a)
(b)
Figure 4.15: Energy of Andreev levels for a single mode with transmission probability T [T = 1 in (a) and T = 0.9 in (b)] through the normal part for ETh = ∆ (red curve), ETh = 2∆ (green curve), ETh = 5∆ (blue curve), and ETh= 10∆(yellow curve). The shaded regions in (b) correspond to variations of the energies with the position of the scatterer along the trajectory (described by the parameterx).
at either superconductor is reflected as a hole traveling towards the normal region from the same side, thus Andreev reflection is described by a diagonal matrix. The condition for a bound state reads
det⇥1≠SNe (E,x)SAhæe(E,Ï/2)SNh(E,x)SAeæh(E,≠Ï/2)⇤= 0. (4.16) The bound-state energies in dependence ofÏare plotted in Fig.4.15for different values of ETh. Without backscattering in the normal region, atÏ = 0 the two Andreev levels are degenerate [Fig. 4.15 (a)]. Taking into account impurity scattering in the normal part [Fig.4.15 (b)] this degeneracy is lifted (the exact curve depends on the position where scattering occurs, described by a parameter x). This results in the characteristic shape of the minigap and the secondary
“Smile”-gap. Figure4.15(b) shows the x-averaged results for Andreev bound states with one scattering event with T = 0.9(weak scattering).
These considerations with a single scattering event in the normal region can be generalized to multiple scattering events along a trajectory through the normal part from one superconductor to the other. In that case the full scattering matrix of the normal region is built up from scattering matrices corresponding to single scattering events. Using the transfer matrix formalism the full scattering matrix is calculated. Instead of a single free parameter x, which describes the position of the scattering event along the trajectory, one such parameter appears for each scattering event.
The results for Andreev levels with up to 3 scattering events along one trajectory are plotted in Fig. 4.16 for the transmission probability of a single scattering event being T = 0.8 (Fig. 4.16 (a)), T = 0.9 (Fig. 4.16 (b)) and T = 0.95 (Fig. 4.16 (c)) and ETh = ∆. The red curves show the energies of Andreev bound states as a function of Ï for paths without impurity scattering and an average dwell time in the normal region of 1/ETh. The green regions show the energies occupied by Andreev levels for different positions of the impurity scattering in case of one impurity scattering event along a trajectory.
Analogously the blue and yellow regions denote the energy intervals occupied by Andreev levels with two and three scattering events averaged over positions of these scattering events.
It is worth mentioning that only channels without scattering contribute to the zero-energy Andreev states at Ï =±fi. For paths with one or more scattering events, these levels are shifted to higher energies. Thus, we have shown that the secondary gap can be understood from the phase-dependence of the Andreev level if the junction length exceeds a length of the order of the superconducting
4.6 Conclusion 65
(a) (b)
(c)
Figure 4.16: Energies occupied by Andreev levels for different numbers of scattering events along a trajectory. The red curve describes Andreev levels without scattering, the green, blue and yellow energies correspond to 1-, 2-and 3 scattering events along a trajectory. The shaded regions correspond to averages over the positions where scattering occurs. The three figures differ in the transmission probability of each scattering event: (a)T = 0.8, (b) T = 0.9, (c)T = 0.95. In all three cases the Thouless energy is fixed to ETh=∆.
coherence length, corresponding toETh &∆. The “smile” shape can be traced back to the effect of backscattering.
4.6 Conclusion
To summarize, we have calculated the local density of states for diffusive Josephson systems for a wide range of contact types with attention to the energy range below∆, in which a secondary gap can appear. We have generalized the calculations of chapter3 for ballistic contacts and shown that the secondary
“Smile”-gap is a robust feature in the proximity density of states for large Thouless energies. We thus suggest that this feature should be accessible to an experimental detection by means of high-resolution scanning tunneling spectroscopy and want to encourage research in this direction.
