Theory of semi-ballistic wave propagation

34 Pages·0.57 MB·

by  A. Mosk

Theory of semi-ballistic wave propagation
A. Mosk, Th. M. Nieuwenhuizen
Van der Waals-Zeeman Instituut
Valckenierstraat 65-67, 1018 XE Amsterdam, The Netherlands
and C. Barnes
Cavendish Laboratory, University of Cambridge
Madingley road, Cambridge CB3 OHE, United Kingdom
(February 1, 2008)
Wavepropagationthroughwaveguides,quantumwiresorfilmswithamodestamountofdisorder
isinthesemi-ballisticregimewheninthetransversaldirection(s)almostnoscatteringoccurs,while
6 in thelong direction(s) thereis so much scattering that thetransport is diffusive.
9 For such systems randomness is modelled by an inhomogeneous density of point-like scatterers.
9 ThesearefirstconsideredinthesecondorderBornapproximationandthenbeyondthatapproxima-
1 tion. Inthelattercaseitisfoundthatattractivepointscatterersinacavityalwayshavegeometric

n resonances, even for Schr¨odinger wavescattering.
a In the long sample limit the transport equation is solved analytically. Various geometries are
J considered: waveguides, films, and tunneling geometries such as Fabry-P´erot interferometers and

1 doublebarrierquantumwells. Thepredictionsare comparedwith newandexistingnumerical data
1 and with experiment. Theagreement is quitesatisfactory.
pacs

1 71.55 Jv (Disordered structures)
v
73.20 Dx (Electrons in low dimens. struct.)
8
73.50 -h (Electron transport in thin films)
3
0 42.81 Dp (Propagationin fiber optics)
1 73.40 Cg (Contact resistance)
0 73.40 G (resonance tunneling)
6
9
/
t I. INTRODUCTION
a
m
As is empirically known from the ancient development of music instruments, cavity resonances determine the
-
d transmission of waves through devices which have dimensions of the order of the wavelength. The resonances give
n rise to transmission peaks of diverse systems such as flutes, organ pipes, Fabry-P´erot interferometers, electronic
o nanostructures [1], and electronic waveguides. In these systems the transmission of waves can drastically increase if
c the wavevectorof the incoming waves allows for a new mode to be resonant. In the case of a very pure or very small
:
v cavity, impurity scattering can be neglected and the transmission is said to be ballistic. Ballistic transport has been
i shown to occur in various systems, including quantum point contacts [2] and narrow optical slits [3].
X
For wave propagation through waveguides or quantum wires with a modest amount of disorder, several regimes
r
occur. For rather clean systems one still has ballistic transportof essentially unscattered waves. In the limit of dirty,
a
but non-absorptive and phase coherent systems, the intensity mainly diffuses through the system. As pointed out
recently by one of us, for long wires or thin films there is a third regime, the semi-ballistic regime. [4] Here disorder
islargeenoughtocausediffusioninthe longdirection(s),butsmallenoughtomaintainballisticmotioninthenarrow
directions. In the present work we shall focus on this regime.
For electronic systems the conductance can be expressed by the Landauer formula,
2e2
G = flux
h Tab
ab
X
in terms of the flux transmissioncoefficients flux of the system, where a and b stand for the incoming and outgoing
Tab
channels,respectively. Intheballisticregimethereisnochannel-to-channelscatteringandthetransmissioncoefficients
are diagonalin mode space. They are oforder unity for the propagatingmodes (the low ordercavity resonancesthat
can be excited at the energy of the incoming wave), and exponentially small for the evanescent modes (the higher
order cavity resonances). Whenever a new cavity mode becomes resonant,the conductance makes a step of universal
heighth 2e2/h.
Itiswellknownthatmultiplescatteringbyalargedensityofimpuritieschangesthisballistictransporttoadiffusive
one. In the diffusive regime all intensity is completely randomly distributed over the system.
