Scattering theory and conductance fluctuations in mesoscopic systems

Physica A 167 (1990) 28-42 North-Holland

SCATTERING THEORY AND CONDUCTANCE FLUCTUATIONS IN MESOSCOPIC SYSTEMS Hans A. W E I D E N M O L L E R Max-Planck-Institut fiir Kernphysik, 6900 Heidelberg, Fed. Rep. Germany A random-matrix model developed originally in the framework of the statistical theory of nuclear reactions is (via the many-channel approximation of Landauer's formula) applied to universal conductance fluctuations (UCF) in a disordered sample connected to two or three leads. The leads are modelled as ideal conductors. For large sample lengths, the results agree with those obtained by other approaches. This shows that UCF can be modelled successfully in terms of random matrices. For sample lengths of several tens of elastic mean free paths, the conductance fluctuations are seen to depend sensitively on the coupling to the leads. This is due to the appearance of a new energy scale, the width F for emission of an electron from the disordered sample into the leads. This quantity F also influences the autocorrelation function of the conductance. Results are presented for the three canonical random-matrix ensembles: The Gaussian orthogonal, unitary, and symplectic ensembles.

1. Motivation

At sufficiently high excitation energies, the spectra of complex atoms, of molecules, and of atomic nuclei display fluctuation properties which are indicative of chaotic dynamics [1]. Cross sections for various reactions in which these systems are formed as intermediate compounds correspondingly display stochastic fluctuations. This fact has led to the development of a statistical scattering theory aimed at the calculation of averages, variances and other moments of fluctuating cross sections. The theory has been based on a random-matrix description of the fluctuation properties of the intermediate compounds [1]. In the present paper, I show that the ideas and techniques developed in the above-mentioned framework can also successfully be applied to the modelling of conductance properties of mesoscopic systems. Such an application is, in fact, suggested by two observations. (i) A formula originally due to Landauer expresses the conductance G of a mesoscopic system in terms of the elements of a scattering matrix. This establishes contact with scattering theory. (ii) The disorder in the mesoscopic system is usually simulated in terms of a random potential. This description is akin to the random-matrix modelling of intermediate compounds in molecular and nuclear physics. It shows that the 0378-4371/90/$03.50 © 1990- Elsevier Science Publishers B.V. (North-Holland)

H.A. Weidenmiiller / Conductance fluctuations in mesoscopic systems

29

scattering matrix appearing in Landauer's formula is also a stochastic or statistical matrix. In section 2, the random-matrix model is introduced for a two-lead geometry (the extenstion to more complex patterns being obvious). The techniques to calculate in this model the moments of the conductance are briefly mentioned in section 3. The results for the two- and the three-lead geometry are given in section 4. Some conclusions are drawn in section 5. The approach described in this paper adds two novel features to the theory [2, 3] of conductance fluctuations in mesoscopic systems: (i) it shows that a stochastic modelling of the Hamiltonian (rather than of the transfer matrix) leads to a satisfactory analytical understanding of these fluctuations; (ii) by accounting for the coupling to the leads in a way which is both realistic and simple, the present approach introduces into the theory a novel energy scale, the width F for emission of an electron from the disordered sample into the leads. Depending on the coupling to the leads, the influence of F on the fluctuation properties of the conductance extends up to length scales of the disordered sample which amount to at least several tens of mean free paths.

2. The model [4, 5] We consider a single electron in a disordered sample of length L x. The sample is connected to two ideal leads (see fig. 1). We disregard inelastic scattering, consider the leads as ideal conductors, and assume that in the sample the product of elastic mean free path l and Fermi wave number k F is much larger than one (metallic regime). The extension of our model to a three-lead and a four-lead geometry should be obvious. With c = L, R labeling the left and right lead, respectively, the transverse modes of the electron in the leads are labeled by a = 1,2, 3 . . . . ; the associated energies are e~. The longitudinal wave number k~ for an electron with energy E and in transverse mode a is given by

h2(k~) 2 E-

2m*

c

+ca

(2.1)

where m* is the effective mass. Each transverse mode defines a channel. With it

left ideal lead L

1

3

4

5

K-1 /J

II K

lead R

Fig. 1. The mode| for a disordered wire coupled to two ideal leads (see text).

