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The Physics of NanoelectronicsTransport and Fluctuation Phenomena at Low Temperatures$

Tero T. Heikkilä

Print publication date: 2013

Print ISBN-13: 9780199592449

Published to Oxford Scholarship Online: December 2013

DOI: 10.1093/acprof:oso/9780199592449.001.0001

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(p.249) D Derivation of the Boltzmann–Langevin noise formula

(p.249) D Derivation of the Boltzmann–Langevin noise formula

Source:
The Physics of Nanoelectronics
Publisher:
Oxford University Press

In this appendix, we derive eqn (6.39) for the noise by adding to the Boltzmann equation (2.6) a Langevin force ξ‎(t) which describes the stochasticity of scattering. In the stationary case, concentrating on low frequencies, we can neglect the time dependence of both the average and the fluctuating part of the distribution function. In this case eqn (2.6) should be replaced by1

v · f ( r , p ^ , E , t ) = I coll [ f ] + ξ ( r , p , t ) ,
(D.1)

where the Langevin forces ξ‎ are

ξ ( r , p , t ) = Ω d 3 p ( 2 π ) 3 [ δ J p , p δ J p , p ] .
(D.2)

Here Ω‎ is the volume of the system, which cancels out at the end of the calculation. This term comes directly from the fluctuating part of I coll[f].

Let us consider what happens in the diffusive limit, where f can be expanded in spherical harmonics in the p ^ -dependence; see eqn (2.14). Now the two included harmonics contain both the average and the fluctuating parts. Proceeding as in Sec. 2.4, we first note that the angular average of the Langevin source terms vanishes,

d p ^ ξ ( r , p ^ , t ) = 0.
(D.3)

This reflects the fact that the number of electrons is conserved also in the presence of fluctuations. Therefore, eqn (2.19) is unaltered by the Langevin term. However, we get an extra term for the p-wave part compared to eqn (2.20). In the static case this is

δ f = v τ f 0 + 3 τ p ^ ξ d p ^ .
(D.4)

Therefore, the full distribution function (average + fluctuations) satisfies an equation analogous to eqn (2.22),

D 2 f 0 = el · p ^ ξ d p ^ + I inel .
(D.5)

(p.250) Assuming that inelastic scattering is much weaker than elastic, we may ignore the collision integral for the fluctuating part, and get

D 2 δ f 0 = el · p ^ ξ d p ^ .
(D.6)

We can calculate the current density by integrating δ f over the energy as in eqns (2.8), (2.24), and using the fluctuating distribution function instead of the average one. We get for the fluctuating part (see eqn (D.4))

δ j = e N F v F d E δ ( δ f ) d p ^ p ^ 2 = e N F d E ( D δ f 0 σ δ μ / e + el d p ^ p ^ ξ δ j s )
(D.7)

Here σ‎ = e 2 N F D is the Drude conductivity. Thus, the total local current fluctuations consist of local potential fluctuations σδμ‎/e, and the fluctuations δ‎j s coming from elastic scattering.

In a quasi-one-dimensional geometry (where the distribution function f only changes in one direction, say, x), we can solve eqn (D.6) and get

δ f 0 = c 1 D x + el D 0 x d x d p ^ p ^ ξ + c 2 ,
(D.8)

where c 0 and c 0 are integration constants. Assuming that the sample is purely voltage-biased, the fluctuations of the distribution function may be set to vanish at the contacts, say at x = 0 and x = L. This implies c 2 = 0 and

c 1 = el L 0 L d x d p ^ p ^ ξ .
(D.9)

Comparing eqn (D.7) and (D.9), we find that the total current fluctuations are obtained as

δ j = e N F d E c 1 = 1 L 0 L δ j s d x .
(D.10)

In the following, we derive an equation for the correlator of δ‎j s and thereby for δ‎j.