1
In the intermediate regime of semi-ballistic transport a moderate amount of scatterers is present. On one hand,
the cavity modes of the pure system are hardly perturbed while on the other hand multiple scattering dominates
transport in the long direction(s).
In such systems, interesting effects appear especially near the onset of new resonances. The conductance of a
quantumwireshowsadipjustbeforeanewmodebecomesresonant.[5][6][7]. CertainGaAs/AlGaAsdoublebarrier
quantum wells show diffusive broadening of their transmission resonance [8].
We study the averagetransmission properties of semi-ballistic systems using the scalar wave approximation in the
limit of point-scatterers. We assume that neither finite-temperature effects nor Anderson localization are relevant.
In ref. [4] one of the authors discussed a model to explain the transportproperties of these systems. In the present
work we will present the derivations of the results in that article and extend the approach beyond the second order
Born approximationto the t-matrix.
Thesetupofthispaperisasfollows: InsectionIIwewillcalculatethetransmissioncoefficientsofalong,moderately
disorderedwaveguide,to supply the derivation of the results presented in [4]. We then apply them to the calculation
of the conductivity of disordered quantum wires.
In section IV the transmission of semi-ballistic double barrier structures is examined. Ballistic double-barrier
structureshaveatransmissionpeakedatcertaindiscretewavevectors. Inthesemi-ballisticregimethereisabroadening
of these transmission resonances, and transmission to all other resonant channels.
In section V we extend the discussion to include resonance effects of the scatterers that may be induced by the
geometry.
In section VI we discuss the comparison between our model and numerical simulations of an Anderson model in
the semi-ballistic regime. We close with a summary.
II. SEMI-BALLISTIC TRANSPORT IN A WAVEGUIDE
We first consider the case of a moderately disordered rectangular waveguide. To simplify the problem we consider
scalar waves instead of vector waves, thus neglecting polarization effects. In this way we model the propagation of
TE modes in electromagnetic waveguides. The same approachalso applies to electron propagationin quantum wires
and, to some extent, sound propagationin long corridors.
Ourgeometryhasalsobeenchosenassimpleaspossible. Itconsistsofaninfinitelylongwaveguidewithinthemiddle
asectionoffinitelengthinwhichamoderateamountofscatterersispresent(seefig. 1). Outsidethedisorderedregion
we have a clean system (perfect leads, for electronic systems) in which ballistic transport occurs. In the disordered
region semi-ballistic transport occurs. Here we consider (longitudinal) transport inside such a geometry, so that the
wall potentials can be considered infinitely high. In later sections we consider (tranverse) tunneling through such a
device and its boundaries. For that apllication the wall potentials must have a finite strength.
A. The pure system
Our waveguide consists of four conducting plates, at x = 0, x = d , y = 0 and y = d , and it is infinite in the
x x
z-direction. The conducting plates impose the boundary condition Ψ=0 at their surface, to describe TE waves:
ψ(0,y,z)=ψ(d ,y,z)=0 (2.1)
x
ψ(x,0,z)=ψ(x,d ,z)=0
y
For electron propagation this would correspond to infinite wall potentials. We assume that monochromatic waves of
frequency ω , and free space wavevector k propagate through the guide. In the scalar wave approximation this is
0 0
described by the following equation:
2 k2 ψ(r)=0. (2.2)
{−∇ − 0}
In the absence of impurities the waves travelling through the guide have the form
iq z
ψ(x,y,z)= Ψ (ρ)e p (2.3)
p
p
X
where ρ=(x,y) is the transversalposition and Ψ (ρ) are the discrete transversal eigenmodes of the system:
p
2
2 2
Ψ (ρ)= sin(p x) sin(p y) (2.4)
p x y
rdx sdy
m π m π
x y
p = p = (m 1;m integer)
x y x,y x,y
d d ≥
x y
Other geometries e.g., cylindrical waveguides, can be described in the same way by substituting the corresponding
mode wavefunctions for Ψ. The modes for which p2 <k2, are ‘free’, i.e., they can propagate in the z-direction, with
0
the mode-dependent wavenumber q = k2 p2. The Green’s function of this pure system, to be denoted as G0, is
p 0−
the solution of the equation
p
2 k2 G0(r,r′)=δ(r r′) (2.5)
{−∇ − 0} −
withtheboundarycondidtions(2.1). Toincorporatetheboundaryconditionsintheequationwestudytheprojections
G0
p
G0(z,z′)= dρdρ′ Ψ∗(ρ)G0(ρ;z,ρ′;z′)Ψ (ρ′) (2.6)
p p p
Z
Since we have translational invariance we can use a Fourier transform to find from (2.5):
1
G0(q)= (2.7)
p q2+p2 k2 i0
− 0−
The extra term i0 ensures convergence of the back-transformation. In real space one has:
iexp i k2 p2 z z′
G0(z z′)= { 0− | − |} (2.8)
p − 2 k2 p2
p 0−
p
B. Adding impurities
Now suppose a small density of scattering impurities is present at random positions R in the region 0 z d .
