Quantum transport and chaos in semiconductor microstructures

PHYSICA® I~[~qFNIFR

Physlca D 83 (1995) 3(1-45

Quantum transport and chaos in semiconductor microstructures Harold U. Baranger AT&T Bell Laboratorte~, 600 Mountam Ave ID-230, Murray Htll, NJ 07974-0636, USA

Abstract It is shown that classical chaotic scattenng has experimentally measurable consequences for the quantum conductance of semiconductor mlcrostructures These include the existence of conductance fluctuations -a sensitivity of the conductance to either Fermi energy or magnetic field- and weak-locahzatlon -a change m the average conductance upon applying a magnetic field We use semlclassacal theory, random S-matrix theory, and numerical results to descnbe these interference effects for mlcrostructures modeled by bilhards attached to leads The semlclass~cal theory predicts that the difference between chaotic and regular classical scattering produces a quahtatlve d~fference m the fluctuation spectrum and weak-locahzat~on hneshape of chaotic versus non-chaotic structures The random S-matnx theory yields results for the magmtude of these interference effects The conductance fluctuation and weak-locahzatlon magnitudes are umversal ff the number of incoming modes, N, is large they are independent of the size and shape of the cavity Of more relevance experimentally, m the hm~t of small N the full d~stnbut~on of the conductance shows a striking dependence on N and magnetic field

1. Introduction

which impurity scattering can be neglected so that only scattering from the boundaries of the conduct-

The effect of quantum interference on the properties of microstructures has been intensively investigated during the last decade and is one of the main subjects of mesoscopic physics [ 1 ] Most of this work has been on disordered systems in which elastic scattering produces diffusive transport In this regime, both microscopic perturbatwe and macroscopic random matrix theories give a good account of the phenomena In the latter case [2], one assumes, for instance, that the total transfer matrix can be built up multiplicatively using transfer matrices chosen from a simple statistical ensemble with only symmetry constraints applied The success of random matrix theory is perhaps the best theoretical demonstration of the ubiquity of mesoscoplc interference effects More recently, interest has focused on quantum properties of ballistic microstructures -structures in 1995 Elsewer Science B V SSDI 0167-2789(94)00248-7

ing region is important [ ! ]

Such ballistic transport

is possible, for instance, m high-mobility semiconductor heterostructures made with the GaAs/A1GaAs materials If one m addition studies these structures at low temperature so that the phase-breaking rate is small, quantum interference effects occur which can only depend on the shape of the boundaries It is natural, then, to ask whether these effects depend on the shape of the boundary and in what way they differ from those in diffusive systems The study of the quantum properties of simple dynamical systems has itself received a great deal of attention m the last decade [3] Of particular interest has been the connection between chaotic or non-chaotic classical dynamics and various quantum properties such as the distribution of energy levels This topic is what is usually meant by the term "quan-

H U Baranger / Ph}*l~a D 83 (1995) 30-4~ turn chaos" This suggests, then, a close connection between certain aspects of "mesoscoplc physics" and "quantum chaos", in particular, it was suggested [4] that low-temperature transport measurements In mlcrostructures could provide a new system for the comparison of theory and experiment in the area of quantum-chaotic scattering In fact, it was shown theoretically that the transport properties of such structures [ 4 - 1 5 ] depend on whether the classical dynamics IS regular or chaotic Recent experiments probing quantum transport in balhstlc mlcrostructures [16-25] have indeed observed [16,211 a difference between the transport properties of nominally regular and chaotic structures Interference effects should be evident in a variety of properties of balhstlc mlcrostructures First, perhaps the most natural property from the dynamical systems point of view is the energy levels of an tsolated mesoscoplc system [ 3 ] This has received a good deal of theoretical attention, especially as regards magnetic field dependence [ 2 6 - 2 8 ] , and some experimental attention [29,30] Mesoscoplc magnetization is a closely related property [31-35] Second, transport through almost closed systems can be considered the tunnehng regime Theoretical work has addressed the distribution of the heights and widths of the conductance peaks [28,36,37] While many experiments on almost closed quantum dots have been performed [38,39 ], the connection with geometrical interference effects, and possibly issues of "quantum chaos", has not yet been made Finally, one can consider larger contacts to the mesoscoplc structure so that many of the individual resonances of the structure overlap This is the regime of transport through an open system and is closely related to the well-known Ericson fluctuation regime [40,41] of nuclear physics The two main effects present are ( 1 ) a decrease in the average conductance due to constructive interference of timereversed traJectories and (2) conductance fluctuations -reproducible fluctuations in the conductance versus magnetic field or Fermi energy Experiments are easier in this regime and have yielded strong evidence for a connection between the classical dynamics and quantum transport [ 16-23] Because of the greater likelihood of experiments, and because of greater fa-

31

mlllarlty on the part of the author, the properties of transport through open systems will be the topic of the rest of this paper In order to treat transport through phase-coherent mlcrostructures, we take the point of view, developed in the work of Landauer and Buttlker [ 42,43], that resistance measurements on such systems are related to simple quantum scattering problems If one measures the conductance o f a mlcrostructure for which the electron transit time is short compared to the inelastic scattering time, the conduction process involves simply the coherent quantum-mechanical transmission of Independent particles The conductance associated with this transmission process can be evaluated by a countlng argument [42,43] or from linear response theory The basic physical idea of this approach is to consider the mlcrostructure as a single phase-coherent unit attached to ideal reservoirs which represent the much larger electrical contacts whose resistance is neghglble In an Ideal two-probe measurement the sample ts attached between two perfect reservoirs with electrochemical potentials/xj and/x2 =/xl - eV (where V is the apphed voltage), and these reservoirs serve as both current sources and voltage terminals In the energy interval between/z~ and /xl electrons are injected into right-going states emerging from reservoir 1, but none are injected into left-going states emerging from reservoir 2 Thus there is a net right-going current given by N

N

rtl=l

tt=l

dnm l=e~_.~Vm~-eeVZTnm,

(1)

where N is the number of propagating channels In the sample, Vm 1s the longitudinal velocity for the ruth channel at the Fermi surface, 7".., is the transmission probability from m to n, and the quasi-onedimensional density of states for non-interacting particles is dnm/de = 1/hvm Eq ( l ) shows that the two-probe conductance is just proportional to the total transmission coefficient of the mlcrostructure. e2 ~

G=~

e2

T..,=_--fiT n ttl= l

(2)

H U Barangel

32

/

Phvst~a D b¢3 (1995) 30-45

It is often convenient to use reflection coefficients, de3

fined analogous to the transmission coefficients N o t e

cl

that the normallzatton condition on these transmission and reflection coefficients is T + R = N (unitarlty o f the S - m a t r i x )

H e n c e conductance measurements

directly p r o b e the q u a n t u m S-matrix describing the multtple-scatterlng f r o m the entire sample treated as a

0

0

single scatterer " c o n d u c t a n c e is transmission" B e f o r e p r o c e e d i n g to a discussion o f the theory, we illustrate m Fig

1

25

~

2 kW/~z kc

1 the main effects that will be

treated numerically c o m p u t e d T ( k ) and T ( B ) curves

3

4

5

b Var(T)

g# 2 0 -

are shown for an a s y m m e t r l z e d stadium in both the classical and q u a n t u m limits The transmitted flux increases hnearly as a function o f k as the Incident flux

( ~ kW/~r) Increases (panel ( a ) ) The average quan-

lO

tum result lies b e l o w the classical curve, largely be-

30

30

32 ~

kW/~ ,

34

6

,

,

cause o f m o d e effects f r o m confinement in the lead However, part o f this offset is sensitive to a weak m a g n e t i c field (panel ( b ) ) , this is the effect o f quantum interference on the average conductance, known as w e a k - l o c a l i z a t i o n in disordered systems [44] Perhaps the m o s t striking feature o f Fig

