The Thouless formula $G = (e^2/h)(E_c/\Delta)$ for the two-probe dc
conductance $G$ of a d-dimensional mesoscopic cube is re-analysed to relate its
quantum capacitance $C_Q$ to the reciprocal of the level spacing $\Delta$. To
this end, the escape time-scale $\tau$ occurring in the Thouless correlation
energy $E_c = \hbar/\tau$ is interpreted as the {\em time constant} $\tau =
RC_Q$ with $RG \equiv$ 1, giving at once $C_Q = (e^2/2\pi \Delta)$. Thus,
the statistics of the quantum capacitance is directly related to that of the
level spacing, which is well known from the Random Matrix Theory for all the
three universality classes of statistical ensembles. The basic questions of how
intrinsic this quantum capacitance can arise purely quantum-resistively, and of
its observability {\em vis-a-vis} the external geometric capacitance that
combines with it in series, are discussed

Jayannavar, A. M.

Kumar, N.

English

arXiv

The Thouless formula (G = (e2/h)(Ec/&#916;) for the two-probe dc conductance G of a d-dimensional mesoscopic cube is re-analysed to relate its quantum capacitance CQ to the reciprocal of the level spacing &#916;. To this end, the escape time-scale &#964; occurring in the Thouless correlation energy (Ec = h/&#964;) is interpreted as the time constant &#964; = RCQ with RG &#8801; 1, giving at once CQ = (e2/2&#960; &#916;). Thus, the statistics of the quantum capacitance is directly related to that of the level spacing, which is well known from the Random Matrix Theory for all the three universality classes of statistical ensembles. The basic questions of how intrinsic this quantum capacitance can arise purely quantum-resistively, and of its observability vis-a-vis the external geometric capacitance that combines with it in series, are discussed