i z
≤ ≤
Their scattering properties can be expressed in a scattering potential V(r)= V (r R ). We replace the single
i s − i
scatterer potential V by a δ-potential with scattering strength u. This is known to be a good approximation for
s
− P
electron scattering off a screened charge. The density of scatterers
n(r) = n(x,y) 0<z <d
z (2.9)
= 0 z <0; z >d
z
needs not be homogeneous in the x,y directions, we will assume though that it is independent of z for 0 z d .
z
≤ ≤
The density of scatterers will be assumed to be so small that the cavity modes (2.4) are still well defined, i.e. the
scatteringmeanfreepathℓ mustbemuchlargerthanthetransversalsizesd ,d . Thewaveequationforthissystem
p x y
is
2 u δ(r R ) ψ(r)=k2ψ(r) (2.10)
{−∇ − − i } 0
i
X
Here R are the random positions of the scatterers, distributed according to a density n(r).
i
C. The t-matrix of a single scatterer
We first consider the case when only one scatterer is present at position r. The t-matrix of the point scatterer is
simply the sum of the Born-series expressing repeated scattering events at the same scatterer:
t(r)=u+uG(r,r)u+uG(r,r)uG(r,r)u+... (2.11)
u
= (2.12)
1 uG(r,r)
−
3
Notethatsincethescattererispointlike,thereisnomomentumdependenceandthet-matrixisdiagonalinrealspace.
Eq. (2.12) depends on the return Green’s function G, which we have not expressed yet. As G is a property of the
(local) environmentof the scatterer it is clear what the physicalsignificance of the t-matrix is: the t-matrix describes
the effect of a scatterer in its local environment. In the case where many randomly positioned scatterers are present,
the return Green’s function G(r,r) in t depends on t itself, which makes eq. (2.12) self-consistent. This means the
t-matrix is not a property that can be taken from literature. It must be calculated explicitly using the appropiate
return Green’s function in the system under consideration.
In one dimension (d = 1), equation (2.12) is well defined. For d 2 the real part of the return Green’s function
≥
divergesanda reinterpretationis needed. Indeed, fora pure systeminthree dimensions the divergencyis wellknown
from the equivalent of Coulomb’s law
eik|r−r′| 1 ik
G(r,r′)= + (2.13)
4π r r′ ≈ 4π r r′ 4π
| − | | − |
As this divergent term appears in the denominator of (2.12), the t-matrix vanishes, strictly speaking. This problem
was discussed for scatterers in free space in ref. [9]. We will examine this problem in more detail for constricted
geometries in section V.
Fornowwewillproceedusingthesimplestwayaroundthisproblem,knownasthesecondorderBornapproximation.