30

H.A. WeidenmiiUer / Conductance fluctuations in mesoscopic systems

E = E v, the Fermi energy, a channel is open if k ] i n eq. (2.1) is real. Only open channels contribute to scattering processes; only open channels are therefore considered in all that follows. The number A of open channels is roughly given by LyLzkF/'rr. 2 The transverse dimensions Ly, L z of mesoscopic systems are typically of the order of l or larger. Therefore, A ~> 1: We deal with a many-channel problem. This fact permits us to use for the dimensionless conductance g = Gh/e 2 the many-channel approximation to Landauer's formula. It reads g = 2 ~'. vLR 2 a,b

~ab

(2.2)

Here, SaLR is an element of the scattering matrix connecting an incoming wave in channel a in the left lead with an outgoing wave in channel b in the right lead. The factor 2 is due to spin. It remains to define a stochastic Hamiltonian for the disordered sample, and to express SaLR in terms of this Hamiltonian. To this end, we consider a (fictitious) longitudinal division of the disordered sample into K = Lx/l equal slices of length I, where K is an integer (see fig. 1). We refer to each slice as a site , the sites are numbered from left to right with a running label j = 1, 2 . . . . , K. The number N of electronic states in each site is roughly given by A l k v >> A >> 1. The theory is ultimately worked out in terms of a two-fold asymptotic expansion: An expansion in inverse powers of N of which only the leading term is kept, and an expansion in inverse,powers of A. This procedure yields approximate analyt___ical expressions for the average conductance ~, for the variance v a r ( g ) = g 2 ~2, and for the autocorrelation function g(EF) g ( E F + AE). We are now in a position to simulate the disorder in the sample by introducing a random-matrix model. With j, l = 1 , . . . , K, labelling the sites (I trust that the running site label I will not be confused with the mean free path) and /z, v = 1 , . . . , N labeling the states in each of the sites, we consider the matrix elements H~J~) of the Hamiltonian in the disordered sample, taken between the bra (/xjl and the ket Ilv). We assume that H~J~) = 0 for IJ - II > 1 (thus allowing only for nearest-neighbour hopping of the electron from site to site). We assume that the elements H ~ ) on a fixed site are members of the Gaussian orthogonal ensemble ( G O E ) of dimension N. This means that ~ = H ~(J~), and that the independent matrix elements (i.e. the ones H~J~) H(JJ) for which/x t> v) are uncorrelated Gaussian distributed random variables with mean value zero and a second moment given by =

t~ 2

HOJ)H .,~, = -~ (6~.,,6~, + 6~,~,6~,,) .

(2.3)

H.A. Weidenmiiller / Conductance fluctuations in mesoscopic systems

31 (ll)

(The bar denotes the average over the ensemble.) Matrices H(~J~) and H~,,v, on different sites j ~ 1 are assumed to be uncorrelated. Nearest-neighbour hopping between sites is described by the N-dimensional real m a t r i c e s H(J/+1) with H(j - - t x u j+X) H,~,(J+I/) . Two choices are possible: (i) the =

elements H ~ / + 1) are uncorrelated random variables with a Gaussian probability distribution, vanishing mean value, and a second moment H ( j j + l ) H ( t l + l ) = V2

(2,4)

This corresponds to Wegner's "gauge-invariant disorder" model. (ii) The matrices H (j.~/+1) = vS,~ are multiples of the unit matrix in N dimensions. This correponds to the "site diagonal disorder" model. In the framework of the present approach, both models (i) and (ii) yield identical results. This supports our belief that our results do not depend on details of the random-matrix model u s e d for the disordered sample. It remains to define the dynamical coupling between the disordered sample and the leads. This is done in two steps. (i) We must impose boundary conditions on the surfaces separating the left lead and site 1 (and the right lead and site K, respectively). This defines a self-adjoint Hamiltonian on each of the leads and the associated continuum (or scattering) states. It turns out that these boundary conditions do not affect the value of g given by eq. (2.2) and are therefore irrelevant. (ii) We must define the (real) coupling matrix elements cj W ~ connecting channel a in lead c = L, R with the state/x = 1 . . . . . N on site c/ j = 1 , . . . , K. In keeping with our choice of --+,~/-/(Jt),we assume that W ~ =0 unless j = 1 ( j = K) for c = L (c RI respectively). We accordingly omit the c index j on the nonzero elements W ~ . Without loss of generality, we can also assume that = /x

(W..)

