X-SHAPED AND ^-SHAPED ANDREEV RESONANCE PROFILES IN A SUPERCONDUCTING QUANTUM DOT
Shuo Mi, D. I. Pikulin, M. Marciani, C. W. J. Beenakker*
Institmit-Lorentz, lhvivcrsitc.it Leiden, P.O. Box 9506 2300, RA Leiden, The Netherlands
Received May 20, 2014
The quasi-bound states of a superconducting quantum dot that is weakly coupled to a normal metal appear as resonances in the Andreev reflection probability, measured via the differential conductance. We study the evolution of these Andreev resonances when an external parameter (such as the magnetic field or gate voltage) is varied, using a random-matrix model for the N x N scattering matrix. We contrast the two ensembles with broken time-reversal symmetry, in the presence or absence of spinrotation symmetry (class C or D). The poles of the scattering matrix in the complex plane, encoding the center and width of the resonance, are repelled from the imaginary axis in class C. In class D, in contrast, a number oc \fN of the poles has zero real part. The corresponding Andreev resonances are pinned to the middle of the gap and produce a zero-bias conductance peak that does not split over a range of parameter values (l"-shaped profile), unlike the usual conductance peaks that merge and then immediately split (.Y-shaped profile).
Contribution for the JETP special issue in honor of A. F. Andreev's 75th birthday
DOI: 10.7868/S0044451014120025
1. INTRODUCTION
Half a century has passed since Alexander Andreev reported the curious retro-reflection of electrons at the interface between a normal metal and a superconductor [1]. One reason why Andreev reflection is still very much a topic of active research is the recent interest in Majorana zero modes [2]: nondegenerate bound states at the Fermi level (E = 0) consisting of a coherent superposition of electrons and holes, coupled via Andreev reflection. These are observed in the différential conductance as a resonant peak around zero bias voltage V that does not split upon variation of a magnetic field B [3 6]. In the (B, V) plane, the conductance peaks trace out an unusual Y-shaped profile, distinct from the more common A'-shaped profile of peaks that meet and immediately split again (see Fig. 1).
It is tempting to think that the absence of a splitting of the zero-bias conductance peak demonstrates that the quasi-bound state is nondegenerate, and hence Majorana. This is mistaken. As shown in a computer simulation [7], the Y-shaped conductance proE-mail: beenakker'ffllorentz. leidenuniv.nl
file is generic for superconductors with broken spinrotation and broken time-reversal symmetry, irrespective of the presence or absence of Majorana zero modes. The theoretical analysis in Ref. [7] focused on the ensemble-averaged conductance peak, in the context of the weak antilocalization effect [8 11]. Here, we analyze the sample-specific conductance profile, by relating the A'-shape and Y-shape to different configurations of poles of the scattering matrix in the complex energy plane [12].
2. ANDREEV BILLIARD 2.1. Scattering resonances
We study the Andreev billiard geometry shown in Fig. 2: a semiconductor quantum dot strongly coupled to a superconductor and weakly coupled to a normal metal. In the presence of time-reversal symmetry, an excitation gap is induced in the quantum dot by the proximity effect [13]. We assume that the gap is closed by a sufficiently strong magnetic field. Quasi-bound states can then appear near the Fermi level (E = 0), described by the Hamiltonian
Conductance resonance
S-matrix pole E — iy
hfi cC
0
0
E
Fig. 1. Left panel: Magnetic field B-dependence of peaks in the differential conductance G = dl/dV. The peak positions trace out an .Y-shaped or l"-shaped profile in the (B.V) plane. Right panel: Location of the poles of the scattering matrix S{s) in the complex energy plane s = E — iy. The arrows indicate how the poles move with increasing magnetic field
The bound states in the closed quantum dot are eigenvalues of the M x M Hermitian matrix H = H^. The M x AT matrix W couples the basis states |//.) in the quantum dot to the normal metal, via AT propagating modes |u) through a point contact. In principle, we should take the limit M —¥ oc, but in practice M AT suffices.
The amplitudes of incoming and outgoing modes in the point contact at an energy E (relative to the Fermi level) are related by the ATxAT scattering matrix [14, 15]
Point, contact.
Superconductor ^
^ Quantum dot.
Fig. 2. Schematic illustration of an Andreev billiard
S(E) = 1 + 2mW^ (H - /-inr1 - E) 1 IT. (2)
This is a unitary matrix, S(E)S^(E) = 1.
A scattering resonance corresponds to a pole e = E — ¿7 of the scattering matrix in the complex energy plane, which is an eigenvalue of the non-Hermitian matrix
il, ,,- Ii ^ /-inr+.
(3)
The positive definiteness of ITH"1 ensures that the poles all lie in the lower half of the complex plane, 7 > 0, as required by causality. Particle hole symmetry implies that e and ^e* are both eigenvalues of Hcjf, and hence the poles are symmetrically arranged around the imaginary axis.
