Mesoscopic Fluctuations Of Orbital Magnetic Response In Level-Quantized Metals

5 Pages·0.12 MB·English

by  R. A. Serota

Mesoscopic Fluctuations Of Orbital Magnetic Response In Level-Quantized Metals
R. A. Serota
Department of Physics
University of Cincinnati
Cincinnati, OH 45221-0011
1
[email protected]
0
0 Weevaluatethedistribution function of mesoscopic fluctuationsof orbital magnetic response in
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finite-size level-quantized metal particles and Aharonov-Bohm rings for temperatures smaller than

n the mean level spacing. We find a broad distribution with the reduced moments much larger than
a themean. Forstrongspin-orbitinteractionwefindverylongtailsduetothermalactivationoflarge
J effective moments of theelectrons at theFermi level.

0
3
I. INTRODUCTION

]
l
l
a The question of mesoscopic fluctuations of orbital magnetic response has been extensively studied within the
h perturbation theory approach which, generally, holds when the temperature and/or level broadening is larger than
-
s themeanlevelspacing∆[1]. In2D(whichweonlyconsiderhere),thevarianceofthemagneticmomentwaspredicted
e to be1,2
m
t. δM2 µ2 (k ℓ)2 φ 2ln Ec µ2 (k ℓ)2 µBH 2ln Ec (1)
a ∼ B F (cid:18)φ (cid:19) (cid:18)T∗(cid:19)∼ B F (cid:18) ∆ (cid:19) (cid:18)T∗(cid:19)
m (cid:10) (cid:11) 0
where
-
d
n T∗ =max T,τ−1 (2)
H
o (cid:8) (cid:9)
c with the following notations: µB is Bohr magneton, kF is the magnitude of the Fermi wave vector, ℓ is the electron
[ mean-free-path, φ is the flux through the sample or the Aharonov-Bohm (AB) flux for rings, φ = 2π/e is the flux
0
quantum in units where h¯ = c = 1, E D/L2, where D = v ℓ/2 is the diffusion coefficient, and L the sample size
1 c ∼ F
v (typically a ring or disk circumference), and
6
5 τ−1 E φ 2 (µBH)2 (k ℓ) (3a)
4 H ∼ c(cid:18)φ (cid:19) ∼ ∆ F
0
1
0 The logarithmic dependence changes its form when
1
0 1/2
T
/ φ φ φ (4)
t ≈ c ∼ 0(cid:18)E (cid:19)
a c
m
Thesignificanceofthefluxscaleφ issuchthatatT ∆itdeterminesthescaleoftransitionfromGaussianOrthogo-
c
- nal(GOE)toGaussianUnitary(GUE)Ensemble. It∼isalsothescaleoflinearresponseintheproblem. (Whileeq. (1)
d
indicates that, aside from logarithmic corrections, the linear response regime extends to φ φ for the fluctuations,
n 0
∼
o itisobtainedinthesocalled”zero-mode”approximation[1]forsystemswithsimplyconnectedgeometries. However,
c the higher modes may need to be summed up to correctly evaluate the fluctuations [4]. Consequently, unless specifi-
: callystatedotherwise,weshalllimitourconsiderationtoφ<φ wheretheresponseislinearbothforthefluctuations
v c
i and the mean, for simply connected and AB geometries.)
X
r
a
1
In an earlier work, Altshuler and Spivak [2] considered current fluctuations in SNS junctions with broken time-reversal
symmetry. This problem is equivalent to the persistent current fluctuations in AB rings. If the time-reversal symmetry is
not broken and the Cooperon contribution is taken into account, their result would be equivalent to Ref. [1], including the
logarithmic term.
2
Insofar as the exact numerical coefficient in eq. (1), both Refs. [1] and [2] used the Euler-Maclaurin method for Matsubara
summation with a resulting integral over the continuous variable x= mT/Ec, dx= T/Ec ≪ 1. However, for small x a more
accurateproceduremustbeused. ItisexplainedinRef.[3]forthemeanresponse(AppendixII)andcanbetriviallyextended
to fluctuations.