Publications of the IAS Fellows

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Statistics of Mesoscopic Fluctuations of Quantum
Capacitance
N. Kumar∗ and A.M.Jayannavar∗∗
International Centre for Theoretical Physics
34100 Trieste, Italy
Abstract
The Thouless formula G = (e2/h)(Ec/∆) for the two-probe dc con-
ductance G of a d-dimensional mesoscopic cube is re-analysed to relate
its quantum capacitance CQ to the reciprocal of the level spacing ∆. To
this end, the escape time-scale τ occurring in the Thouless correlation en-
ergy Ec = h¯/τ is interpreted as the time constant τ = RCQ with RG ≡ 1,
giving at once CQ = (e
2/2pi∆). Thus, the statistics of the quantum capac-
itance is directly related to that of the level spacing, which is well known
from the Random Matrix Theory for all the three universality classes of
statistical ensembles. The basic questions of how intrinsic this quantum
capacitance can arise purely quantum-resistively, and of its observabil-
ity vis-a-vis the external geometric capacitance that combines with it in
series, are discussed.
∗Permanent address: Raman Research Institute, Bangalore 560080, India
∗∗Permanent address: Institute of Physics, Bhubaneswar 751005, India
1
The non-self-averaging nature of quantum conductance, the transmit-
tance and the energy-level spacing of a mesoscopic system, due ultimately
to quantum coherence and the interference, has motivated extensive stud-
ies of the statistics of their sample-to-sample fluctuations.1 For a given
sample, however, these reproducible fluctuations manifest as random func-
tions of some externally controlled parameters such as the Fermi energy,
or the Aharonov-Bohm magnetic flux, etc. Microscopic analysis of the
statistics of these fluctuations in disordered electronic structures having
diffusive transport has been based on the impurity-diagrammatic pertur-
bative technique, while the macroscopic approaches generally appealed to
the random matrix theory.1 Fluctuations of the wave transmission coef-
ficient and of the eigenvalue spacing for ballistic microstructures (classi-
cally chaotic cavities) have been treated through the semiclassical and the
random matrix approaches.2,3 Very recently, quantum capacitance4,5 has
been added to the growing list of these non-selfaveraging quantities, where
the authors6 have calculated the statistics of the mesoscopic capacitance
fluctuations for the case of a chaotic cavity coupled capacitatively to a
backgate. This has involved calculating the Wigner delay time which is
given by the energy derivative of the phase associated with the scattering
matrix. The latter in turn was obtained from a global random matrix
theory for all the three universality (symmetry) classes of the statisti-
cal ensembles. It may be noted in passing that, for the case of a single
channel, the distribution of the time delay for a disordered mesoscopic sys-
tem was obtained earlier using the invariant imbedding approach.7 In the
2
present work we have addressed the problem of calculating the statistics of
the quantum capacitance of a mesoscopic disordered conductor directly in
terms of a generally valid argument predicated on the Thouless expression
for the conductance G = (e2/h)(Ec/∆) for a d-dimensional cube. Here ∆
is the energy level spacing at the Fermi level and Ec = h¯/τ the Thouless
(correlation) energy, where τ is the time scale of escape out of the sample.
Results for the RC-time constant for the ballistic, the diffusive as well as
for the possibly anomalous diffusive regime at the mobility edge are re-
ported. We also address the physical question of how a reactive element
such as the capacitance is determined resistively through the Thouless
conductance expression.
Starting from the Thouless expression for the conductanceG = (e2/h)(Ec/∆),
and recalling that Ec = h¯/τ , where τ ≡ RCQ is to be interpreted as the
RC-time constant with CQ as the effective (quantum) capacitance, and
R(≡ 1/G) the series quantum resistance, we get at once CQ = (e
2/2pi∆).
Here the level spacing ∆ scales as L−d for a d-dimensional sample. Thus,
we notice immediately that the quantum capacitance CQ is a volume ef-
fect a (proportional to L−3 in three dimensions), quite unlike the classical
(geometrical) capacitance that scales as L−1 in three dimensions.
Now the problem of finding the statistics of CQ reduces to that of
finding the level-spacing distribution. Defining the dimensionless capac-
itance x = CQ/C¯Q, with C¯Q as the ensemble averaged capacitance, we
can at once write down the probability distribution Pβ(x) for the three
3
universality classes, labelled by the symmetry parameter β = 1, 2 and 4
corresponding, respectively, to the Gaussian Orthogonal Ensemble (GOE),
the Gaussian Unitary Ensemble (GUE) and the Gaussian Symplectic En-
semble (GSE), as
P1(x) =
1
2pix3
exp[−(1/pix2)] , (1)
with the average capacitance C¯Q = (e
2/4 < ∆ >);
P2(x) =
pi
2x4
exp[−(pi/4x2)] , (2)
with the average capacitance C¯Q = (2e
2/pi2 < ∆ >); and
P4(x) = (81pi
2/128x6)exp[−(9pi/16x2)], (3)
with the average capacitance C¯Q = (16e
2/9pi2 < ∆ >). In all the three
cases above, the quantum capacitance distribution has a single peak and
a long tail. In fact the variance diverges (weakly, i.e., logarithmically) for
the GOE case. For a mesoscopic sample of interest, we have typically ∆ ∼
10 mK, and this gives CQ ∼ 20 fF.
The fact that the quantum capacitance is determined resistively via
the Thouless formula has several consequences for the RC relaxation time
of the mesoscopic sample, coupled capacitatively to a backgate. This can
be seen as follows for the various transport regimes of interest.
Ballistic regime
Here the Thouless energy Ec ≡ (h¯
2/2m)
∫
| ▽ψ |2 ddx, (being the extra
4
kinetic energy associated with the phase-twisting of the wavefunction ψ
from the periodic to the anti-periodic boundary condition) scales as Ec ∼
L−1. Recalling that ∆ ∼ L−d, we get at once G ∼ Ld−1 (the two-probe
Sharvin conductance of an ideal metal with Ld−1 transverse channels).
The corresponding RC-time constant τ scales as τ ∼ L.
Diffusive limit
Arguing as above, we have Ec ∼ 1/Larc, where Larc is the arc- length of
a random walk with the end-to-end distance ∼ L (sample size). Thus,
Larc ∼ L
2 giving RC-time constant τ ∼ L2.
Anomalous diffusion
At and near the mobility edge when the sample size L≪ ξ (the coherence
length), we can expect the diffusion to become anomalous in that Larc ∼
L2η with η 6= 1. Reasoning as above, this would give RC time constant
τ ∼ L2η.
The above clearly shows the characteristically different scaling behaviour
of τ for the three different transport regimes.
We now return to the question of how a reactive element such as the
quantum capacitance can arise concomitantly with the quantum resis-
tance. The whole point is that on the one hand the quantum capacitance
(unlike its classical geometrical counterpart) involves the energy cost of
promoting the piled-up electrons across the energy-level spacing, while
on the other the latter also represents the typical energy mismatch (ob-
structing the electron diffusion) which is involved in determining the con-
5
ductance via the Thouless formula. Indeed, the larger the level spacing
the greater the promotional energy cost, and hence the smaller the ca-
pacitance. Concomitantly, the larger energy spacing implies larger resis-
tance. The two are functionally related, and conspire to give the RC-time
constant that scales with the sample size as derived above for the three
metallic regimes.
As for the observability of the quantum capacitance, we note that it
combines with the classical (geometrical) capacitance in series.5 Thus, it is
effective only for sufficiently small samples inasmuch as CQ = e
2/2pi∆ ∼
L3. while Cclassical ∼ L for a three-dimensional mesoscopic sample.
Finally, our present approach to mesoscopic quantum capacitance,
based as it is on the Thouless conductance formula, shares the latter’s
generality inasmuch as the single electron coupling and the level spacing
may be generalized to the coupling of correlated electron excitations and
the associated excitation energy-spacings.8
In conclusion, we have derived the statistics of mesoscopic capacitance
fluctuations basing directly on the generally valid Thouless conductance
formula by identifying the escape time as the RC-time constant. Inas-
much as this quantum capacitance combines in series with the classical
geometrical capacitance, the speed of a mesoscopic circuit element must
be limited ultimately by this quantum RC-time constant.
References
6
[1] For a general review, see Mesoscopic Phenomena in Solids, edited by
B.L.Altshular, P.A.Lee and R.A.Webb (North-Holland, New York,
1991); C.W.J. Beenakker and H. van-Houten, in Solid State Physics,
edited by Ehrenreich and D.Turnbull (Academic Press, New York,
1991).
[2] H.U.Baranger and P.A.Mello, Phys. Rev. Lett., 73, 142 (1994)
[3] R.A.Jalabert, J.L.Pichard and C.W.J. Beenakker, Europhys. Lett.,
27, 255 (1994)
[4] M.Bu¨ttiker, J. Phys., 5, 9361 (1993)
[5] N.Kumar, Curr. Sci., 68, 945 (1995)
[6] V.A.Gopar, P.A.Mello and M.Bu¨ttiker, cond-mat/9605016
[7] A.M.Jayannavar, G.V.Vijayagovindan and N.Kumar, Z. Phys., B75,
77 (1989)
[8] Y. Imry, Europhys. Lett., 30, 405 (1995)
7