This approximation is commonly used in electronic systems. Indeed, approaches with random potentials that obey
Gaussian statistics are equivalent to the second order Born approxiamtion. One of our aims is to see whether this is
still a good aprroximationin cavities. In this approachone has
t =u+iu2ImG(r,r) iu2ImG(r,r) (2.14)
Born
≈
The real part u gives rise to a small average potential and will be neglected from here on. This approximation
maintains the property of scattering but it does not take into account possible resonant behaviour of the scatterer.
In general,it is a good approximationfor very weak scatterers. In section V we will turn to the problem of including
the full t-matrix in our calculations. This will give rise to interesting resonance effects near the subband edge.
D. The amplitude Green’s function and the selfenergy
The self-energy Σ(r,r′) is defined as the sum of all irreducible scattering events that may be inserted in a Green’s
function line. As the density of impurities is low, we can restrict ourselves to the lowest order approximation to the
average self-energy, which is diagonal in the space coordinates, Σ(r,r′)=δ(r r′)Σ(r) with
−
Σ(r) n(ρ)t (r)=iu2n(ρ)ImG(r,r) (2.15)
Born
≈
The averageGreen’s function G is expressed in the (average) self-energy by the Dyson equation:
G(r,r′)=G0(r,r′)+ d3r′′G0(r,r′′)Σ(r′′)G(r′′,r′) (2.16)
Z
To deal with the multiple scattering problemwe have to averageover impurity positions, weighted with their density
n(ρ). Within thesecondorderBornapproximationwefindanexplicitformfortheGreen’sfunctions G ofthe mode
p
p
1
G (q)= (2.17)
p p2+q2 k2 Σ
− 0 − p
In the second order Born approximationΣ =iΓ , so the resonance width reads
p p
Γp= d2ρn(ρ)Imt (ρ)Ψ2(ρ) (2.18)
Born p
Z
= d2ρu2n(ρ)ImG(ρ,z,ρ,z)Ψ2(ρ)
p
Z
= d2ρu2n(ρ) ImGp′(z,z)Ψ2p′(ρ)Ψ2p(ρ)
Z p′
X
4
The result for Γ does not depend on z, since after averaging we have translational invariance when we are far away
p
from the leads at z =0 and z =d . The form (2.17) for G yields in real space
z
iexp i k2 p2+iΓ z z′
G (z,z′)= { 0− p| − |} (2.19)
p
2 k2 p2+iΓ
p 0− p
iq z z′
= ie p| −p | e−|z−z′|/2ℓp
2q +i/ℓ
p p
with q =Re k2 p2+iΓ (2.20)
p 0− p
q 1
and ℓ = (2.21)
p
2Im k2 p2+iΓ
0 − p
The quantity ℓ is the mode-dependent elastic mean frpee path. We find the selfconsistent equation
p
Γ =ΓD (2.22)
p p
ΓDp ≡ Npp′νp′ (2.23)
p′
X
ν (k ) Re 1 = 12qp (2.24)
p 0 ≡ 2 k02−p2+iΓp |k02−p2+iΓp|
Npp′ ≡u2 pd2ρ n(ρ)Ψ2p(ρ)Ψ2p′(ρ) (2.25)
Z
The number of states (per unit length) in a mode is
p
N
1
= Re k2 p2+iΓ (2.26)
Np π 0− p
q
from which we can see that ν is proportional to the density of states
p
π d
p
νp = N (2.27)
2k dk
0 0
E. The Bethe-Salpeter equation
To describe transport of intensity, electromagnetic energy or the probability of Schr¨odingers particles through the
system, we need the averagedintensity Green’s function,
H(r,r′)=G(r,r′)G∗(r,r′) (2.28)
In our model of discrete eigenmodes we consider the projection of the intensity Green’s function Hpp′. It describes
the propagationof intensity from mode p to p′, and it obeys the following Bethe-Salpeter (BS) equation:
Hpp′(z,z′)=Gp(z,z′)G∗p(z,z′)δpp′ + (2.29)
dz′′G (z′,z′′)G∗(z′,z′′)
p p ×
p′′ Z
X
Upp′′Hp′′p′(z′′,z′)
This equation involves the irreducible vertex Upp′. In our situation it is independent of z for 0 < z < dz, while it
vanishes in the ‘leads’ < z < 0 and d < z < . The irreducible vertex can be shown to be the sum of all
z
−∞ ∞
two-particle irreducible diagrams that can be inserted in the intensity Green’s function. Two-particle irreducible in
this context means that the diagrams cannot be split into two separate diagrams by cutting one propagator and one
complex conjugated propagator line.