.

(2.5)

/.~

This is due to the form (2.2) of g. Our model is now completely defined. The associated scattering matrix can be calculated analytically and is given by SaLr = - 2 i ~ r ~ Wa~,tD L -1 )~,,Wb,. lr R

(2.6)

I,L u

The inverse propagator has the form D~,vit = ESjtS~ v _ H~(m + i_,~(JO~.~%~.

(2.7)

32

H . A . Weidenmiiller / Conductance fluctuations in mesoscopic systems

The matrix O describes the coupling to the leads and has the form W .,.W .~ + '~/K~. Wb,~ W ~ a

•

(2.8)

b

It is useful to exhibit the parameters contained in the model. Aside from the length L x and the associated integer K, they are: (i) The strength parameter A of the GOE on each site, cf. eq. (2.3). This parameter ultimately relates to the mean level spacing d in the disordered sample. (ii) The strength parameter v of the site-site hopping matrix elements, cf. eq. (2.4). This parameter ultimately relates to the diffusion constant D in the disordered sample. c 2 (iii) The strength of the couplings E~, (Wa~,) between leads and disordered sample. Orthogonal invariance of the Hamiltonian for the disordered sample implies that the moments of g depend only on the orthogonal invariants E~ (Wa~,)c2; physical reason suggests that all couplings E~, (Wa~,) c 2 are equal; the remaining parameters are then A, the number of channels, and the strength a (with 0~< a ~< 1) of the channel-site coupling. Here, a is a simple rational function of E~, (Wa~,) 2. We observe that our model contains only a small number of parameters: Lx, A, d are essentially given by the geometry, and ot and D define the transport properties. Our model has several limitations which can be (but have not yet been) removed: It does not allow for ballistic electrons and therefore is only applicable for K ~>3 or 5; it allows only nearest-site coupling. Moreover, the definitions given above apply to time-reversal invariant systems without spinorbit coupling. The extension to time-reversal symmetry breaking systems described by the Gaussian unitary ensemble, and that to systems with spinorbit interaction described by the Gaussian symplectic ensemble, has been worked out, however [6].

3. The method of calculating averages

We employ the method developed by Wegner [7] in the form proposed by Efetov [8] to calculate moments of g in terms of a generating function. The method has been applied to statistical scattering problems in refs. [9, 10]. The special case of multistep-compound reactions considered in ref. [11] is most akin to the present problem [4, 5]. Lack of space limits the presentation to a bare sketch of the procedure.

H.A. Weidenmfiller I Conductance fluctuations in mesoscopic systems

33

Products of S-matrix elements are written as multiple derivatives with respect to suitable auxiliary variables of a generating function Z which is defined as an integral over 8 N K integration variables, half of them anticommuting. The average and higher moments of g are obtained by differentiating the average of the generating function. The average Z can easily be worked out and subsequently be simplified via the introduction to composite variables and the Hubbard-Stratonovitch transformation. This leaves a total of 32K integration variables. For N >> 1, the integral can be evaluated with the saddle point approximation. Because of the site-site and the site-lead couplings, and since A >> 1, the saddle point is unique (absence of Goldstone modes). Expanding the remaining integrals in the vicinity of the saddle point, one generates the asymptotic expansion in inverse powers of A. According to eq. (2.2), g is given in terms of a double summation over channels, and g2 analogously contains a four-fold summation. Since each sum essentially counts like a power of A, the asymptotic series must be carried to the point that the terms omitted in g (or g2, as the case may be) are of order A -1.