Correlator of fluctuations

Now we should make an assumption about the correlator of the fluctuations. Typically one takes the currents J p , p as independent Poisson processes. This means that they are correlated only when the initial and final states and times are the same, and if they are evaluated at the same point. Moreover, in a Poisson process the second-order correlator (~ variance) is directly proportional to the average. Therefore, we may write

δ J p 1 , p 1 ( r 1 , t 1 ) δ J p 2 , p 2 ( r 2 , t 2 ) = ( 2 π ) 6 Ω δ ( p 1 p 2 ) δ ( p 1 p 2 ) δ ( r 1 r 2 ) δ ( t 1 t 2 ) J ¯ p , p ( r 1 , t 1 ) .
(D.11)

(p.251) Using this, we can write a relation for the correlator of the Langevin forces,

ξ ( r , p , t ) ξ ( r , p , t ) = δ ( r r ) δ ( t t ) Ω { δ ( p p ) d p [ J ¯ p , p + J ¯ p , p ] J ¯ p , p J ¯ p , p } ,
(D.12)

where in the latter term the position and time arguments have been omitted.

For elastic scattering it is enough to consider the case when the states lie on the same energy and only the angle p ^ changes in the scattering. We thus assume

J ¯ p , p = 1 N F δ ( E p E p ) J p ^ , p ^ ,
(D.13)

where the density of states N F at the Fermi level has been used for proper normalization. In this case we get

ξ ( r , p ^ , E , t ) ξ ( r , p ^ , E , t ) = 1 N F δ ( r r ) δ ( t t ) δ ( E E ) G ( p ^ , p ^ , r , E ) .
(D.14)

Using eqn (2.12), we can relate this back to the average distribution function by noting that for purely elastic scattering,

G ( p ^ , p ^ ) = d p [ δ ( p ^ p ^ ) δ ( p ^ p ^ ) ] [ W ( p ^ , p ^ ) f ¯ ( p ^ ) ( 1 f ¯ ( p ^ ) ) + W ( p ^ , p ^ ) f ¯ ( p ^ ) ( 1 f ¯ ( p ^ ) ) ] .
(D.15)

Here all functions are evaluated at the same position, the same time and with the same energy E.

In the diffusive limit we then get for the fluctuations of the local current density

δ j s ( r , t ) δ j s ( r , t ) = e 2 N F 2 el 2 d E d E d p ^ d p ^ p ^ p ^ ξ ( r , p ^ , E , t ) ξ ( r , p ^ , E , t ) = e 2 N F 2 el 2 δ ( r r ) δ ( t t ) d E d p ^ d p ^ p ^ p ^ G ( p ^ , p ^ , r , E ) .
(D.16)

Substituting the expansion f ( p ^ ) = f 0 + δ f · p ^ we can perform the integrals over p ^ and p ^ and get

δ j s ( r , t ) δ j s ( r , t ) = 2 σ δ ( r r ) δ ( t t ) Λ ( r )
(D.17)

with

Λ ( r ) = d E f ¯ 0 ( r , E ) [ 1 f ¯ 0 ( r , E ) ] ,
(D.18)

specified via the average s-wave distribution function f ¯ 0 .

(p.252) Fluctuations in a quasi-one-dimensional geometry

Combining the results (D.10), (D.17) and (D.18), we finally get for the full noise correlator

S ( t , t ) 2 δ I ( t ) δ I ( t ) = 2 d y δ j ( t , y ) d y δ j ( t , y ) = 2 L 2 0 L d x 0 L d x d y d y δ j s ( x , y , t ) δ j s ( x , y , t ) = 4 A σ L 2 0 L d x Λ ( x ) δ ( t t ) .
(D.19)

Here we denoted the transverse coordinates by y and the cross-section of the sample by A. We thus find white noise with the power spectral density

S ( ω ω * ) = 4 G N L 0 L d x d E f ¯ 0 ( E , x ) ( 1 f ¯ 0 ( E , x ) ) .
(D.20)

Here G N = Aσ‎/L is the conductance of the wire. The frequency scale ω‎* is related to the (inverse) time scales for particle diffusion through the sample or for charge relaxation, whichever is smaller. This analysis applies if the considered frequencies are much below these.

Notes:

(1) Here we neglect the fluctuation of the electric field, which would add a term on the left-hand side. This is valid if we can ignore the Coulomb-blockade -type effects, and concentrate on the limit of low frequencies.