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Quantum Transport in Mesoscopic SystemsComplexity and Statistical Fluctuations. A Maximum Entropy Viewpoint$

Pier A. Mello and Narendra Kumar

Print publication date: 2004

Print ISBN-13: 9780198525820

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198525820.001.0001

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(p.375) APPENDIX C THE CONDUCTANCE IN TERMS OF THE TRANSMISSION COEFFICIENT OF THE SAMPLE

(p.375) APPENDIX C THE CONDUCTANCE IN TERMS OF THE TRANSMISSION COEFFICIENT OF THE SAMPLE

Source:
Quantum Transport in Mesoscopic Systems
Publisher:
Oxford University Press

Figure C.1 shows the system studied in Chapter 4, consisting of a sample (a cavity, say) and the two expanding horns which, for convenience in the discussion, are assumed to have a constant cross-section (each supporting N′ open channels) within some interval far away from the sample, and to be scatterer-free within that interval. We wish to calculate the scattering matrix S (tot) for the full system, combining the scattering matrices S 1, S and S 2 for the left contact, the sample and the right contact, respectively. The sample is assumed to go into the expanding horns through very long, scatterer-free, quasi-one-dimensional, N-open-channel conductors, so that, according to the discussion presented in Section 3.1.7, we ignore closed channels and combine only the standard, open channel, S matrices. The amplitudes of the various incoming and outgoing waves in the regions assumed to be scatterer-free are indicated in Fig. C.1. Every amplitude represents a column vector associated with open channels only, whose number appears in parentheses next to that amplitude.

APPENDIX C THE CONDUCTANCE IN TERMS OF THE TRANSMISSION COEFFICIENT OF THE SAMPLE

Fig. C.1. The system studied in Chapter 4, consisting of a sample and two expanding horns, which, for convenience in the discussion, are assumed to have constant cross-section within some interval far away from the sample. Shown in the figure are the amplitudes of the incoming and outgoing waves that constitute the most general wave function in the regions assumed to be scatterer-free. Every amplitude represents a column vector, with a dimension given by the number of open channels indicated in parentheses.

(p.376) We follow the same ideas as in Sections 2.1.6 and 3.1.4. The notation is also followed closely, with a few exceptions. Here, for instance, S and S (tot) indicate the scattering matrices of the sample and of the total system, respectively. The scattering matrices S 1, S and S 2 associated with the left contact, the sample and the right contact, respectively, are defined to be

(C.1a)
[ b ( 1 ) ( N ) b ( 2 )′ ( N ) ] = S 1 [ a ( 1 ) ( N ) a ( 2 ) ( N ) ] ,
(C.1b)
[ a ( 2 ) ( N ) b ( 2 ) ( N ) ] = S [ b ( 2 )′ ( N ) b ( 2 )′ ( N ) ] ,
(C.1c)
[ a ( 2 )′ ( N ) b ( 3 ) ( N ) ] = S [ b ( 2 ) ( N ) b ( 3 ) ( N ) ] .
These matrices have the structure (the dimensionality being indicated in parentheses)
(C.2a)
S 1 ( ( N + N ) × ( N + N ) ) = [ r 1 t 1 t 1 r 1 ] ,
(C.2b)
S ( 2 N × 2 N ) = [ r t t r ] ,
(C.2c)
S 2 ( ( N + N ) × ( N + N ) ) = [ r 2 t 2 t 2 r 2 ] .
It is useful to keep the various S matrices in their general form, i.e., without imposing TRI. We combine eqns (C.1a) and (C.1c) as
(C.3)
[ b ( 1 ) ( N ) b ( 2 )′ ( N ) b ( 2 )′ ( N ) b ( 3 ) ( N ) ] = S 12 0 [ b ( 1 ) ( N ) b ( 2 ) ( N ) b ( 2 ) ( N ) b ( 3 ) ( N ) ] ,
where
(C.4)
S 12 0 = [ s 1 0 0 s 2 ] = [ r 1 t 1 t 1 r 1 0 0 r 2 t 2 t 2 r 2 ] .
We reorder the rows and columns of this last matrix and define
(C.5)
S 12 [ r 1 0 t 1 0 0 r 2 0 t 2 t 1 0 r 1 0 0 t 2 0 r 2 ] [ S 12 P P S 12 P Q S 12 Q P S 12 Q Q ] .