4. Results 4.1. Qualitative c o n s i d e r a t i o n s

The results of the calculation depend on the parameters of the model summarized at the end of section 2 mainly via three combinations, all of them having the dimension of energy. They are: (i) The mean level spacing d of the disordered sample. It is related to A, and is inversely proportional to L~, d ~ (Zx) -1 .

(4.1)

(ii) The Thouless energy Ec, related to the diffusion constant D (which in turn relates to v) by ,.tr2D h

'

Eo=

: '~

!

'

(4.2)

Note that E c ~ (Lx) -2. (iii) The decay width F for emission of an electron from the disordered sample into the leads, d F = ~ 2A" oe

(0~< a ~< 1).

(4.3)

34

H.A. Weidenmiiller / Conductance fluctuations in mesoscopic systems

As mentioned above, the parameter a is related to Z~, (Wa~,) c 2 (which is assumed to be the same for all a, c). While E c and d appear in all previous treatments of the problem, F is a novel energy scale, and is therefore briefly discussed. If the disordered sample were C isolated (all Wa~, = 0 and therefore also a = 0 and F = 0), the levels in the sample would be the stationary eigenstates of the Hamiltonian H~J~) with normalizable eigenfunctions. As we allow W~, to differ from zero (a prerequisite for the existence of a nonzero conductance!), the levels in the disordered sample acquire a finite width and become resonances in a scattering problem. This is evident from eqs. (2.6)-(2.8), which are the many-level generalization of the single-level Breit-Wigner formula. For sufficiently weak coupling between leads and sites (so that A a <~ 1), we have F < d: the resonances are isolated a n d / o r overlap only weakly. In this case, ~ is of order unity: We deal with a poor conductor. More interesting is the case where A s >> 1 (although a may still differ from unity). Then ff-> 1 (unless L x becomes large and of the order of the localization length), and we deal with a good conductor. In this case, F >> d: The resonances overlap very strongly. This is the case of physical interest for mesoscopic systems. We note that A and o~ are independent of L x, so that F - L ~ -1. The three energy scales introduced above can also be written as time scales, with r d a f = h / E c the diffusion time through the sample, rde¢ = A / F the decay time for electron emission, and Tp = h / d the Poincar6 recurrence time. This last entity yields a simple and intuitive interpretation of eq. (4.3): d/2~rh is the frequency with which a time-dependent wave packet in the disordered sample returns to its original position, a is the probability with which it escapes from such a position into one of the channels, and F is therefore the total decay width. Eq. (4.3) suggests that the parameters a and A always occur in the combination a A . This, however, is not the case. This is why they were listed separately. Fig. 2 shows a schematic plot 0f d . Lx, F . Lx and E¢ • L x versus the length Lx of the sample. The points of intersection of E¢- L, with d L , and F L x define, respectively, the length scales L 0 and L,o¢, and thereby three domains in which ~, var(g) and the autocorrelation function of g behave in a qualitatively different fashion. For L, > L,oc, we are in the localized regime where ~ is expected to decrease exponentially with increasing L x. For L 0 ~< L, ~< L,o¢, we have rde¢
H.A. Weidenmiiller

Conductance fluctuations in mesoscopic systems

E" Lx" -~E

35

l-'Lx

c'Lx

d'Lx I

I

Lo

Lloc

~ Lx

M--Ohmic regime--~14-1ocalized regime

D

non-Ohmic regime

Fig. 2. Schematic plot of d" L x, of F . Lx, and of Ec" L, versus L x (see text).