The irreducible vertex is not available in closed form so it must be approximated. We are however not free in
choosinghowto approximateit: the approximationmustbe consistentwith the approximationto Σ we made earlier.
5
This can be understood as follows: U describes the emission of diffuse intensity by the scatterers, Σ describes the
intensity extinction due to the scattering. If the two are not balanced, our description of the system will show gain
or absorbtion, which is certainly unphysical in the systems we consider here. Flux conservation is guaranteed by the
Ward-Takahashi identity which can be derived from field theory or by manipulation of diagrams [10], [11].
ImΣDp = Upp′ImGp′(z,z) (2.30)
p′
X
If this identity holds, flux is conservedto everyorder in the scatterer density while most of our other approximations
are valid only in leading order in density. Comparing (2.30) to (2.23) shows that
Upp′ =Npp′ (2.31)
satisfies the Ward-Takahashiidentity. It is, in fact, the ladder vertex,constituting one step in a ladder diagram. It is
known from transport theory that the ladder diagrams describe diffusive transport.
F. Solving the transport equation in a waveguide
Since we cannot solve the Bethe-Salpeter equation analytically we will try to gain as much information as pos-
sible from approximations. The quantities we are interested in are the average longitudinal intensity transmission
coefficients T , where we use a for the transverse momentum of an incoming cavity mode and b for an outgoing
ab
mode.
We consider a wave coming in from z = of the form ψ (r) = Ψ (ρ)exp(iq z). It will be attenuated upon
in a a
−∞
enteringthe disorderedregion. This isdescribedbythe amplitude Green’sfunction. Inthe disorderedsectionitgives
rise to a source intensity
z/ℓ
Sp(z)=δp,ae− p (2.32)
Wehaveneglectedpossiblesurfacereflectionsasweassumethedispersionrelationtobethesameinsidethedisordered
regionas outside. The total intensity present in any mode as a function of position is denoted Φ (z). Using (2.29) it
p
can be shown that this quantity obeys the ladder equation
dz
Φ (z)=S + dz′G (z,z′)G∗(z,z′) (2.33)
p p p p ×
Z0
Upp′Φp′(z′)
p′
X
We can re-express this by expanding the Green’s functions
Φ (z)=S (z)+ (2.34)
p p
νp dzdz′e−|z−z′|/ℓp Upp′Φp′(z′)
2Γ ℓ
p p Z0 p′
X
where we have inserted the relation
1 2ν
p
=
k2 p2+iΓ Γ ℓ
| 0 − p| p p
Equation (2.34) is a linear system of Fredholm integral equations of the second kind. The solution of this type of
equation, for d , is the sum of a homogeneous solution, ΦH, and a special solution Φa which is dependent on
z
→ ∞
the source term S . (In fact, the special solution is defined except for a multiple of the homogeneous solution which
a
we can add to it. We will choose Φa such that it remains finite as z .)