1
The averagemagnetic moment was predicted to be [1]
µ φφ0, φ<φ
B φ2 c
M c (5)
h i∼ µ φ0, φ>φ
B φ c
which, for T ∆, becomes
∼
µ (k ℓ) φ µ µBH (k ℓ), φ<φ
M B F φ0 ∼ B ∆ F c (6)
h i∼ µ φ0 µ ∆ , φ>φ
B φ ∼ BµBH c
Comparing(6)and(1)onthescaleφ<φ ,weseethat M and δM2 1/2 areofthesameorderofmagnitude. While
c
h i
numerically the latter is larger[1], it is clear that the diamagneti(cid:10)c resp(cid:11)onse is much less likely than the paramagnetic
response.
Inthisworkweaddresstheregimeinwhichlevel-quantizationbecomesimportant,namely,T ∆. Inthiscase,the
≪
perturbation theory approach, which uses the level density correlation function and standard thermodynamics, is no
longer applicable and one needs to use a single electron picture in conjunction with thermal occupancy of a two-level
system. In previous papers, we considered the mean response of GOE and Gaussian Symplectic Ensemble (GSE),
the latter being the case for strong spin-orbit (SO) interaction. We have argued the for GOE the mean response can
be formulated in terms of a single-electron van Vleck response at the Fermi level [5] and for GSE in terms of the
effective magnetic moments/persistent currents of electrons in the last occupied (Fermi) state [6]. Here, we apply
these approaches to mesoscopic fluctuations.
II. GOE
As explainedinRef.[5],the meanorbitalresponsecanbe understoodinterms ofthe vanVleck(vV) responsethat
involves virtual transitions from the last occupied level ε to the first unoccupied level ε of the Fermi sea
i f
2 2
i M f H2 M H2
z if
ǫ = (cid:12)h | | i(cid:12) (cid:12) (cid:12) (7)
vV (cid:12)(cid:12) cεi−εf(cid:12)(cid:12) ≡ (cid:12)(cid:12)cεi−(cid:12)(cid:12)εf
whereM isthemagneticmomentalongthemagneticfieldH (perpendiculartothesample). Therearethreepossible
z
sourcesfornon-self-averaging(fluctuations)basedonthispicture. Thefirsttwo,basedoneq. (7),arethefluctuations
c
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of M and the distribution of the level spacing at the Fermi level (that is, the distribution of the values in the
if
(cid:12) (cid:12)
den(cid:12)(cid:12)ocmin(cid:12)(cid:12)ator of (7)). We expect the latter to be dominant and will neglect the former. The third, which may lead to
occasionaldiamagnetism, is the system-dependant nature of the Fermi-sea cancellationbetween the diamagnetic and
paramagnetic contributions to the total magnetic response [5], [7]; it is not studied here.
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ThemeanvalueofthevVresponseisobtainedasfollows. M canbefoundusingthesemiclassicalapproximation
if
(cid:12) (cid:12)
(and, more precisely, using the result for the magnetic dipo(cid:12)(cid:12)lecab(cid:12)(cid:12)sorption) [5] and is given by
2
M µ2 (k ℓ) (8)
(cid:12) if(cid:12) ∼ B F
In what follows, we will use a dimensionless mea(cid:12)(cid:12)scure(cid:12)(cid:12)x for the level spacing at the Fermi level,
ε ε =x∆ (9)
f i
−
Averaging with the GOE [8] distribution function for the nearest energy levels, we find [5]
2 π ∞ πx2 πυ 2
ǫ = s M H2 exp dx= M H2 (10)
vV if if
− (cid:12) (cid:12) 2∆Z0 (cid:18)− 4 (cid:19) − 2 (cid:12) (cid:12)
χ(cid:12)(cid:12)cH(cid:12)(cid:12)2A(k ℓ) τ−1 (cid:12)(cid:12)c (cid:12)(cid:12) (11)
∼−| L| F ∼− H
Here s is the level degeneracy (s = 2, on the account of spin), υ = s∆−1 is the density of levels, χ is the Landau
L
susceptibility [9]andAis the samplearea. As waspointedoutearlier,itisassumedthatτ−1 ∆,with the opposite
H ≪
limit corresponding to GUE.
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To consider the fluctuations, we notice that the second order perturbation theory is used in derivation of (7).
