Coulomb drag in longitudinal magnetic field in quantum wells

0.16 MB·English

by  V. L. Gurevich

Coulomb drag in longitudinal magnetic field in quantum wells
V. L. Gurevich and M. I. Muradov
Solid State Physics Division, A.F.Ioffe Institute, 194021 Saint Petersburg, Russia
(February2, 2008)
The influence of a longitudinal magnetic field on the Coulomb drag current created
4
0 in the ballistic transport regime in a quantum well by a ballistic current in a nearby
0 parallelquantumwellisinvestigated. Weconsiderthecasewherethemagneticfieldisso
2
strongthattheLarmourradiusissmallerthanthewidthofthewell. BothinOhmicand

n
non-Ohmic case, sharp oscillations of the drag current as a function of the gate voltage
a
J orchemicalpotentialarepredicted. Wealsostudydependenceofthedragcurrentonthe

9 voltage V across thedrivingwire, as well as on the magnetic field B.
StudyingtheCoulombdragonecanmakeconclusionsabouttheelectronspectrumand
]
l and electron-electron interaction in quantum wells.
l
a
h
- I. INTRODUCTION
s
e
m
The influence of a magnetic field on the Coulomb drag are investigated in different geometries. The
.
at Coulombdragbetweentwotwo-dimensional(2D)quantumwellsinastrongmagneticfieldperpendicular
m
to the planes of the wells and in the presence of disorder has been investigated in Ref. [1]. In magnetic
- fieldperpendiculartotheplanestheHallvoltagecanbeinducedinthedragquantumwellinthedirection
d
n perpendicular to both direction of the magnetic field and of the current in the drive well [2], [3]. These
o
two geometries can be called transverse.
c
[ Thepurposeofthepresentpaperistostudytheinfluenceofanin-wellmagneticfieldBontheCoulomb

1 dragcurrentinthecourseofballistic(collisionless)electrontransportinaquantumwellduetoaballistic
v
drive current in a parallel quantum well. In other words, we consider the longitudinal geometry, i.e. the
8
3 case where the magnetic field is parallel to the applied electric field E and to the plane of the well itself.
1
Wewillconcernourselveswiththecaseofastrongmagneticfieldthatmakesthemotionofthecarriers
1
0 along the field one dimensional and alters the density of electron states. Moreover,we restrict ourselves
4
with the quantum limit when only the ground Landau oscillator states are occupied by electrons in the
0
/ two quantum wells, so that
t
a
m
¯hω >µ (1.1)
B
- ∼
d
n Here ωB is the cyclotron frequency while µ is the chemical potential. A theory of electronic transport
o
through three-dimensional ballistic microwires in longitudinal magnetic fields at low temperatures has
c
: beendevelopedinRef.[4]. OurgeometryissimilartothatconsideredinRef.[4]. However,inthepresent
v
i paper we consider much simpler situation of a very strong magnetic field satisfying Eq. (1.1). Later on
X
we hope to return to a more generalcase of a weaker magnetic field where severalLandau levels may be
r
a involved.
Themagneticfieldmakingthemotionoftheelectronsinthetransversedirectiononedimensionalmaps
the problem under consideration onto the Coulomb drag problem in two one-dimensional wires already
considered by the authors in Ref. [5] in the Fermi liquid approach. Therefore, our final formulae for the
Coulomb drag current appear to be similar to those obtained in [5]. Physically the magnetic field may
play the following important role. It will suppress the tunneling of electrons between the quantum wells
that, if present, would impede observation of the Coulomb drag.
1
The magnetic field may change the electron quasimomentum relaxation time. Scattering of electrons
byionizedimpuritiesinsufficientlystrongmagneticfieldsmaybeevenweakerthanforB =0. Asforthe
relaxation due to the phonon scattering, a strong magnetic field can alter the density of electron states
and in the quantum limit the relaxation rate may be bigger than for B = 0. We, however, will assume
the temperature to be so low that the transport remains ballistic even in the presence of magnetic field.
We consider the case where the magnetic length
¯hc
a = (1.2)
B
s eB
| |
is much smaller than the distance W between the quantum wells of the width L W each
x
∼
W/a 1. (1.3)
B
≫
Thisinequalityestablisheslowerboundforthevaluesofthemagneticfieldforagivendistancebetween
the quantum wells. For instance, for W 80 nm the inequality requires magnetic fields of the order of
∼
B 1 T, or bigger.