→∞
By differentiating the system of Fredholm equations we find the following equivalent set of differential equations:
1 ν
Φ′p′(z)= ℓ2Φp(z)− Γ pℓ2 Upp′Φp′(z) (2.35)
p p p p′
X
ℓ Φ′ (0)= 2S (0) Φ (0) (2.36)
p p − p − p
ℓ Φ′ (d )=Φ (d ) (2.37)
p p z p z
6
We will first study the solution to the homogeneous form of the system (2.35), that is to say, we take S = 0 and
d . Then we have only the boundary condition (2.36)at z =0 . This will then be used to construct to solution
z
→∞
for finite d .
z
Using eq. (2.23) the Ward identity (2.30) can be written as
Upp′νp′ =Γp (2.38)
p′
X
Thisimpliesthattherighthandsideofeq. (2.35)vanishesifweinsertΦ (z) ν . Thedifferentialequationtherefore
p p
∝
has a solution of the form
ΦH(z)=(z +z)ν , forz >>ℓ (2.39)
p 0 p p
Near the boundary there will be other terms because of the condition (2.36). They are related to the non-zero
eigenvalues of the matrix, so they decay exponentially away from the edges. The asymptotic behaviour of the
homogeneous solution is characteristic of one-dimensional diffusion: the intensity decreases linearly with z. As
expressed by the factor ν in (2.39), the intensity is distributed over the modes according to their density of states.
p
The shift z will be calculated further on, when we take in to account the boundaries.
0
The special solution for an incoming wave of unit intensity, ψin=eiqzzΨ in mode a is called Φa. In the case of a
a
semi-infinite system we can choose it such, that it convergesto a constant awayfrom the boundary. The distribution
over the modes is then given by
Φa(z) C ν , z >>ℓ (2.40)
p → a p p
The coefficient C is different for each incoming mode.
a
We now examine the behavior of the solutions to (2.35) in terms of the eigenvalues of the matrix of the system.
There are exponentially growing solutions, exponentially decaying ones and linear+constant solutions corresponding
to the zero eigenvalue of the system. For a semi-infinite system, the exponentially growing solutions will be absent.
The equation can then be solved formally, yielding
Φp(z)= ciRpie−zλi +(α+βz)Rp0 (2.41)
i
X
where Ri are the right eigenvectors of the system (2.39) and all eigenvalues λ are positive. The linear plus constant
p i
term corresponds to the eigenvalue zero of the system, with the right-eigenvectorR0 =ν .
p p
The boundary condition at z = 0 puts constraints on the coefficients α,β and c . For the homogeneous solution
i
defined in (2.39) the boundary condition is
ℓ Φ′ (0)=Φ (0) (2.42)
p p p
according to this definition, β = 1 and α = z which leads to the equation for the coefficients of the homogeneous
0
solution cH:
i
cHri(λ ℓ +1)+z R0 =R0ℓ (2.43)
i p i p 0 p p p
i
X
For the special solution to the problem of a source intensity in channel a the definition (2.40) leads to α=C and
a
β =0. The resulting equation for the coefficients ca of the special solution reads
i
caRi(λ ℓ +1)+C R0 =2δ (2.44)
i p i p a p a,p
i
X
This equation is very similar to (2.43), if we take the sum R0ℓ ... on both sides of (2.44)the equations become
a a a
identical and we can conclude
P
1
z = ν ℓ C (2.45)
0 a a a
2
a
X
A very useful sum rule can be found from this equation by multiplying (2.44) by R0 on both sides and summing:
a
7
R0caRi(λ ℓ +1)+R0C R0 =2R0 (2.46)
a i p i p a a p p
i,a
X
Taking the inproduct with a vector orthogonalto the leftmost term, but not to R0, we find:
p
ν C =2 (2.47)
a a
a
X
To find another useful constant we study the two-particle Green’s function H. From the time reversal symmetry of
the problem we can derive an useful identity:
Hpp′(z,z′)=Hp′p(z′,z) (2.48)
If we let z′ become large, the Green’s function will behave like the homogeneous solution ΦH near z =0 because all
other contributions are extinguished within a few mean free paths from z′. From the symmetry property (2.48) and