3 25 E ~20 v--

1 is the fine

1

structure in the q u a n t u m curve These are the conductance fluctuations, characterized m terms o f both their magnitude, v a r ( T ) , and their correlation scales k~ for

T ( k ) and Bc for T ( B ) In the rest o f the paper, the w o r k in which the author has participated c o n c e r n i n g these effects will be s u m m a r t z e d [4,9,10,12,45]

Section 2 deals with the

shape o f m a g n e t o t r a n s p o r t effects how the average or un-averaged c o n d u c t a n c e v a n e s as a function o f magnetic field or Fermi energy The main theoretical tool here ts semlclasslcal theory, and this material is drawn f r o m Refs [4,9,10,45]

In Section 3 the m a g n i t u d e o f

Var(T) Bc< 50

>

(c)

I

I

I

L

2

4 6 BA/0 o

o

10

Fig 1 Transmission coefficient as a function of wavevector (a),(b) or magnetic field (c) through the cavity shown m panel (a) In all the panels, the straight sohd hne is the classical transmission T,t, the fluctuating sohd hne ~s the full quantum transmission Tqm, and the dashed (dotted) hne is the smoothed Tqm at B = 0 (BA/qbo = 0 25) Note the following four main effects (1) The smoothed Tqm i n c r e a s e s with the same slope as T,I (a), (2) The smoothed Tqm is smaller than Tct by a substantial amount, (~T) (a), wMch is sensmve to B (b) and therefore consututes an average magneto-conductance, (3) Tqm fluctuates by an amount of order unity as a function of both k and B var(T) ~ 1 (panels (a)-(c)), (4) The spectrum of Tqm has a characteristic scale, k, or B~, which can be related to classical quannties ( W ~s the width of the leads, and A ~s the area of the cavity ) (From Ref [ 10} )

magnetotransport effects is discussed how big is the correction to the average c o n d u c t a n c e at zero field and h o w large are the c o n d u c t a n c e fluctuations Section 4

2. Line shape of magnetotransport effects: semiclassical theory

presents results on the full distribution o f the conductance The main theoretical tool in these two sections

2 1 Serniclasstcal transmission coefficients

is r a n d o m S-matrix theory, and they f o l l o w closely R e f [ 12] Finally, Section 5 contains the conclusions

Ballistic mlcrostructures are a c o n d e n s e d matter system where one may hope to describe q u a n t u m properties in terms o f properties o f the classical phase space

The semlclasstcal Green functions are the

natural link between q u a n t u m and classical physics,

33

H U Baranger / Phvstca D 83 (1995) 3 ( I - 4 5

these Green functions are expressed as sums over classical trajectories whose actions and stabilities determine the associated phases and weights The Green function in which we are interested is that related to the transmission and so connects one opening in the structure with another The transmission (reflection) amplitude from a mode m on the left to a mode n on the right (left) is given by the projection of the Green function onto the transverse wavefunctlons ~bm and ~b. of the modes [46]

t,,,,,

=-,h(,,nv.,)'/2f dyr f

dy qS,*7(y')qSm(y)

x G ( L , f ,O,y,E),

r.m=8,,,,,-th(,',,C'm)l/"-f dyrf dydp~(y') &re(Y) ×G(O, yr,O, y,E)

(3)

The intuitive interpretation o f the above equations as arriving at the cavity in mode m, propagating inside the cavity (through the Green function), and exltmg in mode n is quite straightforward These equations constitute an exact starting point which is also the basis of our numerical calculations As described in Ref [45], the Green function of a discretized Schrodinger equation is obtained by a recursive algorithm and then a projection onto the transverse wavefunctlons in the leads is performed This yields the exact transmission amphtude of the dlscretlzed problem, and the conductance follows from Eq ( 2 ) The semiclassical approximation proceeds by replacing the Green function by its semiclasslcal pathintegral expression [3] Using this expression in Eqs ( 3 ) and performing the integrals by stationary phase, one arrives at a semlclasslcal expression for the transmission amplitudes [47,4,10] The transmission coefficients are the square o f the transmission amplitudes, Z,,m = It.,,,I 2, and so will be expressed as sums over parrs o f trajectories For billiards it is convenient to scale out the energy dependence and use the notation

[9] N

N

n=l t/l=l

77"

tl m

~

II

(4)

F~.m(k) =

exp

[ l k ( L , - L.) + *~d',. ] (5)

The paths in the sum, labeled s and u, are those which enter at (x, y ) with fixed angle sin 0 = +mTr/kW and exit at ( x ' . f ) with angle sin 0' = ±nTr/kW In terms of the action Ss, the phase factor is kL, = S,/h+ ky sin 0 - k f s i n 0 r plus an additional phase. ~b~.., associated with singular points in the classical dynamics [45 ] The prefactor is fi.s = I ( 0 y / 0 0 ' ) o ) ]/( W cos Or) Some special care must be taken for structures in which there are direct trajectories between the incoming and outgoing leads [ 10] In this case the traJectories contributing to the stationary phase evaluation o f the integrals in Eqs (3) are not isolated The contrlbutlon of the family of direct trajectories has a different dependence on h and in fact dominates in the llmlt k --~ ~ , comphcatlng the comparison between experiment or calculation and the semiclassical theory Thus in our simulations we have Introduced "stoppers" in the bilhards which ehmlnate this effect, the experimentalists have also tried to avoid this problem by displacing the leads [ 17] or having an angle smaller than ~- between them [ 16] Even If direct trajectories are avoided, the special character of the semIclasslcal contribution o f a family of trajectories will remain for lntegrable structures like rectangles The contribution o f families o f trajectories in cavities is more complicated to evaluate, but nevertheless one expects the same dependence on h and so expects integrable scattering systems to approach the classical limit differently from chaotic scattering systems The issue of convergence of semlclasslcal expressions is. of course, complicated [3] At the semiclasslcal level, the main difference between the scattering problem and the more famlhar problem of energylevels of an isolated system is that the trace formula for the density of states involves periodic orbits while the transmission amplitude is given by open trajectories that traverse the scattering region The fact that the open trajectories escape from the scattering region implies that the convergence of the transmission sum will be less problematic than that o f the trace formula [3.48] Convergence makes possible the numerical

H U Baranger / Ph~stca D 83 (1995) 3 0 - 4 5

34

evaluation of the semiclassical transmission amphtude by dxrect summation While d~rect summation of semiclassical expressions has not been pursued, note that it has proved valuable in the context of closed btlhards [49] and could be used to clarify some of the issues discussed below

2 2 Average magnetoconductance Although the most dramatic effect of quantum coherence visible m the data of Ftg 1 is the fluctuattons of the conductance, we start our discussion of interference effects in ballistic bllhards by considering the average conductance, G(B), since it is a stmpler quannty to analyze theoretically For a gwen balhstlc mtcrostructure the average must be defined by summlng over some additional parameter (such as energy or magnetic field), roughly such an average wdl ehmmate the aperiodic fine structure seen m Fig 1 leaving any additional smooth dependences on parameters of interest Following closely the discussion in Refs [9,10,45 ], we first show that the leading contribution to the average conductance is classical and then show that there ts a quantum correction sensitive to the magnetic field which is analogous to weak-locahzatlon [44] in disordered conductors Because the classical transmlssmn coefficient ts proportional to kW/qr ( F i g l a ) , one expects a hnear contribution to the average quantum transmission and call the slope 7- By averagmg T(k)/(kW/Tr) over all k, q, +q g ,

(A)

hm (l/q)

q---* :xz

[ dkA(k),

q~W/~>> 1,

(6)

d

q,

one can show [45] that only terms with paired paths, s = u, contribute to 7- The result of averaging ts 1