CHAPTER 5
Universal properties of mesoscopic fluctuations of the secondary “Smile”-gap
5.1 Abstract
We investigate the mesoscopic fluctuations of the secondary “Smile”-gap, which appears in the quasiclassical spectrum of a chaotic cavity coupled to super-conductors. In the universal regime ETh ∫∆we calculate the distribution of gap-sizes for different combinations of the number of levels inside the cavity M and the number of channels towards the superconductorN. The distribution is determined by an intermediate energy scale∆g ≥(∆≠Ec)4/3”s2/3 which is situ-ated between the levelspacing of the cavity ”s and the size of the quasiclassical gap∆≠Ec. From these finite M and N calculations we estimate the first two cumulants of the gap distribution in the limitM ∫N ∫1 by extrapolation.
There is strong evidence that the distribution in this limit is the same like the one found in previous works [Vav01] for the distribution of the minigap in the limit ETh π ∆. We recalculate these distributions of the minigap and find deviations from the results in [Vav01]. Only in the regime M ∫ N ∫1 our results agree with the universal curve of [Tra94;Tra96].
This distribution seems to be a general property of proximity systems at the edge of a continuous spectrum, comparable to the Wigner-Dyson distribution between neighboring levels in the continuous part of the spectrum. In a next step we have a closer look at the parameter rangeETh .∆, where the secondary gap is detached from ∆ and discrete Andreev levels exist above as well as below the gap. We find a continuous violation of Wigner-Dyson statistics by the opening of the secondary “Smile”-gap as a function of superconducting phase-difference.
67
5.2 Introduction
Normal metals connected to one or more superconductors are subject to the so-called the proximity effect [Deu69], which arises due to the penetration of superconducting correlations into the normal metal. Its influence on the normal metal properties is most striking for finite size normal metals, whose characteristic length scales are of the order of the superconducting coherence length ›. While systems with more than one superconductors involved can host equilibrium supercurrents [Jos62], the most eye-catching observable which is strongly modified by the proximity of even a single superconductor is the local density of states (LDOS) [Bel96; Lev08]. Besides the emergence of a gap around the Fermi energy EF [McM68], which is known as the “minigap”, we recently reported another secondary gap for a special class of structures consisting of a chaotic normal cavity connected to two superconductors via ideally transmitting ballistic contacts (see chapter 3).
Such systems are known to have a universal behavior in the sense that their physical properties in the quasiclassical limit (number of levels M in the uncoupled cavity is much larger than the number of channels N, connecting the cavity to the superconductors, which is much larger than 1) do not depend on microscopic details of the system, but only on the presence of fundamental symmetries [Bee97]. Microscopic details are the exact distribution of impurities in a diffusive metal or the shape of a ballistic cavity with chaotic scattering at the boundaries. This universality is only valid, if the system is sufficiently chaotic. Quantitatively speaking the time an excitation spends inside the normal region ·dwell, before reentering a connector towards a superconductor must be much larger than the ergodic time ·erg necessary for exploring the full phase space of the cavity [Bee97]. The only relevant parameter describing the normal metal properties is thus the energy scale related to the inverse dwell-time, which is called the Thouless energy of the system [Tho77]:
ETh =~/·dwell
The fundamental symmetries in the system are time-reversal symmetry (TRS), which can be broken by an external magnetic field and spin rotation symmetry (SRS), which is broken in systems where spin-orbit interaction plays a role [Dys62;Zir11]. In this work we stick to the case where both of these symmetries are present.
This universality makes a description in terms of random matrices, which fulfill these symmetries, possible. These matrices can either be random
Hamilto-5.3 Model 69 nians in the description of finite systems (see section2.3.2), or random scattering matrices in the description of open systems (see section 2.3.3). The theory dealing with this kind of random matrices is called Random Matrix Theory (RMT). It has turned out to be a powerful tool in the description of average properties [Mel97] as well as in the description of mesoscopic fluctuations of these averages [Vav01]. So far most interest was attributed to the minigap and its statistical properties [Bee05; Vav03]. In the regime ETh π ∆the system can be described by an effective Hamiltonian [Mel96], whose smallest eigenvalue indicates the minigap. This eigenvalue was found to be distributed according to a universal distribution function, which is generally valid for random hermitian matrices with a gapped spectrum [Tra94; Tra96;Vav01].