∼
It is convenient to break our calculations into severalparts. In the first part we will give the principal
equations of our theory based on the Boltzmann treatment of the transport. We will consider a linear
response in Sec. III. Next we will discuss a non-Ohmic case in Sec. IV. Comparison of our results with
the 1D Coulomb drag results in the longitudinal geometry and 2D Coulomb drag results for B = 0 will
be given in Summary.
II. BOLTZMANN EQUATION
We consider two parallel quantum wells perpendicular to x axis. The eigenfunctions and eigenvalues
for a one electron problem in a magnetic field along the z axis in the ith quantum well is [we use the
gauge A=(0,Bx,0)]
1 x x
ψ = ϕ − py exp(ip y/¯h+ip z/¯h), (2.1)
0pypz LyLz 0(cid:18) aB (cid:19) y z
p
¯hω p2
ε =U + B + z . (2.2)
0pz i 2 2m
Here m is the effective electron mass, ϕ (x x )/a is the wave function of a harmonic oscillator in
0 − py B
the ground state oscillating about the point x = a2p /¯h = p /mω . The wave function ψ
(cid:2) py − (cid:3)B y − y B 0pypz
describes a state for which the electronprobabilitydistribution is largeonly within the slabof the width
a symmetrically situated about the plane x = x and falls off exponentially outside the slab. As
≈ B py
we consider the case L a we will assume the wave function to be equal ψ if x is within
x ≫ B 0pypz py
the quantum well and zero otherwise. In what follows we will need the matrix elements of the functions
exp( iqr) between two stationary states. We have
±
h0p′yp′z|e±iqr|0pypzi=e±iqx(xpy+xp′y)/2
×e−a2Bqx2/4e−(xpy−xp′y)2/4a2Bδp′z,pz±h¯qzδp′y,py±h¯qy. (2.3)
ThediagramrepresentingCoulombdrageffectisillustratedinFig.1. Theexternaldrivingforceenters
2
0 0
(2);py;pz
0 0
qx0;qy;qz py(cid:0)qy;pz(cid:0)qz qx;qy;qz
(1);py;pz
py+qy;pz+qz
py+qy;pz+qz
(cid:1)
J
FIG.1. Coulomb drag diagram. Herethelabels 2, (1) stand for the drive(drag) quantumwells.
the diagram through nonequilibrium distribution function represented by the solid lines marked by the
symbol 2 indicating that they represent the drive quantum well.
Now we embark on analysis of the conservation laws for the collisions of electrons belonging to two
different quantum wells. We have
ε(1) +ε(2) =ε(1) +ε(2) (2.4)
0pz 0p′z 0pz+h¯qz 0p′z−h¯qz
where ε(1,2) =U +h¯ω /2+p2/2m.
0pz 1,2 B z
The solution of Eq.(2.4) is
¯hq =p′ p . (2.5)
z z− z
The δ function describing energy conservation can be recast into the form
−
m
δ(ε(1) +ε(2) ε(1) ε(2) )= δ[h¯q (p′ p )]. (2.6)
npz lp′z − npz+h¯qz − lp′z−h¯qz ¯h qz z − z − z
| |
Therefore,theinitialquasimomentap andp′ afterthecollisionbecomep +h¯q =p′ andp′ ¯hq =p ,
z z z z z z− z z
i.e. the electrons swap their quasimomenta as a result of collision.
Following [5,6] we assume that the drag current in the quantum well 1 is much smaller than the drive
ballisticcurrentinthequantumwell2andcalculateitbysolvingtheBoltzmannequationforthequantum
well 1. We have
∂∆F(1)(p ,z)
v 0py z = I(12) F(1),F(2) , (2.7)
z
∂z − { }
where F(1,2) are the electron distribution functions in the quantum wells 1 and 2 respectively, and the
collision integral I(12) F(1),F(2) takes into account the interwell electron-electronscattering
{ }
I(12) F(1),F(2) = W1pz+h¯qz,2p′z−h¯qz(q′,q ,q ,p ,p′) . (2.8)
{ } 1pzn,2p′z x x y y y S
p′p′q′q
zXy x
In this expression the sum over p′ should be determined by the requirement that the x-center of the
y
oscillator function is within the second quantum well. The requirement imposes the constraint
3
¯h L ¯h L
W +L x < p′ < W +L + x , (2.9)
a2 x− 2 y a2 x 2
B (cid:18) (cid:19) B (cid:18) (cid:19)
and the product of distribution functions is
S
=F(1)F(2) 1 F(1) 1 F(2) F(1) F(2) 1 F(1) 1 F(2) . (2.10)
S 0pz 0p′z − 0pz+h¯qz − 0p′z−h¯qz − 0pz+h¯qz 0p′z−h¯qz − 0pz − 0p′z
(cid:16) (cid:17)(cid:16) (cid:17) (cid:16) (cid:17)(cid:16) (cid:17)
F(1) =θ[v ]f(ε µ1L)+θ[ v ]f(ε µ1R)+∆F(1) (2.11)
0pz z 0pz − B − z 0pz − B 0pz
where
1 for v >0
z
θ[v ]= .