1

7-= ~ -1

-1

s(O 0')

Using the definition of J~, one can change variables from sin 0 ~to the transverse mltlal position y and arrive at the usual expression for the classical probability of transmission

l

7-=

W

(8)

l f d(slnO) f wdv- f(v ,o), -1

0

where f(y, O) = 1 if the trajectory with initial conditions (y, O) Js transmitted and f(y, O) = 0 otherwise Thus, as expected, the leading order term in the average quantum conductance as the Fermi energy goes to infinity IS just the classical conductance [45] The quantum correction to the conductance at zero magnetic field is most easily discussed in terms of the reflection coefficient, R = N - T The quantum corrections to R are ~R =

F,~:~ + Z

Z

F;)~

,

(9)

n # m ~4.u

where the terms diagonal m mode number, 6RD -}-~..U1 8Rnn, are separated from the off-diagonal terms From results m disordered conductors [44], one expects that coherent backscattermg wtll influence (SRD(B) In fact, from previous work on both disordered systems [44.50] and quantum-chaotic scatterlng [7,15], it is now well known that a typical diagonal reflection element is twice as large as a typical off-diagonal element when the system ts time-reversal mvarlant The ratto of mean diagonal to off-diagonal elements ts known as the elastic enhancement factor. or in the disordered context the coherent backscattering peak For the discussion of the magnetic field dependence of 6R in this section, we restrict our discussion to ~RD which is patterned after prevtous semtclasstcal treatments of the elastic enhancement factor [7,15,51] There is a natural procedure [9,10] for finding the average of (SRo over all k 1 denoted (SRD) the sum of N reflection elements each with I sln0l = I stn0'l can be converted to an integral over angle as N --. oo, (rr/kW) y'~.. ---, f d(smO) Then, the only kdependence is In the exponent so that the average eliminates all paths except those for which /., = L. exactly 2 In the absence of symmetry L~ = L. only tf I It is important to average over an mfimte w i n d o w m k, see Eq

(6) 2 T h e r e are several subtle aspects o f th~s a r g u m e n t w h i c h we address briefly ( 1 ) The paths mvolved m the s u m c h a n g e as

H U Baranger / Physwa D 83 (1995) 30-45

35

s = u, but with time-reversal s y m m e t r y ( B = 0) L~ =

lead on that direction, sin 0' H e n c e one can assume

Lu also if u is s time-reversed

the o u t g o i n g distribution is uniform In sin 0 t for an ar-

I n t r o d u c i n g a m a g n e t i c field changes both the clas-

bitrary distribution o f i n c o m i n g trajectories Classical

sical paths traversed and the action along a given path,

simulations confirm that this is a p p r o x i m a t e l y o b e y e d

however, for low-fields o n l y the phase difference is

for the structure in Fig

~mportant For time-reversed pairs this phase difference arises f r o m the " e n c l o s e d flux", ( & - S . ) / h =

backscattered paths in Eq ( 1 0 ) can be replaced by an

20~B/(9o where O~ - 2~r f~ A

average over all sin 0 t, and the resulting expression for

dl/B ts the effec-

1 and improves if the open-

lng to the leads ts m a d e smaller Thus, the sum over

tive "area" e n c l o s e d by the path (times 27r) and qS0 =

(6RD(B = 0)) is the same as that for ~ (defined in

hc/e Thus,

the same way as T in Eq ( 7 ) ) dence o f

1

Z "4~e'2("~B/4"°' (6RD(B)) = I f d(slnO) -1 s(O,O),~(O,--O)

To obtain the depen-

6Ro on B, group the backscattered paths by

their effective area and average over the distribution o f this area,

N(O),

(10)

(6RD(B)} cx f

dON(O) exp[t2OB/qho]

(11)

which yields an order unity ( k - i n d e p e n d e n t ) c o n t n b u --

tton to

r'X3

G(B) Since the average over all k eliminates

all but the s y m m e t r y - r e l a t e d paths, the contribution to

The assumption that the distribution is u n i f o r m in sin 0

(6R} which is diagonal in m o d e s only depends on the

implies that

interference o f paths which are precise time-reversed pairs This result is

exact to leading order in k Eq

N(O) IS independent o f 0

B e f o r e proceeding note that this coherent backscatterlng effect does not contribute at all to the off-

( 1 0 ) contains only classical quantities, at thts point

diagonal

a plausible but a p p r o x i m a t e a s s u m p t m n about the dy-

partner will only interfere in

namics permits an e x p h c l t evaluation o f

(6RD)

reflection

a path

and

its time-reversed

6Ro We cannot show,

however, that breaking time-reversal s y m m e t r y has no

For a chaotic system in which the m m n g time for

effect on the off-diagonal contribution we k n o w o f no

particles within the cavity ~s m u c h shorter than the es-

analytic procedure for evaluating the second term In

cape time. the probability o f scattering out with any

Eq (9) exphcltly However, numerical results b e l o w

angle 0' is j u s t proportional to the projection o f the

demonstrate that the naive guess that the absence o f exact time-reversed pairs in this term w o u l d m a k e it

a functmn of k because the boundary conditions on the angles change Tht,, effect is. however, not essential and can be removed by either averaging using a discrete set of kn = 2"k0 for which paths always remain in the considered set (though new paths enter. of course), or by inserting a delta-function when introducing the integral over angles and performing the integrals more carefully (2) Because we are consldenng the part of R whtch is diagonal in mode index, the number of paths contnbutmg to the sum increases only hnearly with k. and there is no interference between paths whmh start contnbutlng at some large ~-2 and tho~e which were already contributing at kl < k2 For instance the paths contnbutlng to ~ , z ~v~nR,,,(2k) are exactly the same as those contnbutlng to ~~,tll n R,m(k) whale the paths contributing to ~-~n odd R,m(2k) are new Thus 6RD is a simpler quantity than the full two-point correlatmn functmn considered, for instance in the statistics of energy level~ m Ref [31 It is this simphclty whmh allows an exact evaluation (3) We assume, of course, that the ,,em~classlcal expressmns for rnm and Rnm converge 148] Further, we assume that the expression for t~Ro converges absolutely m order to exchange the limit q ~ ~ and the sum over paths

insensitive to the presence o f time-reversal s y m m e t r y is w r o n g H a v i n g related

(6RD ( B ) } to the classical area dis-

tribution, we must now evaluate this distribution and the related distributions o f lengths B o t h analytic arguments and numerical calculations [ 52,4,6,7.10,11,53 ] have found that the area distribution for l o n g orbits takes on a universal form for simple chaotic bllhards

N(O) ~ exp ( - a c t [ O [ ) ,

(12)

where the only parameter characterizing the classical phase space is the inverse o f the typical area enclosed by a scattering trajectory, act This universal f o r m may be understood qualitatively by the f o l l o w i n g a r g u m e n t [6,7.52] First, the n u m b e r o f orbits w h i c h remain in the scattering region long e n o u g h to c o v e r a dis-