(Ec/F)Lx(4/~r2), can be obtained from simple estimates of level densities and from D = LVF appropriate for quasi one-dimensional models. It is given by L o = 161/a. While the factor 16 may be a result of the simplicity of the estimate and, hence, not reliable, the factor a - ~shows that for weak site-lead coupling, the influence of this coupling extends deeply into the disordered region. Therefore, the effects discussed below do possess physical significance, even though the factor 16 may not be reliable. 4.2. The average conductance ~, We introduce the dimensionless parameter 3' = ½LxlF/hD, note that 3' is independent of the length L x of the sample, and have [4, 5], for a = 1,

=2+(K-1)3"

2 1+

= [2+(/~'- 1)7]"

+~7(A-1)'

(4.4)

where a, are numerical coefficients independent of K. The aysmptotic expansion is expected to be reliable as long as the second term on the r.h.s, of eq. (4.4) is smaller than the first. Equating both terms we find approximately K ~ A, suggesting that the asymptotic expansion for ~ is good up to the localization length Lloc~Al. We hope (but have not checked) that the asymptotic expansions for var(g) and for the autocorrelation function have the same domain of validity. We observe that the factor - { multiplying the square bracket is the well-known weak-localization correction to ~. We turn to the first term and note that f o r / ( 7 >>1, this term attains the well-known Ohmic form 8,rrhD/L~d = (8/'rr)(Ec/d) which is independent of F, and thus of the coupling

36

H.A. Weidenmiiller / Conductance fluctuations in mesoscopic systems

to the channels. (This last statement holds also for a < 1.) As we expect in view of the remarks made in section 4.1, this form is determined entirely by the diffusion time %if. For Ky ~ 1, on the other hand, this term is nearly independent of K. This "non-Ohmic" behaviour is understood easily: For Zdie < ~'d~, the probability density for the electron is spread uniformly over the sample before electron emission into the leads occurs. Therefore, ~ attains half of its maximum value 2A because emission into both leads is equally probable. The value 2A, equal to the quantized contact resistance, is not attained even for K = 1 because our model in its present form does not allow for ballistic electrons. We have also derived [5] the form of ff for a < 1 but do not discuss it here. 4.3. The variance o f g

Fig. 3 shows var(g) (full line) versus K for a = 1. We observe that var(g) is nearly constant, equalling 0.5 for K --- 1 and ~ for L x ---> Lto c. The latter value is the canonical value [2] for quasi one-dimensional systems. Actually, var(g) consists of two contributions, a "bulk" contribution which is independent of the coupling to the leads, and a "surface contribution" which enbodies this coupling. The two have very different K-dependence and fortuitously add to a nearly constant function. Fig. 4 shows var(g) versus K for three values of a. We recall that L o ~ a - 1 this explains the different rates at which the three curves approach the asymptotic value 8 . We also note that reducing a increases the surface contribution to var(g).

0°6

f

0°4 '

//

13)

/

/

I

bulk

;:> 0.2

surface

~ 0°0

'

0

20

40

60

80

100

I<-1 Fig. 3. The variance of the conductance versus length K = Lx/1 for the model of fig. 1 (see text). (Taken from ref. [5].)

H.A. WeidenmiiUer / Conductance fluctuations in mesoscopic systems

37

1.0

..... ____

"',

.

a=0,1 a=0.3 a--1.O

0.8

> 0.6 f

0.4" o

I

I

I

20

40

60

80

100

K-1 Fig. 4. The variance of the conductance versus length K --- Lx/l for three coupling strengths to the leads (see text). (Taken from ref. [5].)

All results given so far apply for the G O E . The fluctuations are reduced by a factor 2 (4) if the G U E (the GSE, respectively) are considered instead [6]. 4.4. The autocorrelation function C(AE) = [g(EF) g(E F + AE) - ~2]

This function is for a = 1 shown [12] in fig. 5 in a three-dimensional plot, the x-axis giving the length scale and the y-axis the scale for the energy shift AE.

z 0,53

0I

MHIIfJIHf llllllllllllllVo.ooo IIIIIllIp, x

13

~ 4

0.050

Fig. 5. The autocorrelation function C(E) versus length Lx and energy shift AE (see text). (Taken from ref. [12].)

