Pis'ma v ZhETF, vol. 101, iss. 5, pp. 390-395
© 2015 March 10
Oscillations of magnetoresistance in a clean hollow cylinder with
fluctuating radius
A. S. Ioselevich^
Landau Institute for Theoretical Physics of the RAS, 119334 Moscow, Russia Moscow Institute of Physics and Technology, 141700 Moscow, Russia Submitted 22 January 2014
We consider magnetic oscillations of resistivity of a clean (mean free path I R) hollow cylinder with fluctuating (with an amplitude of fluctuations AR <C R) radius R, threaded by magnetic flux We demonstrate, that for weak fluctuations (AR <C jJp1) the oscillations have a standard period 2$o, characteristic for oscillations in a clean system, while for AR p^1 they become <&o-periodic, which was expected only for dirty systems with I <C R. The work is motivated by observation of predominantly <&o-periodic magnetic oscillations in very clean Bismuth wires.
DOI: 10.7868/S0370274X15050148
Introduction. Oscillations of resistivity p of a hollow conducting cylinder with change of magnetic field H, parallel to the axis of the cylinder, is a well known phenomenon, manifesting the quantum nature of conducting electrons in metals. The basic reason for these oscillations is the interference of electrons traversing the cylinder along paths with different topological winding numbers. In the simplest case of clean cylinder (with the circumference 2-nR <C I) it is actually a direct realization of the Aharonov-Bohm effect [1]; the resistance is a periodic function of the flux $ = -kR2H with the period 2$o, where = nchj2e is the flux quantum (see, e.g., [3, 4]). The general arguments (c.f. [2]) show that the same periodicity should be observed for any thermodynamic or kinetic property of a cylinder - for arbitrary relation between 2-nR and I. In particular, for the resistivity
P = Po
1 + An cos(7rn$/<I>o
(1)
In dirty cylinders, where 2ttR I the odd-n harmonics of the AB-oscillations are washed out due to strong variations in the length of different diffusive trajectories that lead to randomisation of the non-magnetic part of the phases of electronic wave-functions. In a seminal paper by Altshuler, Aronov and Spivak [5] it was shown, however, that the even-n harmonics - the oscillations, associated with a special sort of trajectories (the ones, containing closed topologically non-trivial loops on the cylinder) survive the randomisation. These even-n oscil-
e-mail: aioselevich@gmail.com
lations arise due to the weak localisation corrections and manifest the interference between the contributions of paths, corresponding to the same trajectories, but with opposite directions in which the loop is traversed. In such an interference the randomised nonmagnetic part of the phase does not enter, while the magnetic part of the phase is doubled and, therefore, the AAS-oscillations occur with the period The AAS-oscillations were observed experimentally (see [6, 7]), although their amplitude is small: ~ (ppl)-1.
Thus, a simple correspondence seemed to be established: in clean cylinders with 2ttR <C / the AB-oscillations with the period 2$o should dominate, while dirty cylinders (with 2nR I) is the domain of the AAS-oscillations with the period
Recently, however, the dominance of the ^o-periodic (AAS-like) oscillations was experimentally observed [8] in a manifestly clean Bismuth wires, where the AB-oscillations would be natural to expect. Note, that in this system the surface states are believed to be mainly responsible for the conductivity and the spin-orbit interaction is important for these states. In this situation the oscillations of resistivity were discussed in [9], where it was shown that in the case of strong surface disorder the odd harmonics in (1) should be suppressed, and the oscillations should be indeed ^o-periodic, like in the AAS-case. However, the model, used in [9] seems to be not applicable to the experiments [8], since it implies strong disorder, which is apparently absent there.
Trying to explain this unexpected experimental observation of [8], one should address the following question: suppose that there is some mechanism of suppres-
sion of odd-n harmonics, then how is it possible, that the same mechanism does not suppress also the mean free path I and does not show up in the monotonic part of the resistivity?
On the other hand, under the condition 2nR <C I the standard method of finding even-n AAS-oscillations, based on the diffusion approximation, does not apply, and even the qualitative physics underlying the $0-periodic oscillations may be different from the weak localisation effect, considered in [5].
In this Letter we propose a simple mechanism for this effect, related to weak and smooth fluctuations of the radius of the wire. Here we will discuss it only for the simplest case of weak short range impurities, that can be treated within the Born approximation; we also do not take into account the spin-orbital interaction. The latter effects will be considered in a separate publication, here we only mention, that they may, in some cases, considerably alter the amplitude of the even-n harmonics, but they do not change the principal result: odd harmonics are suppressed much stronger than even ones.