Statistics of mesoscopic fluctuations of quantum capacitance

Restricted Access.The Thouless formula (G = (e2/h)(Ec/Δ) for the two-probe dc conductance G of a d-dimensional mesoscopic cube is re-analysed to relate its quantum capacitance CQ to the reciprocal of the level spacing Δ. To this end, the escape time-scale τ occurring in the Thouless correlation energy (Ec = ℏ/τ) is interpreted as the time constant τ = RCQ with RG ≡ 1, giving at once CQ = (e2/2π Δ). Thus, the statistics of the quantum capacitance is directly related to that of the level spacing, which is well known from the Random Matrix Theory for all the three universality classes of statistical ensembles. The basic questions of how intrinsic this quantum capacitance can arise purely quantum-resistively, and of its observability vis-à-vis the external geometric capacitance that combines with it in series, are discussed

Jayannavar, A.M.

Raman Research Institute Digital Repository

Modern Physics Letters B

Statistics of Mesoscopic Fluctuations of Quantum Capacitance

Indian Academy of Sciences

N. Kumar

A. M. Jayannavar

Crossref

http://dx.doi.org/10.1142/S0217984997000086

Statistics of Mesoscopic Fluctuations of Quantum Capacitance

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Raman Research Institute Digital Repository

Indian Academy of Sciences

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