the behaviour at large z we find the mode distribution at z′ must be proportional to ν :
p
z′l→im∞Hpp′(z,z′)=C0νp′ΦHp(z) (2.49)
To findanexpressionforthe coefficientC we considerapoint sourceinmode a ata largedistance z >>ℓ fromthe
0 1
boundary, so that all contributions that correspond to the nonzero eigenvalues of the system will have damped out
there. It follows we only have to consider the contribution that corresponds to the zero eigenvalue, which yields, by
considering the jump in the derivative,
−1
C = ν ℓ Γ (2.50)
0 p p p
( )
p
X
Using this coefficient we find an expression for C :
p
∞
Cp = dz e−z/ℓpC0 Upp′ΦHp′(z) (2.51)
Z0 p′
X
leading to the interesting relation
ν
ΦH(0)= p C (2.52)
p 2Γ ℓ C0 p
p p
G. Transmission coefficients
If the sample is finite a certain fraction of the intensity that enters the sample at z = 0 will be transmitted to
z =d . Wecancalculatethetransmissioncoefficientsfromchannelatochannelbfor‘opticallythick’samples(length
z
of many mean free paths) by matching the solution of the ladder equation near both boundaries:
For0 z butz d >>ℓ thesolutionwilbethesumofthespecialsolutionandamultipleofthehomogeneous
z p
• ≥ −
solution
Φ (z)= cΦH(z)+ (2.53)
p − p
∞
dz′Hpp′(z,z′)Up′ae−z′/ℓa
p′ Z0
X
For z d but z >> ℓ the problem can be considered from z = L. There is no incoming intensity and only
z p
• ≤
the homogeneous solution will be present,
Φ (z)=cΦH(d z) (2.54)
p p z −
8
In the bulk both solutions have a linear+constant form. This makes it possible to match a (special+homogeneous)
solution at z =0 to a (homogeneous) solution at z =d .
z
Φbulk(z)=(C c(z+z ))ν (2.55)
p a− 0 p
Φbulk(z)=c(d z+z )ν (2.56)
p z − 0 p
It follows that
1
c= C (2.57)
a
(d +2z )
z 0
SincetheaverageGreen’sfunctionextinguishesinfewmeanfreepaths,wedonothavetotakeintoaccounttheprecise
behaviour of the intensity at z = 0 for calculating the transmission to z = d . As usual, since the sample’s length is
z
many mean free paths, the transmitted fraction of the unscattered intensity is negligible. The intensity transmission
coefficient for transmission from channel a to channel b is then equal to the intensity Φ (d ),
b z
dz
T = dzG (z,d )G∗(z,d ) U Φ (z)
ab b z b z bp p
Z0 p
X
C
a ΦH(0)
≈ (d +2z ) b
z 0
C C
= a b Γ ℓ2ν
4(d +2z )q2 p p p
z 0 b p
X
We havenow derivedformula(4) ofref.[4]. It holds for the transmissionof scalarwavesthroughwaveguidesand can
mutatis mutandis be applied to the propagationof EM wavesor Schr¨odinger waves. Below it is used to calculate the
conductance of a quantum wire.
III. CONDUCTANCE OF ELECTRONIC SYSTEMS
We nowapplyourresultto the electroniccaseofadisorderedconductingchannel. TheLandauerformulagivesthe
average zero temperature conductance of a sample of arbitrary dimensions connected to two reservoirs of electrons,
in terms of the averageflux transmission coefficients:
2e2 2e2 q
R−1 =G= flux = bT (3.1)
h Tab h q ab
a
a,b a,b
X X
where2e2/histhequantumofconduction,thefactor2comesfromspindegeneracy. Thefluxtransmissioncoefficients
differ in our case from the intensity transmission coefficients by a factor q /q , where the q ’s stand for the z-
b;0 a;0 0
wavenumbers of the incoming and outgoing waves outside the disordered region. To a good approximation it holds
for the propagating modes,
1
q q2+ℓ−2 (3.2)
a;0 ≈ a a ≈ 2ν
a
q
With these approximations and relation (2.47), we find
2e2 4
G= Γ ℓ2ν (3.3)
h (d +2z ) p p p
z 0 p
X
¿From this formula, it is easy to see that Ohm’s law is valid for the average conductance of samples in the semi-
ballistic regime. The resistance, defined as the inverse of the average conductance, reads [12]
d 1
z
R= +2R (3.4)
c
d d σ
x y
with the conductivity
9
8e2
σ = Γ ℓ2ν (3.5)
hd d p p p
x y
p
X
and the ‘contact resistances’ at z =0 and z =d
z
z
0
R = (3.6)
c
σ
Note, however,that R as well as σ are complicated functions of k .