We argue, that even relatively small fluctuations of the cylinders radius R(z), measured at different cross-sections z, are able to exponentially suppress the oscillations. Namely
A„(Q, q) = a„
=-2 «V
\J imQ
cos(2nQ + 7r/4),
6 JO (n- odd),
a'n = / n\3 1 / A W
7r(nQII (n - even),
where we have introduced the parameters
Q=TrRpF, q = TrARpF <g, (3)
R being the cylinders radius, averaged over the positions of the cross section, and AR = (R — R)2 R, be-
ing the relatively small variance of R. The quasiclassical parameter Q we will assume to be large: Q ^ 1, while q may be either small, or large.
We demonstrate that at q 1 the fluctuations exponentially suppress all the amplitudes An; however, while the odd harmonics are suppressed to zero, the even ones saturate at some small (in parameter Q ), but finite values an. Thus, at q —> oo the oscillations become periodic, as in the dirty case.
On the other hand, if the fluctuations of R(z) are adiabatically smooth (their correlation radius £ R) then this kind of disorder leads to only exponentially small contribution to the scattering of electrons and, therefore, practically does not affect the mean free path I.
It should also be noted that our result (2), in contrast with the result of [5], is not proportional to small parameter (ppl)^1: indeed it does not involve any weak localisation effects and does not depend on I at all. Although the derivation of expression for the residual amplitude an requires taking into account quantum corrections to the standard Drude formula, these corrections are not associated with the interference of different scattering events. It does not mean that the localisation corrections do not exist in our case - they do, but for low concentration of impurities the corresponding corrections to An are small.
A homogeneous tube. We start with considering a hollow metallic cylinder with a fixed radius R and length L, threaded by a magnetic flux $ (the magnetic field is oriented along the axis of a cylinder z). The effective two-dimensional metal, constituting the tube, is characterized by an isotropic spectrum of electrons, pf and «f being, respectively, the momentum and the velocity at the Fermi level Ep. There are rare impurities in the system - with the two-dimensional concentration nimP' the corresponding mean free path being I. In this paper we restrict our consideration to the case of low concentration <C R~2 and consider the semiclassi-cal and clean case, when
pFl < 2ttR< Azimp,
I < L,
(4)
where Az;,
(2 nRn^p)
1 ^ R is a characteristic
separation between impurities.
In an absolutely clean system the eigenfunctions of the hamiltonian at the Fermi surface are
ripm((i>,z) = exp(ikmz + im(f>), (5)
km = ±v/p2F-(m + <S>/2<S>0)2/R2, (6)
where m G Z is the azymuthal quantum number.
Under the condition (4) the system can be treated as a multichannel quasi-one-dimensional one (the number of channels A^h = {2-kR)(p^/2-k) = ppR 1), so that the localization length C\oc = Nchl/2 I is large. We will assume that it is larger than the length of the wire:
/ « L < Aoc = kpfr)/2
(7)
and the specific one-dimensional localisation effects are relatively small, and can be treated perturbatively (weak localisation). From this assumption it immediately follows, that, in the leading semiclassical approximation, the conductivity of the wire can be described by the standard Drude formula
r(l d)
(2tt R40))e2D
e2Aoc, D0 = vFl/2, (8)
where Do is the diffusion coefficient, and z/p ' = pp/2-Kvp is the density of states of an infinite two-dimensional metal at the Fermi surface (without "spin two"). The factor 2-nRv^ = {pyR)/vy = N^/vp is nothing else but the effective one-dimensional density of states in the wire.
Certainly, there is no place for the dependence on $ in the result (8), such a dependence can only show up in the corrections to the semiclassical approximation. There are three types of these corrections:
1) "thermodynamic" corrections. Here we gather all the oscillatory effects, that arise already in the ideal tube without any impurities at all - corrections to the density of states and similar. In the leading order in the semiclassical parameter Q 1 the largest of these corrections are ~ Q~ll2 1, but we will need also the smaller ones;
2) "kinetic" corrections - they arise due to oscillatory corrections to the single-impurity scattering amplitude. Usually the single-impurity effects only lead to an irrelevant renormalisation of parameters of an impurity. In our case, however, due to nontrivial topology of the system, these corrections are highly sensitive to the magnetic flux $ and exhibit oscillatory behaviour. The relative amplitude of kinetic corrections (as well as the thermodynamic one
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