z
0 for v <0
(cid:26) z
Here we assume that the electrons move ballistically within the quantum well and the electrons moving
from the left and rightreservoirshave the chemical potentials µ1L =µ eV /2 and µ1R =µ +eV /2
B B− d B B d
respectively. We introduce also the drag voltage V induced across the drag quantum well due to the
d
quasimomentum transfer from the driving quantum well, i.e. we assume an open circuit for the drag
quantum well.
The solution of Eq.(2.7) is (here we omit the equilibrium part)
L 1 p > 0,
∆F(1) = z I(12) F(1),F(2) , for z (2.12)
0pz − ± 2 v { } p < 0.
(cid:18) (cid:19) z z
Using the particle conserving property of the scattering integral
I(12) F(1),F(2) =0 (2.13)
{ }
pXzpy
we get for the total current in the drag quantum well defined as
e
J = v F(1), (2.14)
L z 0pz
z
pXypz
the result
1
J = e I(12) F(1),F(2) +e v [f(ε µ1L) f(ε µ1R)]. (2.15)
− { } L z 0pz − B − 0pz − B
z
py,X(pz>0) py,X(pz>0)
Inthese equationsthe sumoverp isrestrictedbythe requirementthatthex-centerofLandauoscillator
y
must be within the quantum well, so that ¯hL /2a2 < p < ¯hL /2a2. Introducing the density of
− x B y x B
states (including spin) per unit quasimomentum interval
L ¯hL L
z x y
N(p )dp =2 dp (2.16)
z z (2π¯h)2 a2 z
B
we have
2eVd ¯hLxLy ∞ ∂f(ε µB)
J = e dε − . (2.17)
Ohm − (2π¯h)2 a2 − ∂ε
B ZU1+h¯ωB/2 (cid:18) (cid:19)
For the degenerate electron gas this expression can be written as
e2 L L
x y
J = V . (2.18)
Ohm −π¯h2πa2 d
B
4
Here the number of Larmour circles covering the cross section of the quantum well L L /2πa2 appears
x y B
instead of the number of open channels in the 1D situation.
We assume that only the ground Landau oscillator state is occupied, so that
1 3
U + ¯hω < µ < U + ¯hω . (2.19)
1 B B 1 B
2 2
Taking into account Eq.(2.6) we obtain for the Coulomb scattering probability Eq.(2.8)
W11ppzz,+2ph¯′zqz,2p′z−h¯qz(qx′,qx,qy,py,p′y)= ¯hmqz δ[h¯qz −(p′z −pz)]2¯hπUqUqx′qyqz
| |
0pzpy e−iqx′x−iqyy−iqzz 0pz+h¯qzpy+h¯qy 0pz +h¯qzpy+h¯qy eiqxx+iqyy+iqzz 0pzpy
×h | | ih | | i
×h0p′zp′y|eiqx′x+iqyy+iqzz|0p′z−¯hqzp′y −¯hqyih0p′z −¯hqzp′y−¯hqy|e−iqxx−iqyy−iqzz|0p′zp′yi. (2.20)
Here we use the unscreenedCoulombpotentialpostponing discussionasto whenthis approximationcan
be justified until the last section.
To calculate the drag current we iterate the Boltzmann equation in the interwell collision term that
we assume to be small. Therefore one can choose the distribution functions in the collision term to be
equilibrium ones, e.g. F(1) =f(ε(1) µ ) for the first quantum well.