3~

H U Baranger / Physlca D 83 (1995) 30-45

tance L decays exponenually with characteristic rate Yd Crudely, the probability of escaping at each encounter with the boundary is a constant, given by the ratio of the s~ze of the opening to the penmeter, and is uncorrelated with previous bounces A more sophisticated argument relates Yd to the Lmpunov exponent and dimension of the manifold of infinitely trapped trajectories [53] Second, the area O, of a given trajectory is essentially its winding number around a central point in the billiard times a typical area per c~rculatlon A0 For a chaotic system the area accumulates diffusively, as both senses of circulation occur with roughly equal probability The distrIbuUon of areas for orbits of length L is then Gausslan with varmnce A 2 ~ ( L / d ) A o [52,6,7] Finally, the integral of this against the exponential distribution of lengths yields the full distribution of areas, the leading behavior of which may be obtained by the method of steepest descent [6] One finds then the exponential law for ]O I with act ~ yx/~dd/Ao Numerically, the distribution of both lengths and effective areas for the stadium bllhard are clearly exponenual [4,10] The square root dependence of a d on Yd has been confirmed numerically as well [6,11] In contrast to the chaotic case, one does not expect universal forms for the length and area distributions in the regular case For example, m circular bilhards the area grows linearly with the length of the trajectory because a trapped particle never reverses it sense of circulaUon, whereas in polygonal billiards the areas cancel more completely than m the chaotic case One generic feature of regular bflhards is the existence of families of trajectories which sample phase space very differently over time, and hence one does not expect the length and area distributions to be characterized by a single scale as in the chaotic case Thus generically one expects power law behawor for these distributions [8,11,54,55], and this is confirmed numerically for both the rectangular and circular bllhards Returning to (¢3RD(B)), first consider the chaouc case and assume that the exponentml form of the area distribution holds for all O (an accurate approximation for structures similar to those studied here [4,5]) Performing the integral in Eq (11) and combining this result w~th that for the magnitude, we obtain a

Lorentzlan B-dependence ( 6 R D ( B ) ) = "/¢/[ 1 + (2B/o~dfbo) 2]

(13)

Note that this is obtained from the semlclass~cally exact equation (10) using two controllable approximations, uniformity and the exponential area dlStrlbuUon Now consider the regular case, focusing here on polygonal bllhards The uniformity assumption can no longer be used, and so Eq (10) must be evaluated by direct simulation The resulting variation of (cos (2BO/qbo)) with B shows a large interval over which the dependence is roughly hnear Some insight Into this unusual linear behavior of weak-localization can be obtained from the simpler problem of ballistic motion between two planes In this case, all complete traversals between the two planes give zero contribution to the accumulated phase [56], hence, the appearance of parallel walls combined with balhstlc motion produces a flux cancellation effect, as noted earlier in connection with conductance fluctuations [57] We believe that th~s flux cancellation is responsible for the linear weak-localization behavior which we find to be generic for polygonal billiards The semlclassical results suggest analyzing the numerical data by averaging the change m T(k) as shown in Fig 2 The top panel demonstrates the difference between chaotic and regular structures predicted by the semlclasslcal theory the curves for the half-stadia (chaotic) flatten out while that for the halfasymmetric-square (regular) increases linearly (except for very small B where ~t is quadrauc) Thus the difference In classical dynamics between the chaotic and non-chaotic btlhards -ergodtc accunmlatlon oJ flux versus flux cancellanon- producea a quahtattve difference m the dependence of the average conductance on B Note however that not all of our chaotic structures show a clear saturation, nonetheless all do have rapidly changing magneto-conductance at small field followed by a more gradual rise We attribute the lack of saturation to the small size of our structures The lower panel of Fig 2 shows that average magneto-conductance is present even for structures without stoppers It ~s interesting, and important for experiments, that the direct paths do not mask the

H U Baranger / Physzca D 83 (1995) 30-45 8

i

F

/,F------I I

i

/

I

37

magneto-reststance of a mmrostructure can be used to dtstmgutsh chaottc from regular dynamtcs

6 //

2 3 Conductancefluctuattons

// /

O4

g

/

E 02 F 0 _c

I

I

I

&03

i

i

i

///~-i

x:

002

--II L____

>~01

i~

I-r

~

<:

00 -0

1

I

~

i

1

2

3

__

a / Ucl O 0

Fig 2 Average magneto-conductance (weak-locahzatmn)for the six structures shown The magmtude ~s obtained from (T( l~, B) - T(k, B = 0))~ with kW/~r E 14, I 11 Note the difference between the chaotic and regular structures, as well as the sensmwty to ~ymmetry m the lower panel Gd ~s the inverse of the typtcal area enclosed by classmal paths (From Ref [91 )

weak-localization effect completely and the difference between the chaotic and regular cavities is still clear Finally, we comment on the relation between these theoretical results and the experiments [16-21] Extraction o f the weak-locahzatlon effect from the conductance fluctuations requires an averaging procedure In experiments this has been achieved either by varying the Fermi energy with a gate [ 17] or by averaging the characteristics of many similar structures [21] In both cases, the average experimental magneto-resistance shows a maximum at B = 0, in basic agreement with the theory for the existence of weak-localization in balhstlc cavities The agreement between the theoretical hneshape and the results of these two experiments is remarkable [ 1 0 , 2 l ] Most interesting, the experiment of Chang et al [21] shows a very clear difference in hneshape between stadium and circular cavities -Lorentzlan versus a hnear cusp This is the most convincing demonstratmn to date that the predictions of "quantum chaos" can actually be observed in the electronic properties of a solid state system Thus theoo; numertcal calculattons, attd expertment all concur that the average

Fluctuations about the average conductance are characterized by both their magnitude and their power spectrum Here we treat the spectrum o f the fluctuations, returning to their magnitude in the next section We follow closely the calculation of the fluctuations as a function of magnetic field in Refs [4,10] which was based on the semiclasslcal approach to S-matrix fluctuations as a function o f energy introduced by Gutzwaller [59] and extensively developed by Blumel and Smllansky [13] and Gaspard and Rice [60] The main results are that the power spectrum can be directly related to properties of the classical phase space and, as was the case for the average conductance, this implies a quahtatlve difference in the form of the spectrum of chaotic and non-chaotic structures The fluctuations in the transmission intensity are defined by their deviation from the average value studied in the last section, in the absence of any symmetries,

fiT = T - TkW/Tr

(14)

The spectrum o f fluctuations In k is characterized by the correlation function,

C ( A k ) = (6T(k + h k ) 6 T ( k ) ) k ,

(15)

and the corresponding Fourier power spectrum,

d(x) -

fd(Ak)C(Ak)e

,

(16)

defined by averaging over k (see footnote 1 ) The semlclasslcal analysis proceeds by an argument s i m i l a r to that for the average magnetotransport above First, only part of the correlation function can be treated analytically in this case the diagonalapproximation consists in the correlation of transmission coefficients between the same modes

Using the semlclasslcal expression for the transmission amplitudes, we replace the sums over modes by

H U Baranger / Phvsua D 83 (1995) 3 0 - 4 5

38

integrals over angles because the modes become dense for very large k This yields an expression with k-

may be deviations at small L Using the exponential form for all lengths, yields

dependence only in the exponent CD(X)

( e x p [ t k ( L , - L,, + L, - L, )])a

The infinite k average implies that the only contribution is for L, - L. + L, - L, = 0 exactly Because of the definmon of CD, all four paths satisfy the same boundary condmons on angles, and hence they are all chosen from the same discrete set ot paths In the absence of symmetry, the only contribution is v = ~ and t = u (The terms with ~ = u and t = v are excluded because they give the average value ) Thus one finds the exact semmlasslcal expression 1

CD(Ak)=¼fd(slnO).fd(slnO l) x Z

which implies that the wavevector correlation function is Lorentzlan [ 13] The argument for the magnetic field correlation function [4] is very similar The correlation function is again defined as an average over k, C(AB) ~

(~T(k,B + AB)6T(k,B))~,

"d"~'*e'ak(L-L")"

1

(19)

~(t),O' ) u(# O')(u 4: ~)

whmh is an order unity contrlbutmn to the fluctuatmns (it is independent of k) In the chaotic case. assume, in a s~mflar spirit to our treatment of the average conductance, that ( 1 ) the trajectories are uniformly distributed in the sine of the angle, (2) the angular constraints hnking trajectories u and s can be ignored, and (3) the constraint u 4= v can be ignored because of the prohferation of long paths By averaging over the angles associated with trajectory u. one finds that the only remaining constraint on the trajectories is through thmr lengths Introducing the classical distribution of lengths I

