ANDREEV BOUND STATES. SOME QUASICLASSICAL REFLECTIONS
Y. Lin* A. J. Leggett
Dept. of Physics, University of Illinois at Urbana-Champaign 61 SOL Urbana. USA
Received June 18, 2014
We discuss a very simple and essentially exactly solvable model problem which illustrates some nice features of Andreev bound states, namely, the trapping of a single Bogoliubov quasiparticle in a neutral s-wave BCS superfluid by a wide and shallow Zeeman trap. In the quasiclassical limit, the ground state is a doublet with a splitting which is proportional to the exponentially small amplitude for "normal" reflection by the edges of the trap. We comment briefly on a prima facie paradox concerning the continuity equation and conjecture a resolution to it.
Contribution for the JETP special issue in honor of A. F. Andreev's 75th birthday
DOI: 10.7868/S0044451014120049
1. INTRODUCTION
This year, 2014, marks not only the 75th birthday of Sasha Andreev but also the 50th anniversary of what is probably his most famous single piece of work fl], that on the reflection of an electron at a normal superconducting boundary by conversion into a hole. Over the last half-century, the phenomenon of "Andreev reflection" has of course emerged as one of the key notions in mesoscopic physics, with applications which range far beyond the original context of the thermal conductivity of type-I superconductors in the mixed state. In this paper, we briefly discuss a "toy" problem which we feel illustrates some features of the idea in a particularly simple and intuitive way. The problem is indeed so simple that we suspect that, even if it has not been explicitly solved in the published literature in connection with a specific experimental setup, it must have been set more than once as a student exercise; nevertheless, in the present context of celebration of Sasha's work, we find it is worth a brief commentary. As a matter of history, our interest in this problem was motivated by a desire to understand whether results obtained by the standard mean-field method for some rather subtle questions concerning Berry's phase can be replicated by a strictly particle-number-conserving forE-mail: yiriolin'flillinois.edu
malism, an issue which to our knowledge has received little discussion in the existing literature [2]. However, we do not attempt to address this issue here, and hence the level of this paper is essentially pedagogical.
Before we start, one general remark: in the original paper fl] and in much of the subsequent work on it, the phenomenon of Andreev reflection occurs as a consequence of a variation in space of the superconducting order parameter ("gap"). However, it is actually a much more general phenomenon, which crudely speaking occurs in a dense Fermi system whenever quasiparticles of a given energy are allowed in one region of coordinate space and forbidden in another, and the system is dense on the separatrix surface. This is easiest to see in the quasiclassical limit, by which we mean that all physical quantities (potential, density, order parameter, ... ) are slowly varying on the scale of the mean particle separation. We consider a quasiparticle with an (initial) momentum k propagating from the "allowed" region towards the "forbidden" region. Since it cannot enter the latter, it must reverse its velocity. The most obvious way to do so is to reverse the momentum k ("normal" reflection). But it cannot do this gradually (in many small steps) because this would involve going through states of the Fermi sea which are already occupied; and it cannot do it (with any appreciable probability) in one shot, because this requires using a q ~ 2kf Fourier component of the potential (etc.) and by our definition of the quasiclassical limit any such components are ex-
ponontially small. Hence the only option is Androov rofloction (or sonic analog of it in nonsuporconducting systems, cf. Rof. [3]). Of courso, if wo introduce abrupt spatial variations in the physical parameters, then normal reflection is no longer necessarily excluded and the situation is more complicated (cf. Rof. [4]). Wo make the above considerations more quantitative in Sec. 3 and the Appendix.
2. DEFINITION OF THE PROBLEM
We consider a system of 2Ar neutral fermions, initially in zero magnetic field and at T = 0, constrained to move in an annular container of circumference L and transverse dimensions d\ for notational convenience only, we replace this geometry by a rectangular tube of length L in the ¿-direction and impose periodic boundary conditions in all three dimensions. (As we see in what follows, the imposition of such boundary conditions in the transverse directions is mainly a matter of convenience, but that in the longitudinal direction is crucial to our argument.) We take the fermions to interact via a short-range, spin-independent, weakly attractive potential.