In this chapter we investigate regimes with ETh &∆, in which no effective Hamiltonian description of energy levels below the superconducting gap edge∆ is obviously possible (see appendix C). For ETh ∫∆ the quasiclassical density of states has a universal shape which in rescaled units doesn’t depend on ETh
(see section 3.4.1). One of the questions we want to answer is whether the fluctuations of the secondary gap have a similar universal behavior and which distribution determines these fluctuations.
The chapter is structured as follows: First we introduce the model we use and describe how discrete Andreev energies as well as the average density of states are calculated from ensembles of random Hamiltonians. To check our numerics we recalculate the fluctuations of the minigap and compare them to the results of [Vav01].Using RMT we then calculate the full density of states in the regimeETh ∫∆, which also accounts for leakage of Andreev levels into the superconductors, and show that the system we consider is indeed the same like the one considered in chapter 3. In a next step we investigate the mesoscopic fluctuations of the secondary gap by numerically calculating the distribution of Andreev levels. By extrapolation we estimate the first two cumulants of the secondary gap distribution in the quasiclassical limit M ∫N ∫1. In the final section the violation of Wigner-Dyson statistics in the regimeETh .∆ is considered, when the “Smile”-gap is detached from ∆ and continuously opens/closes as a function of phase.
5.3 Model
In order to go beyond the quasiclassical average density of states and approach the statistics of single Andreev levels we use Random Matrix Theory, which is known to be in terms of average results equivalent to quasiclassical Green’s
function methods. However, it has the advantage to give access not only to the average density of states, but also to single Andreev levels, and thus can be used to investigate their statistics. Andreev levels in a chaotic cavity, which is coupled via ballistic contacts to superconductors, are determined by the condition [Bee91;Naz09]
deth
1≠Sˆe(E) ˆSAhæe(E,{Ïi}) ˆSh(E) ˆSAeæh(E,{Ïi})i
= 0. (5.1) The set {Ïi} accounts for superconducting phases of all involved superconduc-tors. Hole scattering in the normal region is related to the scattering matrix of electrons viaSˆh(E) = ˆSeú(≠E). At the interfaces with the superconductors elec-trons and holes are converted into each other, which is described by the Andreev scattering matrices [Hei02;Naz09] SˆAeæh(E,{Ïi}) = exp[≠iarccos(E/∆)≠iÏ]ˆ and SˆAhæe(E,{Ïi}) = ˆSAeæh(E,{≠Ïi}). ψis a diagonal matrix assigning to each outgoing channel from the cavity the phase of the corresponding superconductor.
In this chapter we consider only the case of two superconductors S1 and S2, where a symmetric choice of phases Ï1 =Ï/2 andÏ2 =≠Ï/2 is possible. In the short junction limit the energy-dependence of Sˆe can be neglected and for a chaotic normal region it is described by a random unitary matrix, belonging to one of the circular ensembles of RMT [Bee97; Blü90]. Here we consider only time-reversal invariant systems, which are described by the circular orthogonal ensemble (COE). For the statistics of the secondary “Smile”-gap at Thouless energies ETh &∆the energy-dependence of the scattering matrix cannot be neglected. Information on correlations between scattering matrices at different energies is contained in the Hamiltonian of the system. Expressing the scatter-ing matrix through this Hamiltonian, it has been shown [Bee05; Fra96] that equation (5.1) can be transformed to
deth
Eˆ≠Hˆ + ˆW(E,Ï)i
= 0. (5.2)
The matrices H and W are defined by Hˆ =H·ˆ3,W(E,ˆ Ï) = fi
where W W† describes the coupling of M energy levels inside the cavity via N =N1 +N2 transport channels of transmission Tn to the superconductors S1
andS2. ψis in this case a diagonal matrix of dimensionN, withN1 entries Ï/2
5.3 Model 71 corresponding to modes towardsS1 andN2 entries ≠Ï/2for modes connecting the cavity toS2. The matrix W is given by Wmn =”mnp