P(L)=-Jd(slnO) f d(smO ') 0

x Z

"7~"8(L- Lu),

(20)

CD(X) oc / d L P( L + x)P( L)

(21)

u(O ~')

one finds finally OO

J

0

As discussed above, the dlstrlbutmn of lengths is exponentml for chaotic billiards for large L, while there

1

CD(AB) = I /d(slnO) /d(slnO t) o o x Z Z "4"4"e'aB(r"'-°")/'l'°'

(24)

s(0 ~)') u I 0 , ~ ' ) ( u ÷ ~)

CD(B) oc f dON(O+rl)N(O)

(25)

--OO

Using the exponential form of the distribution of effective area for all values of O yields CD(r/) oc e-'~"l'fl (1 + a~d[r/])

(26)

for the power spectrum or

CD(AB)

I

0

(23)

but the difference in action in the phase factors comes from the difference m B Expanding the change in

0

~

(22)

action to first order in field [ S , ( B + AB) - S.(B + AB) + S,,( B) - S,( B) ] /h = ( 0~ - Ou)AB/dpo, one finds exactly analogous to Eqs (19) and (21).

1

0

~X e -r'flq,

(18)

= CD(0)/[1 +

(AB/cedqbo)2]2

(27)

for the correlation functmn Note for consistency that the field scale of the fluctuatmns is twtce that of the average magneto-conductance (Eq (13) ) because the relevant phase revolves the difference of two "areas" whereas weak-locahzatmn involves the sum As m the case of the average magneto-conductance, the non-diagonal terms in the correlation functmn are much more difficult to treat Thus we simply conjecture that the form of the full power spectrum will be the same as that given by the dmgonal term The conjecture is intuitively reasonable since a given property

H U Baranger / Phvstca D 83 (1995) 30-45

of a chaotic system should be characterized by a single scale which hence should be the same for the full correlation function The semiclassxcal theory of scattering in nonchaotic structures is presumably related to the treatment of direct trajectories but is less well developed than that of chaotic bilhards There are at least two basic differences in the non-chaotic case the traJectories group into families and there is a great deal of angular correlation between lnctdent angle and outgoing angle In the absence of a careful semlclasstcal treatment, we develop a conjecture for the spectrum by analogy with the chaotlC-bllhard results above From Eq ( 2 1 ) , the power spectrum C D ( a ) is evidently related to the distribution of two distinct paths with a difference m length of x In the non-chaonc case, the two &stinct paths must come from different families of trajectories and the constraint that the two paths satisfy the same boundary conditions on angles must be retained Thus we conjecture that ~c

1

Co(a) ,:x dL

d(slnO)

P2(L+ r,L,O),

(28)

39

~" -4 "6 el_

-8

g -to

u_ ~

I

20 ~'2"

-2

~

,

i

. . . .

®m

40 ×/L d i

'

'

60 '

i

. . . .

80 i

,

(b)

-4

q

F

5

'"% "%N~

10

50

100

I

I

x/L d I

I

i

"%. o

-6

m~

-8-

-~

_

0

2o

q ,~,,

4o

' ..........

6o

80

I

F ,,,, ~ilh~iill

loo

120

11/A (I

o

Fig 3 (a) Power spectra of T(k) for the chaonc (squares) and

where P2 ( L + x, L, 0) Is the classical distribution for t w o d i s t i n c t t r a j e c t o r i e s at a n g l e 0, o n e w i t h l e n g t h L and the other w i t h length L+x In the c h a o t i c c a s e , this two-particle distribution factorizes into the product of

length distributions appearing in Eq (21) but in the n o n - c h a o t i c c a s e P2 d o e s n o t f a c t o r i z e

The integral

on the right-hand-side o f Eq (28) has been computed numerically for the rectangular bflhard tt decays as 1/x ? for large x in contrast to the 1 / L 3 decay of P ( L ) For the power spectrum of T(B), we assume that a similar expression holds in terms o f the classical dtstrlbutton for two trajectories at angle 0 with effective areas O and O + r/, j

0

(squares) and regular (triangles) structures shown N = 6 The frequency-m-field scale is normahzed to the area of the structure, A The d~fference between the chaotic and regular structures ~s apparent m the spectrum of their magneto-fingerpnnts (From

Ref [ 10] )

up to a cutoff which is less than the geometric area of

CD(rl) ~ f dO f d(slnO) N2(O+rl, O,O) --2

regular (triangles) structures shown N = 33 The regular structure has more power at large frequencies because of more trajectone~ with large enclosed areas The line is a fit to the spectrum m the chaotic case the exponential decay pre&cted semlclasslcally hold~ over 6 decades (b) Power spectra of T(k) for a second parr of chaonc (squares) and regular (triangles) structures shown on a log-log plot N = 21 Again the transport through the regular structure produces more high-frequency power The ~pectrum of the regular structure is consxstentwith the l / r 2 spectrum pre&cted semlclasslcally and is certainly very different from the spectrum of the chaotic structure (c) Power spectra of T(B) for the chaonc

(29)

The result o f classical simulation for the rectangular bllhard shows that this distribution is roughly constant

the rectangle The predictions of these analyttc arguments can be compared to exact numerical results for the correlation functions obtained by the recurslve Green function method [45] with no free parameters Fig 3 shows results for systems with 4 - 1 5 modes at EF All three

40

H U Baranger / Phy~lca D 83 (1995) 30-45

panels show that the non-chaotic structures have more high-frequency power than the chaotic structures the fluctuations in the non-chaotic case are finer than in the chaotic case [5] This is consistent with exponential (power law) decay in the chaotic (non-chaotic) structures Note that one needs at least 2-3 decades of sensitivity to reliably distinguish the spectra of chaotic and non-chaotic billiards, this indicates the kind of sensitivity necessary for experimental tests The spectrum of T(k) in the chaotic case is indeed exponential note the excellent fit over nearly six decades in Fig 3a In the non-chaotic case the spectrum is indeed power-law the predicted 1/x 2 at large x is obtained in Fig 3b within the statistical error Thus the validity of our semlclassical argument and conjectures for C o ( x ) is confirmed The difference

m the power spectra of chaotic vs non-chaotic btlhards (exponential vs power-law decay) ts explained by the dtffermg classical two-particle distributions of lengths Turning to the spectrum of T ( B ) , Fig 3c, again the properties of the chaotic cavity are different from those of the non-chaotic rectangle The spectrum of the stadium fits the semxclassical theory for three decades of decay Because the classical distribution of area is bounded in the case of the rectangle, the conjecture Eq (29) predicts that the spectrum of T ( B ) should be limited to low frequencies This is clearly not the case in Fig 3c which looks qmte similar to the result for the spectrum of T(k) m panel (a) The reason for the failure of the non-chaotic conjecture is not understood but may well be connected to special propemes of the rectangular bllhard such as the extreme flux-cancellation effect. Finally, note that recent experiments have observed different power spectra In ballistic mlcrostructures fabricated with circular vs stadium-shaped cavities [ 16,20] By fitting the numerical power spectra to the semiclassical theory, we can extract the characteristic scales Tqm, Olqm and compare them to the classical quantities Yct, Cect The quantum and classical scales are in excellent agreement with each other for a variety of two and four probe structures [4,10] Thus tt ts posszble to

predict quantttattvely measurable properties of these balhsttc quantum conductors from a knowledge of the

chaotic classical scattermg dynamics Indeed, recent experiments [ 17,19 ] have Investigated the variation of 3/qm and Ceqmwith the degree of opening of the cavity and find the correct trends and quantitatively reasonable values Throughout this discussion of conductance fluctuations, we have concentrated on the role of long trajectories -those which are trapped in the cavity for some time- which therefore may exhibit some universal properties However, in any structure there are. of course, short non-universal trajectories which may play an important role In isolated systems, the strong influence of periodic orbits on the density of states is well known from the Gutzwlller trace formula [3] Recently the role of periodic orbits in open systems has been studied both experimentally and theoretically Experimentally, periodic conductance fluctuations have been observed at frequencies corresponding to Aharanov-Bhom interference around short periodic orbits [21-25] The theoretical treatment starts from the Kubo formula and develops an expression for the conductance in terms of periodic orbits [ 22,61-64]