0p 0p − B
We assume, in the spirit of the approach developed by Landauer [7], Imry [8] and Bu¨ttiker [9] the
drive quantum well to be connected to reservoirs which we call ’left‘ l and ’right‘ r. Each of them is
in independent equilibrium described by the shifted chemical potentials µl = µ eV/2 and µr =
B B − B
µ +eV/2,whereµ istheequilibriumchemicalpotentialinthe magneticfield. Therefore,the electrons
B B
enteringquantum wellfromthe ’left‘(’right‘) andhavingquasimomentap′ >0 (p′ <0)aredescribedby
z z
F(2) = f(ε(2) µl ) [F(2) = f(ε(2) µr )] and we see that the collision integral Eq.(2.8) is identically
0p′z 0p′z − B 0p′z 0p′z − B
zero if the initial p′ and final p′ q quasimomenta in the drive quantum well are of the same sign. This
z z−
means that only the backscattering processes contribute to the drag current.
Due to Eq.(2.6) we are left only with p′ < 0 (since we are restricted according to Eq.(2.15) by the
z
constraint p′ ¯hq =p >0) and obtain in view of the δ-function in Eq.(2.20) the following product of
z− z z
distribution functions in the collision term
=F(1)F(2)r 1 F(1) 1 F(2)l F(1)F(2)l 1 F(1) 1 F(2)r , (2.21)
P 0pz 0p′z − 0p′z − 0pz − 0p′z 0pz − 0pz − 0p′z
(cid:16) (cid:17)(cid:16) (cid:17) (cid:16) (cid:17)(cid:16) (cid:17)
or
=f(ε(1) µ )f(ε(2) µr )[1 f(ε(1) µ )][1 f(ε(2) µl )]
P 0pz − B 0p′z − B − 0p′z − B − 0pz − B
f(ε(1) µ )f(ε(2) µl )[1 f(ε(1) µ )][1 f(ε(2) µr )]. (2.22)
− 0p′z − B 0pz − B − 0pz − B − 0p′z − B
This equation will be analyzed in the following sections.
III. LINEAR RESPONSE
Inthis caseeV/T 1 (weassumethe Boltzmannconstanttobe equal1)andEq.(2.22)canbe recast
≪
into the form
eV
= f(ε(1) µ )f(ε(2) µ )[1 f(ε(1) µ )][1 f(ε(2) µ )]. (3.1)
P T 0pz − B 0p′z − B − 0p′z − B − 0pz − B
Shifting the integration variable p′ p′ +h¯(W +L )/a2 we have for the drag current
y → y x B
5
eV 2π 4πe2 2 1 ∞ 2L dp ∞ dp′ m
J = e z z z
drag − T ¯h κ 2π¯h 2π¯h 2π¯h(p +p′)
(cid:18) (cid:19) Z0 Z0 z z
f(ε(1) µ )f(ε(2) µ )[1 f(ε(1) µ )][1 f(ε(2) µ )]
× 0pz − B 0p′z − B − 0p′z − B − 0pz − B
h¯Lx/2a2B 2Lydpy dp′yg (p +p′)/¯h,(p p′)/¯h (3.2)
×Z−h¯Lx/2a2B 2π¯h 2π¯h 00 z z y− y
(cid:2) (cid:3)
where
g00(kz,ky)= ∞ dqye−a2Bqy2A2(kz,ky,qy), (3.3)
2π
Z−∞
∞ dqxe−ia2Bqx[ky−(W+Lx)/a2B+qy]e−a2Bqx2/2
A(k ,k ,q )= (3.4)
z y y 2π (k2+q2)
Z−∞ z ⊥
q2 =q2+q2. The last integralover the centers of the Larmourcircles (since x = a2p /¯h) in Eq.(3.2)
⊥ x y p − B y
playsthe roleofaneffectiveCoulombinteractionpotentialbetweentheelectronsfreelymovingalongthe
direction of applied magnetic field.
h¯Lx/2a2B 2Lydpy dp′yg pz+p′z,py−p′y
00
2π¯h 2π¯h ¯h ¯h
−h¯LZx/2a2B (cid:20) (cid:21)
Lx/a2B
2L p +p′ L p +p′ L
= y dk k g z z, x k +g z z,k x (3.5)
(2π)2 y y 00 ¯h a2 − y 00 ¯h y − a2
Z (cid:26) (cid:20) B (cid:21) (cid:20) B(cid:21)(cid:27)
0
We keeponlythefirstterminthisexpressionsincethesecondtermincludesafasteroscillatingexponent
exp(iq (W +L )) as compared to the oscillating exponent in the first term exp(iq W).