Consequently, the limit of its applicability [10] is, per (7) and (9),
τ−1
ǫ H x∆ (12)
vV
∼ x ≪
that is
x τ−1/∆ 1/2 or ε ε τ−1∆ 1/2 (13)
≫ H f − i ≫ H
(cid:0) (cid:1) (cid:0) (cid:1)
Animportantobservationabouttheaboveconditionsisthattherangeoflinearresponseissample-specificanddepends
on the Fermi level spacing of a particular particle. On the other hand, when describing the statistical distribution of
the magnetic energies, it is only meaningful to consider their values at a given field (or a range of fields).
An estimate of the fluctuation can be obtained by substituting the r.h.s. of (13) into eq. (7)
ǫ τ−1∆ 1/2 ǫ (14)
vV ∼ H ≫ vV
(cid:0) (cid:1)
Below we will show that (14)) corresponds to the result for the reduced higher cumulants for the distribution of
magnetic energies and is due to the systems with small level spacing at the Fermi level.
Since the reduced moments grow with the order of the moment, the cumulants and the moments should be of the
sameorderofmagnitudeanditissufficienttoevaluatethelatter. Webeginwiththeevaluationofthevariance/second
moment of the vV response
π dx πx2 ∆
ǫ2 τ−2 exp τ−2ln (15)
∼ 2 H Z(τH−1/∆)1/2 x (cid:18)− 4 (cid:19)∼ H (cid:18)τH−1(cid:19)
(omitting subscript ”vV”). The higher moments and the reduced moments can be evaluated similarly and are given
by
π πx2
ǫn τ−n x−n+1dxexp (16)
∼ 2 H Z(τ−1/∆)1/2 (cid:18)− 4 (cid:19)
H
n√ǫn τ−1∆ 1/2 τH−1 1/n n→∞ τ−1∆ 1/2 (17)
∼ H (cid:18) ∆ (cid:19) → H
(cid:0) (cid:1) (cid:0) (cid:1)
in agreement with the estimate (14). Alternatively, eqs. (15)-(17) can be evaluated using the distribution function
for the fluctuations which is found as
P (ǫ)= π dxδ ǫ τH−1 xexp πx2 = πτH−2 exp πτH−2 , ǫ< τ−1∆ 1/2 (18)
2 Z(τH−1/∆)1/2 (cid:18) − x (cid:19) (cid:18)− 4 (cid:19) 2 ǫ3 (cid:18)− 4ǫ2 (cid:19) (cid:0) H (cid:1)
with the case ǫ> τ−1∆ 1/2 (or, equivalently, x< τ−1/∆ 1/2) addressed below.
H H
A comparison s(cid:0)hould b(cid:1)e made between (15) and(cid:0)the pe(cid:1)rturbative result (1). First, assuming for the latter that
T ∆(thelimitofapplicabilityoftheperturbativeapproximation),wefindthatthecorrespondingenergyfluctuation
∼
is
ǫ2 τ−2ln Ec (19)
pert ∼ H (cid:18)∆(cid:19)
The difference inlog factorsis because (15)is obtainedin a two-levelapproximation,whereas(19) takesinto account
contribution of levels within E of the Fermi level; otherwise, the results are in qualitative agreement. (The mean
c
values ǫ are also in agreement [5]). Although the higher moments had not been evaluated perturbatively, it is
vV
expected that n ǫn τ−1 and are smaller than obtained in a two level picture. This should be anticipated since
q pert ∼ H
the two-level response is very sensitive to the variations of the energy level spacing at the Fermi level.
For x < τ−1/∆ 1/2 the perturbation theory evaluation of ǫ (7) is no longer valid. This applies to the fraction
H vV
τ−1/∆ o(cid:0)f all sys(cid:1)tems with sufficiently small Fermi level spacing. There are two possible approaches to this case
∼ H
that yield, essentially, the same result. In the first approach, the Fermi level and the first unoccupied state can be
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viewed as, effectively, a doubly degenerate state with respect to the magnetic field perturbation. Using the secular
equation [10], we find that the magnetic field will split the levels in the amount
M 2H2 τ−1∆ 1/2 (20)
r(cid:12) if(cid:12) ∼ H
(cid:12)(cid:12)c (cid:12)(cid:12) (cid:0) (cid:1)
(Thesecondordertermisstillgivenbyeq. (7)butwiththefinalstatef beingthenextunoccupiedstateatadistance
greater than τ−1∆ 1/2 from the Fermi level.). Alternatively, one can argue that such systems are effectively in the
H
GUE regime.(cid:0)The p(cid:1)erturbative expression for the mean value is
ǫ τ−1∆ 1/2 (21)
pert(GUE) ∼ H
(cid:0) (cid:1)
which is obtained from the second eq. (6) using the substitution ∆ τ−1∆ 1/2 , where the latter is the upper
→ H
bound of the level spacing in the systems that have effectively undergone(cid:0)the GO(cid:1)E GUE transition.