3. Magnitude of magnetotransport effects: random S-matrix theory In this section we turn from the treatment of the shape of magnetotransport effects in microstructures to consideration of their magnitude For this purpose we have not been able to use the semiclassical theory our inability to evaluate the off-diagonal contributions to either the average conductance or the fluctuations precludes obtaining a prediction for the magnitude We therefore turn to the other main theoretical tool for the study of "quantum chaos", random matrix theory In fact, it has been proposed [ 13,14] that chaotic scatterlng in the quantum regime [ 3 ] should be described by a random matrix theory for the S-matrix The emphasis in both that work and work on the S-matrix of disordered structures [65] is on the eigenphases of S The elgenphases, however, are not directly connected to transport because they involve both reflection and transmission In contrast, very recently the lmphcataons of a random S-matrix theory for the quantum

H U Baranger / Phvstta D &~ (1995) 30 45

transport p r o p e m e s have been derived [ 12,66] In this section and the next we closely follow the treatment of Ref [ 12] m presenting this work

41

2 ( N + &/~)( 1 + 6,,caba) -- aa~ -- abd 2 N ( 2 N + 1 ) ( 2 N - 1 + 48~/~) Performing the trace over channels in Eq (2) ymlds

A quantum scattering problem is described by its S-matrix For scattering involving two leads each with N channels and width W, one has

(T) - N / 2 = - & ~ N / ( 4 N

s=(r

var(T) =

t')

t r'

"

1 + 2) --~ -~6a/~,

(31)

N ( N + 1 )2

(30) 4 ( 4 A ~ - - 1)

where r, t are the N × N reflection and transmlssmn m a m c e s for particles from the left and r', t' for those from the right Current conservation lmphes S is unitary, SS t = I, and for time-reversal symmetry (B = 0) S as symmetric We concentrate on s~tuatlons where the statlstms of the scattering can be described by assigning to S an "equal a priori dastnbutlon" once the symmetry res m c u o n s have been imposed In pamcular, the posSlblhty of "'direct" processes, caused, for instance, by short trajectories and giving rise to a nonvamshlng averaged S-matrix [67,68], is ruled out The appropriate ensembles are well-known m classmal random m a m x theory [41] and are called the Circular Orthogonal Ensemble ( C O E , / 3 = 1 ) m the presence o f tame-reversal symmetry and the Circular Umtary Ensemble ( C U E , / 3 = 2) m its absence These ensembles are defined through thmr mvariant measure the measure on the matrix space which is lnvanant under the appropriate symmetry operatmns To be precise [41 ],

d # ( S ) = d l z ( S ' ) , where S ' = UoSVo, and U0, V0 are arbitrary fixed unitary m a m c e s in the case of the CUE wath the restnctmn V0 = U~" m the COE Numerical evidence for the validity o f this random matrix theory lor describing quantum-chaotm scattermg can be found m Refs [ 13,68] We start by deriving (T) and var(T) where we use an antegratlon over the mvarlant measure as the average [12] Such integrals have been evaluated previously [69], and one finds

f

~ - 2 N +1 3Jl~' d~( S) It~,bl-

d/.t( a) lt~,hl-lt~d l- =

1 -~ ~,

COE,

--' ~ ,

CUE,

(2NN+21)2(2N + 3)

(32) where the hmlt is as N --, cx~ We make several comments concerning these results (1) The semmlasslcal theory and numerical calculations suggested that the average magnetoconductance, (T) - N/2, and the magnitude o f the conductance fluctuations, v a r ( T ) , are Independent o f the size of the system for chaotic billiards [9,10] This is the analogue of the "universality" o f the conductance fluctuations in the diffusive regime [1]

Since the number of modes ts proporuonal to the ~lze of the system (N = mt[ kW/rr]), our N --+ oo results show that the conductance fluctuatlom and weaklocahzatton ate umversal wtthm the random S-matrix theoo' (2) In the large N hmlt, var(T) m the presence of t~me-reversal symmetry is twice as large as in its absence, as in the diffusive regime, demonstrating the universal effect of symmetry ( 3 ) Both quantities show some variation in the small N regime typmal of the experiments, for instance, the rano of var(T) m the presence and absence o f symmetry is not 2 (4) The values obtained in the N --, cxD hmlt are the same as those from an eqmvalent random m a m x theory for the Hamxltonian in which the bllhard is described by the Gaussmn Ensembles and the conductance follows from couphng the bflhard to leads in a random way [14,15] (5) Note that the contribution of the terms which are off-diagonal m mode number is o f the same order as that of the diagonal part in the COE (6RD) ~ 1/2 while (6R) ~ l / 4 as U ---+ oo

Thus tt ~s clear that the off-diagonal terms difficult to evaluate semtclasstcally are cructal in obtamtng the magmtude of these quantum interference effects The predictions of the random matrtx theory are compared to the conductance of a stadium bllhard

H U Baranger / Physwa D 83 (1995) 30-45

42 01

(a)$

I

P

oo z

I

+

+

4. Full

v -0 2

distribution

of T

M o r e detailed predictions of the r a n d o m S-matrix theory can be derived, such as the full distribution o f T

+++++

-o 1

-03 015

I

for small N and the statlsttcs o f the elgenvalues o f tt t, denoted {7-} These results are obtained by e x p r e s s m g

I

I

I

I

I

I

I

I

~,

(b)

the lnvartant measure in terms o f a set o f variables that includes the {7-} [ 12] A n y unitary matrtx o f the form in Eq ( 3 0 ) can be written as [70]

010 t-"L" >

__.__.__.__.__.___.__

v (2~

o 05

x/7

~

o

~,(4~ • (33)

00 0

i 2

~ 4

J 6

i 8

N

Fig 4 The magnitude of (a) the weak-localization correction and (b) conductance fluctuations as a function of the number of modes in the leads, N The numencal results for B = 0 (squares with statistical error bars) agree with the prediction of the COE (dotted line), while those for B ¢ 0 (triangles) agree with the CUE (dashed line) The inset shows a typical cavity The numencal results involve averaging over (1) energy at fixed N (50 points), (2l 6 different cavities obtained by changing the stoppers, and (3) 2 magnetic fields for B 4=0 (BA/(bo = 2, 4 where A is the area of the cavity) (From Ref [12] ) in Fig 4 In these calculations, the S-matrix varies

where 7- is the N x N dmgonal matrix o f the {7-} and the v o) are arbttrary unitary matrices except that e (3) = (v(l~) r and v (4) = (vl2)) r in the presence o f ninereversal s y m m e t r y It is a general property o f measures on vector spaces that a dlfferenttal arc-length written in the form d o "2 = ~ab g~bdx~dxb l m p h e s that the volu m e measure is d # ( V ) = ~ [ I , ,

Substituting for S the form in Eq ( 3 3 ) , one finds (/9 = 1,2)

dtz(S) = p 3 ( { r } ) H d T - a H d t z ( v ( ' ) ) ,

T h e basic assumption (ergodtc hypotheszs)