f(Tn)M”s/fi, with f(Tn) = (2≠Tn±2Ô1≠Tn)/Tnand”sis the average levelspacing of the isolated cavity. ·ˆi are the Pauli matrices in Nambu space, H is the cavity-Hamiltonian of dimension M for electrons. For ballistic coupling (Tn = 1) W is given by Wmn =”mnÔ
M”s/fi. In this work we focus on the ballistic case, since it was found in chapters3 and 4that for this kind of coupling the secondary gap has its maximum extend. For a normal region with the shape of a chaotic cavity, this Hamiltonian is a member of the Gaussian orthogonal ensemble, whose probability distribution is defined by [Meh04]
P(H)≥exph
≠ fi2
4”2sM TrH2i
. (5.3)
Equation (5.2) is no eigenvalue equation, but it rather describes a system of Hamiltonian Hˆ, which is coupled via an energy-dependent self-energyWˆ(E) to an external part. For this system a Green’s function can be defined in the following way [Bee05; Bro97; Mel97]:
G(z) =ˆ D h
zˆ1≠Hˆ + ˆW(E)i≠1E
GOE, (5.4)
where the average means with respect to the Gaussian orthogonal ensemble (5.3). This Green’s function determines the average density of states as follows:
fl(E) = ≠1 fiImn
Trh
(ˆ1+dW/dE)ˆˆ G(E+i”+)io
. (5.5)
The factor (ˆ1+ dW/dE)ˆ has to be included for the full density of states and accounts for the coupling to the superconductors. The local density of states in the cavity is defined without this prefactor. From a physical point of view this factor accounts for evanescent leaking of Andreev levels into the superconductors.
Andreev levels are calculated by solving the determinant equation (5.2).
Dealing with energies E . ∆, there is no obvious possibility to reduce this problem to an eigenvalue problem of an effective Hamiltonian, which was possible in similar studies [Vav01] of the level statistics of the minigap in the limit ETh π∆. More on an effective Hamiltonian description of the secondary gap can be found in appendixC. However, it is possible to reduce the dimension
of Eq. (5.2) by a factor 2. How this is achieved is described in appendix D.
5.4 Results
In chapters 3 and 4it was shown, that a chaotic cavity, which is coupled to a superconductor, besides the usual minigap can have a secondary “Smile”-gap, which appears at Thouless energies ETh &∆and is situated slightly below the supercondcuting gap-edge ∆. This secondary gap has a universal behavior at ETh ∫∆, which is opposite to the usual minigap whose universal regime is at ETh π∆, where it is up to a numerical prefactor given byETh. It was found [Vav01] that not only the average density of states, but also the fluctuations of the minigap have a universal behavior in this limit, and the energy scale of the fluctuations is determined by the average density of states above the gap. In this chapter we consider mesoscopic fluctuations of the secondary gap. The two main points we focus on are on the one hand the universal limit ETh ∫∆atÏ = 0for which we check universal properties of these fluctuations and which energy-scale determines them. From a mathematical point of view we deal with a different problem compared to the minigap, since we do not calculate eigenvalues of random hermitian matrices. The hypothesis concerning the behavior of eigenvalues of a random matrix ensemble at the edge of the spectrum applied in [Vav01] cannot be applied in our case. But also from a physical point of view there are differences: Whereas the minigap is situated symmetrically around zero energy (electron-hole symmetry), the secondary gap is at its upper edge attached to the continuous spectrum above∆, which should affect levels below the gap via repulsive interaction of energy levels (see section 2.3.2 and Fig. 2.2 therein).
The second point we consider is the Thouless energy range ETh .∆, where the secondary gap is detached from ∆. In this limit there are discrete An-dreev levels above the secondary gap as well as below. As a function of superconducting phase difference Ï there appears a continuous transition from Wigner-Dyson statistics towards a gapped distribution as the secondary gap opens.