The differential conductance G(V) = dl/dV of the quantum dot, measured by grounding the superconduc-
tor and applying a bias voltage to the normal metal, is obtained from the scattering matrix via [7]
G(V) =
h
Ar
1
Tr S(eV)t~_SHeV)t~_
(4)
in the electron hole basis, and via ~2 rN ,
G(V) =
h
Tr S(eV)TySHeV)Ty
(5)
in the Majorana basis. The Pauli matrices ry and tact on the electron hole degree of freedom. The two bases are related by the unitary transformation
s^usuK t/ = yir \
(6)
2.2. Gaussian ensembles
For a random-matrix description, we assume that the scattering in the quantum dot is chaotic, and that this applies to normal scattering from the electrostatic potential as well as to Andreev scattering from the pair potential. In the largo-M limit, we can then take a Gaussian distribution for H.
Pill) x oxp (--^-Tr//-)
(7)
By taking the matrix elements of H to be real, complex, or quaternion numbers (in an appropriate basis), one obtains the Wigner Dyson ensembles of nonsuper-conducting chaotic billiards f 16 19]. Particle hole symmetry then plays no role, because normal scattering does not couple electrons and holes.
Altland and Zirnbauer introduced the particle hole symmetric ensembles appropriate for an Andreev billiard [20]. The two ensembles without time-reversal symmetry are obtained by taking the matrix elements of i x H (instead of H itself) to be real or quaternion. When iH is real, there is only particle hole symmetry (class D), while when iH is quaternion, there is particle hole and spin-rotation symmetry (class C).
Both the Wigner Dyson (WD) and the Altland Zirnbauer (AZ) ensembles are characterized by a parameter 3 € {1,2,4} that describes the strength of the level repulsion factor in the probability distribution of distinct eigenvalues of H: a factor f]^ . |£",
i»
the WD ensembles and a factor ni<j l-^f — P AZ ensembles. (The prime indicates that the product includes only the positive eigenvalues.)
In the WD ensembles, the parameter 3 also counts the number of degrees of freedom of the matrix elements of H: 3 = 1. 2, 4 when H is real, complex, or quaternion, respectively. In the AZ ensembles, this connection is lost: 3 = 2 in the class C ensemble (iH real) as well as in the class D ensemble (iH quaternion).
The coefficient c can be related to the average spacing ¿>o of distinct eigenvalues of H in the bulk of the spectrum,
3ir2
( 8<*o X
in the WD ensembles, in the AZ ensembles.
(8)
The coefficient in Eq. (8) for the AZ ensembles is twice as small as it is in the WD ensembles with the same /?, on account of the ±E symmetry of the spectrum (see Appendix A).
Because the distribution of H is basis independent, we can without loss of generality choose a basis such that the coupling matrix W is diagonal,
Wmn = wnSmn, !<//>< .1/. I < // < A". (9)
The coupling strength w„. is related to the tunnel probability F„ € (0,1) of mode n into the quantum dot by [14, 15]
U'n =
MS0
5T2r„
2 — r„ — 2 \/l — rn
(10)
2.3. Class C and D ensembles
We summarize the properties of the 3 = 2 AZ ensembles, symmetry class C and D, that we need for our study of the Andreev resonances. (See Appendix B for the corresponding 3 = 1,4 formulas in symmetry classes CI and Dili.) Similar formulas can be found in Rof. [21].
When Andreev scattering operates together with spin-orbit coupling, we can combine electron and hole degrees of freedom from the same spin band into a real basis of Ma.jorana fermions. (This change of basis amounts to the unitary transformation in Eq. (6).) In the Ma.jorana basis, the constraint of particle hole symmetry is given simply by
H = -H*
(H)
and we can therefore take H = iA with ,4 a real antisymmetric matrix. In the Gaussian ensemble, the upper-diagonal matrix elements Anm (n < m) all have identical and independent distributions,
M
P({A nm })oc JJ oxp
1=n<m
2M&1
(12)
(see Eqs. (7) and (8)). This is the 3 = 2 class-D ensemble, without spin-rotation symmetry.
The 3 = 2 class-C ensemble applies in the absence of spin-orbit coupling, when spin-rotation symmetry is preserved. Andreev reflection from a spin-singlet superconductor couples only electrons and holes from different spin bands, which cannot be combined into a real basis state. It is then more convenient to stay in the electron hole basis and to eliminate the spin degree of freedom by considering a single spin band for the electron and the opposite spin band for the hole. (The matrix dimensionality M and the mean level spacing So then refer to a single spin.) In this basis, the particle hole symmetry requires that
H = —TyH*Ty,
(13)
where the Pauli matrix ry operates on the electron and hole degrees of freedom.
Constraint (13) implies that H = iQ with Q a quaternion anti-Hermitian matrix. Its matrix elements are of the form
Q nm — (hnu To \ ibnmTx + i('nmTy l idln!, T: .
n.in 1,2,... , M/2,
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