We assume that the ground state \P-2A',o is well approximated by the particle-conserving version of the standard BCS state, i.e., apart from normalization,
aL = "¿«L
• <T'-'/." /.. >T
CK
(2.3)
^ 2A',0 —
N
vac), Ct = y^
t
(2.1)
where |vac) denotes the vacuum state and the coefficients c/, are given by
(-k
Vk_ Uk '
i'k Vk
1
1 ±
Ek
(2.2)
I AI2)1/'2
where e^ = h2(k2 — hp) and E/, = (e2 with I'f being the Fermi wave vector and A the (isotropic) BCS energy gap, which is given by the usual self-consistent gap equation and is assumed to be -C Ep = li'lq j'ltn. The only low-energy (E < 2A) excitations of this system are the long-wavelength density fluctuations (Anderson Bogoliubov modes), which in the present context are of no interest to us. If we now consider the ground state and low excited states of the (2Ar + l)-particle system, these correspond to "single fermion" (Bogoliubov quasiparticlo) excitations with the wave vector k (momentum fik), spin ±1/2, and en-orgy Ek• The operator which, acting on the 2Ar-particlo ground state, creates such a Bogoliubov quasiparticlo while leaving the system in a (2Ar + l)-particle number eigenstate is given by
where is the operator which, acting on the 2Ar-par-ticlo ground state, creates the (2Ar + 2)-particle ground state, i.e., apart from normalization, it is just the in Eq. (2.1). Although in other contexts it may be essential to remember the presence of the operator C^, it does not play a significant role in the arguments in this paper, and we mostly do not write it explicitly in what follows, simply assuming implicitly that it is always added when necessary to preserve particle number conservation. We call a Bogoliubov excitation with Ek > 0 a "quasiparticlo" and one with Sk < 0 a "quasihole".
We now add a weak magnetic field B(z) that is a function only of z and which is coupled to the spin via the Zeeman effect (only: we recall that the system is neutral!). It is convenient to take B(z) to be smooth, uniformly positive, and symmetric around î = 0 (the middle of the tube) and to have some characteristic extension in space R -C L and characteristic magnitude B0 which we specify below. Thus, if H0 is the original Hamiltonian of the system including (spin-independent) interactions, the complete Hamiltonian is
now
H = H,
o
V(z
-/iB(z
(2.4)
where //. is the magnetic moment of the particles and <7, is the projection of the spin of the ¿th particle on the axis of B. We now ask: What are the wave functions and energies of the ground state and low-lying energy eigenstates of the (2Ar + l)-particle system?
We can immediately say a few things. First, the ground state must certainly have a positive value of the total spin S = X^*7«- Second, it must be possible to choose it to be invariant under reflection in the plane i = 0 (or equally under time reversal of the orbital coordinates alone: we note that we have assumed that H0 does not contain any spin orbit interactions). Third, the qualitative behavior is intuitively obvious in the two limits of both B0 and R large and both very small: in the former case, a substantial slab of the system becomes normal, while in the latter, the (2Ar + l)-particle ground state and low excited states correspond to spin-up single-fermion excitations which extend far beyond the region of the "trap" (the region |i| < R). In this paper, we are not interested in either of these limits but rather in a particular case, where intuitively speaking, in the (2Ar + l)-particle ground state, the potential V(z) efficiently "traps" a single Bogoliubov quasiparticlo. Moreover, we are interested in
the "Ginzburg Landau limit" in which all the relevant quantities vary slowly in space not merely over a distance kp1 but also over where £ ~ he/ /A is the Cooper-pair radius. What constraints do those requirements place on B0 and f??
It is obvious that to be in the Ginzburg Landau limit, we need and the requirement that the
Zeeman coupling should not destroy superconductivity then enforces the condition fi.B0 < A; we shall be more conservative and require fi.B0 = I'o -C A- While we should expect (in view of the ID nature of the potential) a weakly bound state to exist for any B0, the condition that it be well localized within the range of the potential, i. e., that the extra kinetic energy derived from the confinement be not too close to the binding energy, requires, as we see below, that ftvF/R < Vq -C A, which fortunately is already guaranteed by the condition Thus the necessary conditions on the parameters B0 and R are
£ •€ 1! •€ /.. V0 « A.
(2.5)
We note that we still have freedom to adjust the ratio
n> = (l'o/A)(i?/0 (2.6)
which essentially determines the (order of magnitude of) the number of bound states in the well, in the range ~ 1 00.
With these conditions, B(z), or equivalent ly V(z) = —fiD(z), can be just about any smooth function: when we need a specific example we use the convenient form
V(z) = -Vt) seclr'(://•,').
(2.7)
The Zeeman term in (2.4), with a potential V(z) satisfying (2.5), is possibly the "minimal" nontrivial
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