(34)

a

as a function o f energy because o f the cavity resonances

dxa Here the

differential arc-length is simply d o -2 = Tr {dStdS}

where the j o i n t p r o b a b l h t y distribution o f the {7-} is

is that through these fluctuations S covers the matrix space with u n i f o r m p r o b a b i h t y This should apply to bllhards m which the effect o f short non-chaotic paths is m i n i m i z e d We therefore use a stadium bllhard in which ( 1 ) a stopper blocks any direct transmission be-

P2({7-}) = C2 H

lT-a - 7-hi2,

(35)

a~b

p~({~))=c,l-IIT-.-~lII1/v~2~, a
(36)

c

tween the leads, ( 2 ) a stopper blocks the whispering gallery trajectories w h i c h hug the half-circle part o f the stadium, and (3) the stadium ts asymmetrized to

dtz(v (')) is the lnvarlant ( H a a r ' s ) measure on the unitary group, and C3 are normalization constants 3

break all reflection s y m m e t r i e s E x c e l l e n t agreement

The distribution o f T = ~au__l 7-a follows by mtegra-

between the energy averages found numerically and

tlon over the j o i n t p r o b a b l h t y distribution This can

the invarlant-measure e n s e m b l e averages introduced

be carried out for small N, for mstance, in the triv-

above is o b t a m e d In particular, both the vartatlon at

ial case N = 1 [ 1 2 , 6 6 ] , w(T) = 1 for the C U E and

small N and the ratio o f v a r ( T ) in the presence and

w(T) = 1/2v/-T for the C O E For N = 1 - 3 the w ( T )

absence o f time-reversal s y m m e t r y are verified Thus

the random S-mamx theoo' provides accurate predtcnons for the magmtude of quantum transport effects m mtcrostructures stmtlar to those which can be studied experimentally

3 In Eq (34) we have omitted phase factors which reflect the arbltranness of the decomposition Eq (33) for the CUE case None of the results reported here are affected, see Ref [ 70 ] for a full discussion Also note that I-L runs from t = 1 to 2 for the COE and g = 1 to 4 for the CUE

H U Baranger / Phystca D 83 (1995) 30-45

I

I

small, and would provide a clear test of the appllcablhty of random S-matrix theory to experimental mlcrostructures

Be0

B=0

4

I

I

I

I

I

I

N=I

N=I

3

43

g'2 5. Conclusions 1

0

I

I

I

04

0 I

2O

t

1

I

08

I 04

0 I

I

I

I 08

I

I

N=2

N=2

\

0

05

10

15

0

05

1+,

10

I

I

2

I

TF\

j/

t !

H

00

15

~,

10

20 T

0

I

]

I

II I

I

'

[

~N=3

4,

~

_

[

10

[

I

20 T

Fig 5 The distribution of the transmission intensity at fixed N = 1, 2, or 3 both m the absence (first column) and pretence (second column) of a magnetic field The numencal results (plusses with statistical error bars) are in good agreement w~th the predictions of the circular ensembles (dashed lines) Note the striking difference between the N = 1 and N = 2 results and between the B = 0 and B :~ 0 results for N = 1 For N = 3 the dtstnbution approaches a Gausslan (dotted lines) The cawnes used are the same as those m F l g 1, for B 4: O, BAld)o=2, 3, 4, a n d 5 were used (From Ret [ 121 )

derived analytically from the random matrix theory are plotted in Fig 5 and compared to numerical data for bllhards Note the dramauc dzfference between the CUE and COE m the single mode case, and the difference wtthm each ensemble between the N = 1 and N = 2 cases The results for N = 3 are close t o a Gausslan distribution The agreement between the numerical and theoretical results in Fig 5 is very good in terms of the dependence on both B and N These effects should be observable in experiments, for which N is typically

We have summarized the work of [4,9,10,12,45] concerning the quantum transport properties of balhstic mlcrostructures First, semIclasslcal theory and numerical calculations show that quantum Interference effects in the transport coefficients of ballistic mlcrostructures reflect properties of the underlying classical dynamics If the scattering dynamics IS classically chaottc the average conductance and conductance fluctuations exhibit universal behavior characterized by a single dynamical scale If the scattering dynamics xs non-chaotic then these effects exist but have non-universal features and multiple scales In the chaotic case, semlclasslcal theory within the diagonal approximation is able to predict the dynamical scales quantitatively from properties of the classical phasespace The semlclassIcal theory, numerical results, and experiments show a qualitative difference between the shape of magnetotransport effects m chaotic versus non-chaotic microstructures Second, random matrix theory predicts the magnitude of quantum transport effects The magmtude of both the conductance fluctuatmns and the quantum correction to the average conductance are universal in the large N limit The small N limit is more relevant experimentally, and here the full distribution of T shows a striking dependence on both N and magnetic field (Fig 5) Note that these results are obtained for systems characterized by the simple circular ensembles The "direct" scattering due to short paths ((S) = 0) is neglected, since such scattering is important in many chaotic cavities, the effect of these processes remares an important open question These two complementary theoretical approaches, then, give a complete description of the quantum transport properties of balhstlc mlcrostructures It would be interesting, of course, to extend the two approaches so that their region of applicability overlaps Extensions of random matrix theory in order to describe the shape

H U Baranger / Phvstca D 83 (1995) 3 0 - 4 5

44

o f m a g n e t o t r a n s p o r t effects are. m fact, m p r o g r e s s [71-73]

T h e result for the h n e s h a p e o f the average

m a g n e t o - c o n d u c t a n c e is m very g o o d a g r e e m e n t with the semlclasslcal t h e o r y [71]

In contrast, e x t e n d i n g

the semlclasslcal theory to a c c o u n t for the m a g n t t u d e o f t h e s e i n t e r f e r e n c e effects has not been achzeved bec a u s e o f the difficulty m e v a l u a t m g the o f f - d t a g o n a l terms, thts is p e r h a p s the m o s t f u n d a m e n t a l r e m a i n i n g theoretical p r o b l e m tn balhstJc q u a n t u m transport

Acknowledgements I gratefully t h a n k R A A D

Jalabert, P A

M e l l o . and

S t o n e for their c o l l a b o r a t i o n m p e r f o r m i n g the

r e s e a r c h w h i c h is s u m m a r t z e d here

In addition, I

have benefited f r o m helpful d t s c u s s l o n s w t t h M J Berry. H Bruus. A M C h a n g , E D o r o n , M W Keller. C H L e w e n k o p f . C M M a r c u s , U Smllansky, and S Yomsovlc

References I 1 I For reviews of mesoscoplc physics, tee C WJ Beenakker and H van Houten. m Sohd State Physics. Vol 44. H Ehrenrelch and D Turnbull eds (Academic Press. New York. 1991) pp 1-228. B L Altshuler. PA Lee and R A Webb. eds Mesoscoplc Phenomena m Sohds (North-Holland New York. 1991 ). S Washburn and R A Webb. Adv Phys 35 (1986) 375 121 For a rewew, see AD Stone. PA Mello. K Muttahb and J-L Plchard. m Sohd State Physics. Vol 44. H Ehrenrelch and D Turnbull. eds (Academic Pre~s. New York. 1991) pp 1-228 [3] For revlews, tee M C Gutzwdler. Chaos m Classical and Quantum Mechamcs (Spnnger. New York. 1991 ). M -J Gmnnom. A Voros and J Zlnn-Justm. eds. Chaos and Quantum Physics (North-Holland. New York. 1991) 141 R A Jalabert. H U Baranger and A D Stone. Phys Rev Lett 65 (1990) 2442 [51 R B S Oakeshott and A MacKmnon. Superlat Mlcrostruc I1 (1992) 145 [61 R Jensen. Chaos I (1991) 101 [7] E Doron. U Smllansky and A Frenkel. Physlca D 50 ( 1991 ) 367 [8] Y-C Lm. R Blumel. E Ott and C Grebogl Phys Rev Lett 68 (1992) 3491. and reference~ thereto 191 H U Baranger. R A Jalabert and A D Stone. Phys Rev Lett 70 (1993) 3876 [ 101 H U Baranger. RA Jalabert and A D Stone. Chaos 3 (1993) 665