→
The physical interpretation of the above results may be as follows. In a particular systems, the linear response
regime is determined by the Fermi level spacing in this system, as per eq. (12),
µ2H(k ℓ)
M B F (22)
∼ x∆
As a result of linear response, the magnetic moment approaches the value
M µ (k ℓ)1/2 (23)
B F
∼
with little change for larger fields (until τ−1 approaches ∆ - see below). The statistical distribution of the response
H
implies the need to consider the same range of fields for all systems. The mean magnetic moment is given by
M µ2BH(kFℓ) τH−1∼∆µ (k ℓ)1/2 (24)
B F
h i∼ ∆ →
The slowly decaying distribution function
P(ǫ) τH−2, τ−1 <ǫ< τ−1∆ 1/2 (25)
∝ ǫ3 H H
(cid:0) (cid:1)
is due to systems with small Fermi level spacing, where the magnetic moment may become as large as (23).
At τ−1 >∆, the single-level ansatz is no longer valid and the results of Ref. [1] (as described in the Introduction)
H
should be applied instead; at τ−1 ∆ the latter are consistent with the present results.
H ∼
III. GSE
It was argued [11], [6] that in a GSE the electrons at the Fermi level have a magnetic moment
M µ (k ℓ)1/2 (26)
B F
∼
(which is, incidentally, of the same order of magnitude as given by (23)) induced by the spin-orbit interactions. For
a system with even number of electrons, the magnetic moments of the two Fermi-levelelectrons canceleach other. A
system with an odd number of electrons, on the other hand, should have a permanent magnetic moment [6] and a
Curie-like susceptibility. This is in incomplete analogy with the purely spin magnetism [12], the sole difference being
thatM =µ inthelattercase. Consequently,werefertoRef.[6]forthediscussionofthedistributionfunctionofthe
B
susceptibility obtainedin the two-andthree-levelapproximationforthe even-andodd-electronsystems respectively;
in both cases, one finds long tails due to thermal activation of the magnetic moments.
It should be noted that the single-electron magnetic moment (26) can be attributed to the large electron g-factor,
g (k ℓ)1/2, induced by the spin-orbit interaction[6], [13], [14]. Furthermore,it was argued[13], [14]that the latter
F
∼
can fluctuate by as much as the order of magnitude. We point out, however, that in the temperature regime T ∆
≪
considered here, the thermal activation effects should be dominant as they may produce much larger fluctuations.
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IV. DISCUSSION
The two-level picture considered here predicts the large fluctuations of the orbital magnetic response due to the
possibility of small level separations at the Fermi level. Obviously, the possible observation of these effects imposes
restrictions on the temperature and the magnetic field. Moreover, such restrictions are sample specific since the
magnetic energy and the temperature must be compared with a particular Fermi level separation of the nearest
states. (See, for instance, eqs. (12) and (13)). On the other hand, a statistical description of the fluctuations implies
a common (range of) magnetic field and temperature for all systems. Consequently, while in a given system one can
observethelinearresponseregimeduetothevVenergy(7)bysufficientlyreducingthemagneticfield,theFermi-level
spacing in this system may be smaller than the magnetic energy corresponding to the field at which all systems are
analyzed. The latter is critical for understanding of the effective cut-offs in the analysis of the distribution function.
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The issues that remain to be understood are the fluctuations of the matrix elements M and the details of
if
(cid:12) (cid:12)
the cancellation between the van Vleck paramagnetism and the precession diamagnetism o(cid:12)(cid:12)vcer t(cid:12)(cid:12)he Fermi sea. These
will require further, largely numerical, studies. Also, our formalism should be applicable, with minor changes, to
orbitalmagnetismofintegrablesystems,suchasarectanglewithincommensuratesides[15]. We hopeto addressthis
problem in a future work.
V. ACKNOWLEDGMENTS
I wish to thank Bernie Goodman for many helpful discussions. This research was not supported by any funding
agency.
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