c 0
In ref. [4] an analytic result was obtained for the conductance in a 2D film, with the width d , in the limit
y
→ ∞
of weak disorder:
σ(d ) 3N 1 (d∗)2
x
= N(N +1) (3.7)
σ 2N +1 − 2 d2
bulk x
where d∗ = π/k is the resonant width, N is the number of open channels and σ = 2e2k2ℓ is the Drude
0 bulk 3πh f bulk
conductivity of a 3D bulk sample. This result is reproduced here in figure (2), together with the result for a 1D
quantum wire. Both curves have been scaled with the bulk conductivity.
It is seen that these curves exhibit remarkable drops in the conductivity whenever a new cavity mode becomes
resonant. These drops are explained mathematically by the density of states that grows large near the subband
bottom, thus causing the second order Born t-matrix to become larger. This expresses more efficient scattering and,
therefore, less conductance. The physical explanation is: When the ‘new’ mode is not yet resonant it does not yet
contribute to conduction. There is scattering to this mode however, and the scattered waves interfere destructively
with the wavespresent in the other modes. In section V we will study the analogue of this effect for the full t-matrix
of the pointscatterers.
A different approach for calculating the conductivity was followed by Surke and Wilke in ref [13]. These authors
calculate the conductivity directly from the Kubo formula and derive expressions different from ours. Their results
involvetheaveragescatteringtime,ratherthantheinverseoftheaveragescatteringrate(Γ inourwork). Ittherefore
p
seems to us, that the latter results are unphysical. Indeed, one can consider cases where the scatterer density goes
to zero locally, such that the average scattering time diverges. Then the resulting prediction for the conductivity
diverges, while our result (3.5) is quite insensitive to such limits, as it should be.
IV. TRANSPORT THROUGH A DOUBLE BARRIER STRUCTURE
In sectionII, transmissioncoefficients were derivedfor transportofwaveintensity alongthe length of a waveguide.
In some systems, like the Fabry-PerotInterferometer (FPI), or its electronical analogue,the double barrier quantum
well (DBQW), transport occurs in the transversaldirection due to ‘tunneling’ through the barriers.
In the absence of random scattering these devices transmit only waves for which the perpendicular component of
thewavevectorisresonantwiththecavity. Thelinewidthisverysmall,usuallytheQ-factorofthesedevicesisseveral
thousands. The pure FPI transmits a light beam that meets the resonance condition without changing its direction.
The devices we are interested in contain a small density of impurities, such that the width d of the device is still
x
much smaller than the mean free path, but the width multiplied by the Q factor is much larger than the mean free
path. The problem of multiple scattering in such devices was first considered on a fundamental level by Berkovits
and Feng [14]. Their approach is valid for the situattion of one resonant mode, well away from the onset of further
cavity resonances.
The behaviour near resonances in the multimode situation was later discussed by one of us [4]. In the present
section we will discuss the derivation of these results. It will be seen that multiple elastic scattering will broaden
the resonance linewidth and cause transmissionof the energy into all availableoutgoing channels, independent of the
incoming channel.
A. Double barrier system in one dimension
First, for simplicity, we will consider a one-dimensional double barrier quantum well, like in fig. (3), with the
potential between the barriers equal to that outside. We will describe these barriers,which are imperfect mirrors,by
strong δ-function potentials. Their strengths are allowed to have an imaginary part to model absorbtion of waves by
the mirrors.
10