[11] WA Lm. JB Delos a n d R V Jensen. Chaos3 (1993) 655 [12] HU Barangerand PA Mello Phys Rev Lett 7:~ (1994) 142 13] R Blumel and U qmdan~ky. Phys Rev Lett 60 (1988) 477 64 (1990) 241. Physlca D 36 (1989) 111 14] S hda. H A Weldenmuller and J A Zuk. Ann Phys (NY) 200 (1990) 219 151 C H Lewenkopfand HA Weldenmuller Ann Phys (NY) 212 (1991) 53 16] CM Marcus. AJ RJmberg. RM Westervelt. PF Hopkins and A C Gossard. Phys Rev Lett 69 (1992)506 17] MW Keller. O Mdlo. A Mlttal. DE Prober and RN Sacks Surf Scl 305 (1994) 501 18 ] M J Berry. J H Baskey. R M Westervelt and A C Gossard. prepnnt (1994). M J Berry. J A Katme. C M Marcus. R M Westervelt and AC Gossard. Surf Scl 305 (1994) 495 [19] CM Marcus. RM Westervelt. PF Hopkins and AC Gossard. Phys Rev B 48 (1993) 2460 [20] C M Marcus. R M Westervelt. PF Hopkins and A C Gossard Chaos 3 (1993) 643. Surf Scl 305 (1994) 480 [21] AM Chang. HU Baranger. LN Pfelffer and KW West Phys Rev Lett (1994). submitted [22] D Weiss. K Richter. A Menschlg. R Bergmann. H Schwelzer K yon Khtzmg and G Welmann. Phys Rev Lett 70 (1993) 4118. D Welts K Richter. E Vaslhadou and G Lutjenng. Surf Scl 305 (1994) 408 [23] R Schuster. K Ens~lln. D Wharam S Kuhn. J P Kotthaus. G Bohm W Klein. G Trankle and G Welmann. Phys Rev B 49 (1994) 8510 [241 TM Fromhold. L Eaves. FW Sheard. M L Leadbeatel. TJ Foster and PC Mare. Phys Rev Lett 72 (1994) 2608 [25] G Muller. H Mathur. G Boebmger. LN Pfefffer and K W West, Phys Rev Lett (1994), submitted 126] B D Stmons and B L Altshuler, Phys Rev Lett 70 (1993) 4063, and references therein 1271 S Oh. AYu Zyuzm and RA Serota, Phys Rev B 44 (1991) 8858. R A Serota, Sohd State Commun 84 (1992) 843 [281 H Bruus and AD Stone, Phys Rev B (1994), submmed Physlca B 189 (1993) 43. Surf Scl 305 (1994) 490 [29] RC Ashoon et al. Phys Rev Lett 68 (1992) 3088. Surf Scl 305 (1994) 558 [30] U Swan et al. Europhys Lett 25 (1994) 605 [31] K Nakamura and H Thomas. Phys Rev Lett 61 (1988) 247 [32] L P L6vy. D H Reich. L Pfelffer and K West. Phys~ca B 189 (1993) 204 1331 O Again. prepnnt (1994) [34] D Ullmo. K Richter and R A Jalabert. prepnnt (1994) [351 F yon Oppen. prepnnt (1994) [36] RA Jalabert A D Stone and Y Alhas~d Phys Rev Letl 68 (1992) 3468 [37] VN Pngodm KB Efetovand S hda. Phys Rev Lett 71 (1993) 1230 [38] PL McEuen et al. Phy~ Rev Lett 66 (1991) 1926 [39] For a rewew, tee M Kastner. Rev Mod Phys 64 (1992) 849

H U Baranger / Phvstca D 83 (1995) 30-45 ]401 T Ericson. Phys Rev Lett 5 (1960) 430 [4l ] M L Mehta, Random Matrices (Academic Press. New York, 1991). C E Porter, Statistical Theories of Spectral Fluctuations (Academic Press. New York. 1965) 142] R Landauer. Phd Mag 21 (1970)863 [431 M Bumker. Phys Rev Lett 57 (1986) 1761 1441 For reviews of weak-locahzatlon m disordered metals, see PA Lee and T V Ramaknshnan. Rev Mod Phys 57 (1985) 287, G Bergmann. Phys Rep 107 (1984) 1 [451 H U Baranger. DP DIVmcenzo. RA Jalabert and A D Stone. Phys Rev B 44 (1991) 10637 1461 DS Fisher and PA Lee. Phys Rev B 23 (1981) 6851 1471 WH Mdler. Adv Chem Phys 25 (1974) 69 1481 Convergence of semlclasstcal propagators m chaotic scattenng problems hat recently been addressed by J H Jensen. prepnnt (1993) 1491 See. e g EH Heller. m M-J Gmnnom. A Voros and J Zmn-Justm, eds, Chaos and Quantum Physics (NorthHolland, New York, 1991) pp 547-663 [501 PA Mello and AD Stone, Phys Rev B 44 (1991) 3559 [51] R Blumel and U Smdansky, Phys Rev Lett 69 (1992) 217 [52] MV Berry and M Robmk, J Phys A 19 (1986) 649 [53] For rewews of classical chaotic scattenng, tee T "I"61. in DlrecUon m Chaos. Vol 3, Hao Bal Lm, ed (World Scientific. Smgapore. 1990) pp 149-211. U Smdansky. m M-.I Gmnnom. A Voros and J ZmnJustin eds, Chaos and Quantum Physics (North-Holland. New York. 1991) pp 371-441 1541 W Bauer and G F Bertsch. Phys Rev Lett 65 (1990) 2213 1551 O Legrand and D Sornette. Europhys Lett 11 (1990) 583.

45

Physlca D 44 (1990) 229 [56] P-G DeGennes, Superconductwlty of Metals and Alloys (Addison Wesley, New York, 1966, 1989) pp 255-259 [57] CWJ Beenakker and H van Houten, Plays Rev B 37 (1988) 6544 [58] MB Hastings, AD Stone and HU Baranger. Phys Rev B, in press [591 M C Gutzwdler. Physlca D 7 (1983)341 [60] P Gaspard and S A Rice, J Chem Phys 90 (1989) 2225. 90 (1989) 2242, 90 (1989) 2255 [611 M Wdkmson, J Phys A 20 (1987) 2415 [62] G Hackenbrolch and F von Oppen, prepnnt (1994) 1631 N Argaman. preprlnt (1994) [64] K Richter. private commumcatlon [65] R A Jalabert and J -L Plchard, unpubhshed [661 R A Jalabert, J-L Plchard and C W J Beenakker, Europhyslcs Letters. m press [671 PA Mello. P Pereyraand TH Sehgman, Ann Phys (NY) 161 (1985) 254. WA Friedman and PA Mello. Ann Phys (NY) 161 (1985) 276 [68] E Doron and U Smllansky. Nucl Phys A 545 (1992) C455 [69] PA Mello and TH Sehgman, Nucl Phys A 344 (1980) 489, PA Mello, J Phys A 23 (1990) 4061 1701 PA Mello and J-L Plchard. J Phys I1 (1991) 493, PA Mello and AD Stone. Phys Rev B 44 (1991) 3559 [711 A Pluhar, HA Weldenmuller, J A Zuk and CH Lewenkopf, prepnnt (1994) [72] A M S Macedo, prepnnt (1994) [ 73 ] J -L Plchard, private communication