38

H.A. Weidenmiiller / Conductance fluctuations in mesoscopic systems

We observe that the fall-off with increasing AE becomes more rapid as L x increases. This is qualitatively what we expect: The fwhm of the autocorrelation function should be given by F or Ec, whichever is smaller, and should therefore decrease first (Lx ~< L0) as L~-~ and, for large L~, as L x 2. To check this expectation, we have fitted a Lorentzian to the AE-dependence of C(AE) at various values of L~. Two such fits are shown [12] in fig. 6. The first (K = 10) is excellent, the second (K = 40) fails for large AE. The reason is known [2]: For L x >~ Lo, the fall-off of C(AE) is asymptotically (large AE) given by (AE) -3/2, and not by (AE) -2 as the Lorentzian would predict. Here, we are concerned with the fwhm as determined by the Lorentzian fit. In fig. 7, we .6 Smooth

Lines:

Lorentzion

Fit

._o 2.4

-

la_

-o

~.~ .0

t

.000

.005

o

~

°

o

o

o

o

o

o

o

o

o

i* ° o

.010 .015 .020 .025 Energy Shift (orb. units)

.030

Fig. 6. Lorentzian fits to C(E) for two different lengths (see text). (Taken from ref. [12].) 100

I

I

c

.d 10-~

~

10 -2

g

Correlation

Width

Width

'6 10 -~ 0 (D

10 -4 10 0

I

I

i0 ~

102

103

(5ornple Length)/~ Fig. 7. T h e fwhm of C(E) versus the length of the sample, together with F and Ec. (Taken from ref. [121. )

H.A. Weidenmiiller / Conductance fluctuations in mesoscopic systems

39

present the results of the fit for the fwhm versus L x and compare them with the curves for F and for E~. We note that asymptotically (L~---> 0 and Lx---> L~oc), the fitted values approach F and Ec, respectively, as we expect. We also note, however, that the cross-over point between the two aysmptotes ( F and E~) does not occur at L 0 = 161, but rather at L 0 ~ 80l. This shows that the influence of the coupling to the leads on the fluctuations penetrates more deeply into the disordered region than that on ~. The reason is that the cross-over points are determined by different conditions, i.e. L = L 0 for ~ and F = E c for the autocorrelation function. 4.5. The three-lead p r o b l e m

This problem has also been fully worked out [13]; I confine myself here to the presentation of some of the results. The three leads are attached to a disordered wire each, the other ends of the three disordered wires are joined together. As in fig. 1, we divide the three disordered wires labelled x, y, z respectively into slices, the numbers of slices are Kx, Ky, K z, respectively. A central slice connects the three disordered wires. We take the numbers of channels in the three leads to be the same and denote the strengths of the three couplings by ax, ay, a~, respectively. Fig. 8 shows the variance of the conductance through the wires labelled x and y versus the length K = K~ = Ky = K z for various strengths a = a x = ay and a z of the couplings. For large K, the three curves approach the bulk value known from previous work [14]. The curve for a = a z = 1 shows that a third lead reduces the fluctuations as compared to the two-lead geometry. Reducing the coupling a z in the third wire

var(g~) 0,5

^\

0.4" 0,3" ......--

0.2" ,

0,1" 0.0 0

--/'~

ct= 1 00tz---0.1 ct=otz= 1.0

/

.

.

.

a=0.1 at= 1.0

.

i

I

I

I

2O

40

60

80

100

K-1 Fig. 8. The variance of the conductance for a three-lead geometry versus length for various coupling strengths (see text). (Taken from ref. [13].)

40

H.A, Weidenmiiller / Conductance fluctuations in mesoscopic systems

var(gxv)

13-1.0

11,4

13-0.5

0.2

"~, 13-0.1

0.0 0.0

i

i

0.1

0.2

0.3

O(Z Fig. 9. The variance of the conductance for a three-lead geometry versus the coupling strength in the third lead, for various lengths of the third lead (see text). (Taken from ref. [13],)

enhances the fluctuations towards the two-lead,value. Reducing the coupling a in the two wires x, y reduces the fluctuations. We also note that decreasing any one of the ai's from unity increases the length over which the influence of the surface term is important. In fig. 9, the variance of the conductance through wires x, y is plotted versus a s for a = 1, Kx = Ky = 50, and for three values of B = K z / 2 K . We note that for az---> 0, all curves approach the two-lead value, while different values are attained for az---> 1: The shorter the third lead, the more the fluctuations are reduced.