(p.377) This matrix has the property

(C.6)
[ b ( 1 ) ( N ) b ( 3 ) ( N ) b ( 2 )′ ( N ) b ( 2 )′ ( N ) ] = [ r 1 0 t 1 0 0 r 2 0 t 2 t 1 0 r 1 0 0 t 2 0 r 1 ] [ b ( 1 ) ( N ) b ( 3 ) ( N ) b ( 2 ) ( N ) b ( 2 ) ( N ) ] .
Just as in Sections 2.1.6 and 3.1.4, P projects unto the ‘external’ regions (far left and far right regions with constant cross-section in Fig. C.1) and Q unto the ‘internal’ regions (the scatterer-free, quasi-one-dimensional regions in Fig. C.1 that connect the sample to the horns). Thus S 12 P P is the submatrix that connects the external region to itself, S 12 P Q connects the internal region to the external region, S 12 Q P connects the external region to the internal region, and S 12 Q Q connects the internal region to itself. Equation (C.6) can be rewritten as
(C.7)
[ b P ( 2 N ) S 1 c Q ( 2 N ) ] = [ S 12 P P S 12 P Q S 12 Q P S 12 Q Q ] [ a P ( 2 N ) c Q ( 2 N ) ] .
We have used eqn (C.1b) that defines the matrix S, and we have defined
(C.8)
a p ( 2 N ) = [ a ( 1 ) ( N ) a ( 3 ) ( N ) ] , b p ( 2 N ) = [ a ( 1 ) ( N ) a ( 3 ) ( N ) ] , c Q ( 2 N ) = [ a ( 2 ) ( N ) a ( 2 ) ( N ) ] .
From eqn (C.7) we obtain the pair of coupled equations
(C.9a)
b p = S 12 P P a P + S 12 P Q c Q ,
(C.9b)
S 1 c Q = S 12 Q P a P + S 12 Q Q c Q
Eliminating c Q from this pair of equations, we obtain
(C.10)
b P = S 12 P P a p + S 12 P Q 1 S 1 S 12 Q Q S 12 Q P a P .
The resulting matrix S (tot) satisfies, by definition, the relation
(C.11)
b P = S ( tot ) a P ,
and is thus given by
(C.12)
S ( tot ) = S 12 P P + S 12 P Q 1 S 1 S 12 Q Q S 12 Q P .

(p.378) Now suppose that the sample goes into the horns following a very smooth pro-file, so that waves going from the sample into the horns are almost not reflected, i.e.,

(C.13)
r 1 = r 2 = 0.
The above ‘adiabaticity’ will hold if N′N. (On the other hand, under the same conditions, r 1 and r′ 2 are not required to vanish.) This implies that S 12 Q Q = 0 and hence
(C.14)
S ( tot ) = S 12 P P + S 12 P Q S S 12 Q P ,
or
(C.15)
[ r ( tot ) ( N × N ) t′ ( tot ) ( N′ × N ) t ( tot ) ( N × N ) r′ ( tot ) ( N × N ) ] = [ r 1 0 0 r 2 ] + [ t 1 0 0 t 2 ] [ r t t r ] [ t 1 0 0 t 2 ] = [ r 1 + t 1 r t 1 t 1 t t 2 t 2 t t 1 r 2 + t 2 r t 2 ] .
The trace needed in eqn (4.181) is thus
(C.16)
Tr [ t ( tot ) t ( tot ) ] = Tr [ t 2 t t 1 t 1 t t 2 ] = Tr [ ( t 2 t 2 ) t ( t 1 t 1 ) t ] .
Using the conditions (C.13), unitarity of S 1 and S 2 implies that
(C.17)
S 1 S 1 = [ r 1 t 1 t 1 0 ] [ r 1 t 1 t 1 0 ] = [ r 1 r 1 + t 1 t 1 r 1 t 1 t 1 r 1 t 1 t 1 ] = [ I ( N × N ) 0 ( N × N ) 0 ( N × N ) I ( N × N ) ]
and
(C.18)
S 2 S 2 = [ 0 t 2 t 2 r 2 ] [ 0 t 2 t 2 r 2 ] = [ t 2 t 2 t 2 r 2 r 2 t 2 t 2 t 2 + r 2 r 2 ] = [ I ( N × N ) 0 ( N × N ) 0 ( N × N ) I ( N × N ) ] .
We have thus found that
(C.19)
t 1 t 1 = t 2 t 2 = I ( N × N ) ,
so that eqn (C.16) gives
(C.20)
Tr ( t ( tot ) t ( tot ) ) = Tr [ t t ] ,
which is the relation needed to obtain eqn (4.182) in the text.