5. Conclusions In all cases and for all variances considered so far, the bulk limit of our random-matrix modelling of conductance fluctuations agrees with results obtained previously in other ways [2, 3]. We conclude therefore that fluctuation properties of mesoscopic systems can be modelled successfully in terms of random matrices. The use of the many-channel approximation to Landauer's formula and the connection thereby established between the conductance of disordered systems and statistical scattering theory, has enabled us to investigate in some detail the influence of the coupling to ideal leads on the conductance. This coupling defines a novel e n e r g y scale, the decay width F of the disordered sample, and an associated length scale L 0, approximately given by the condition that F = E c, for L x = L 0. At first sight one may question whether L 0 (which for a = 1 has the value 16l and is only numerically, but not

H.A. Weidenmiiller / Conductance fluctuations in mesoscopic systems

41

parametrically larger than the mean free path l itself) is sufficiently different from l to have independent physical significance. The results displayed in figs. 4, 7, 8 and 9 do show, however, that this is the case: Even for a = 1, the autocorrelation function for the two-lead geometry is affected by F up to length scales of 80/ or so. And the influence of the surface coupling affects even bigger lengths of the sample as a, or, in the three-lead case, one of the a's, is reduced. I wish to stress that the method leads to analytical expressions for the various observables of interest. Technical tools have been developed that greatly simplify the evaluation of these expressions. For this reason, I am confident that the approach outlined above can be applied successfully to a number of further problems. I have in mind the four-lead problem (relevant for a comparison with data), the inclusion of magnetic fields and the treatment of rings (relevant for the Aharanov-Bohm oscillations), the treatment of the electron-phonon interaction (relevant for the understanding of how coherence is lost in inelastic processes), and the inclusion of ballistic electron transport (necessary for the understanding of the transition from ballistic to diffusive transport).

Acknowledgements I am grateful to my collaborators S. Iida, J.A. Zuk, and A. Altland for many insights, and for permission to use the results of refs. [6, 13] prior to publication. I have benefitted from discussions with J. Imry, P.A. Lee and M.R. Zirnbauer.

References [1] For a review and further references, see: O. Bohigas and H.A. Weidenmiiller, Ann. Rev. Nucl. Part. Sci. 38 (1988) 421. [2] P.A. Lee, A.D. Stone and H. Fukuyama, Phys. Rev. B 35 (1987) 1039. [3] B.L. Altshuler and B.I. Shklovski, Soy. Phys. JETP 64 (1986) 127; J.-L. Pichard and M. Sanquer, Physiea A 167 (1990) 66, these Proceedings (the latter reviews the transfer matrix approach). [4] S. Iida, H.A. Weidenmiiller and J.A. Zuk, Phys. Rev. Lett. 64 (1990) 583. [5] S. Iida, H.A. Weidenmiiller and J.A. Zuk, Ann. Phys. (NY), in press. [6] A. Altland, Diploma Thesis (1989); Z. Phys. B, submitted. [7] F.J. Wegner, Phys. Rev. B 19 (1979) 783. [8] K.B. Efetov, Adv. Phys. 32 (1983) 53. [9] H.A. Weidenmiiller, Ann. Phys. (NY) 158 (1984) 120.

42

H.A. Weidenmiiller / Conductance fluctuations in mesoscopic systems

[10] J.J.M. Verbaarschot, H.A. Weidenm~ller and M.R. Zirnbauer, Phys. Rep. 129 (1985) 367. [11] H. Nishioka, J.J.M. Verbaarschot, H.A. Weidenmfiller and S. Yoshida, Ann. Phys. (NY) 172 (1986) 67. [12] A. Altland, private communication; to be published (1989). [13] S. Iida, preprint MPI Heidelberg (1989). [14] C.L. Kane, P.A. Lee and D.P. DiVincenzo, Phys. Rev. B 38 (1988) 2995.