x x x
∼ ∼
As W/a 1 we can sufficiently simplify the expression for g . We obtain
B 00
≫
g00(kz,ky)=ea2Bkz2 ∞ dqyA2(kz,ky,qy). (3.6)
2π
Z−∞
dqxeiqx[(W+Lx)−a2Bky] e−|W+Lx−a2Bky|√qy2+kz2
A(k ,k ,q ) = (3.7)
z y y ≃ Z 2π qx2+qy2+kz2 2 qy2+kz2
q
Finally, the interaction term acquires the form
Z−h¯h¯LLxx//22aa2B2B 2L2yπd¯hpy 2dπp′y¯hg00 kz,(py −p′y)/¯h = 4aB(2LπayBkz)3ea2Bkz2Φ(2Wkz) (3.8)
(cid:2) (cid:3)
where
∞ e−αξ
Φ(α)= dξ , (3.9)
Z1 ξ3 ξ2−1
For α 1 p
≫
π
Φ(α) e−α. (3.10)
≃ 2α
r
This result for the effective interaction (3.8) can be explained as follows: Larmour circles within the
quantum wells of the width L 1/(k L ) near the surfaces contribute to the interaction. The number
x z x
·
of interacting circles from two quantum wells is
2
L L 1/(k L )
y x z x
· .
a a
(cid:18) B B (cid:19)
6
The sumoverq ,q′ k ,q k /W leadsto afactor(k L )2 L k /W. The exponentialdecayof
x x ∼ z y ∼ z z x · y z
the drag with the distance between the quantum wells W is a consequence of one-dimensional character
p p
of the drag in the strong longitudinal magnetic field. Combining all these factors and multiplying the
result by U2 (4πe2)2/(L L L )2k4 we arrive at Eq.(3.8).
∼ x y z z
The product of the distribution functions in Eq. (3.2) is a sharp function of p and p′ at small tem-
z z
peratures, acquiring nonzero values only at p ,p′ pB T/vB. We assume that the quasimomentum
z z ∼ F ± F
interval T/vB is much smaller than h¯/W
F
¯hvB
T F . (3.11)
≪ W
Here we wish to note that the Boltzmann treatment of transport phenomena requires that the uncer-
tainty in longitudinal momentum must be smaller than the same momentum interval h¯/L T/vB.
z ≪ F
ThesetworequirementsautomaticallyleadtotheinequalityW L . Weassumethatthelastinequality
z
≪
holds.
According to our assumptions we can regard the interaction term in Eq. (3.2) as the slowly varying
function and obtain
2 −2
eV T U U
12 12
J =J sinh (3.12)
drag 04εB εB 2T 2T
F F (cid:18) (cid:19) (cid:20) (cid:18) (cid:19)(cid:21)
where
J0 =−κ2(e45πm¯h)3Laz2Ly (a 1kB)2e(2aBkFB)2Φ 4WkFB (3.13)
B B F
(cid:0) (cid:1)
Here we introduced notations U =U U and mvB =pB = 2m[µ U ¯hω /2], kB =pB/¯h.
12 1− 2 F F B − 1− B F F
We assume that the electrons remain degenerate in the magnetic field
p
¯hω
εB µ U B T. (3.14)
F ≡ B − 1− 2 ≫
We consider the quantum limit, i.e. the case when all electrons belong to the first Landau level
εB < ¯hω . (3.15)
F B
Since the electron concentration N under this condition is related to the chemical potential by the
B
equation
m¯hω pB
N = B F (3.16)
B π2¯h3
Eq.(3.14) and Eq.(3.15) lead to
(pB)2 π2¯h3 N
T F < ¯hω , pB = B . (3.17)
≪ 2m B F m ¯hω
B
The first inequality in this relation is weaker than Eq.(3.11) if ε ¯hω and Wk 1. Introducing
F B F
∼ ≥
the electron concentration N and the chemical potential µ for B =0 given by
(2mε )3/2
F
N = , ε =µ U (3.18)
3π2¯h3 F − 1
one can rewrite Eq.(3.17) as
2 2
4 N ε
B F
T ε < ¯hω . (3.19)
F B
≪ 9 N ¯hω
(cid:18) (cid:19) (cid:18) B(cid:19)
7
Note that the second inequality in this expressiondoes not depend on the electron mass and can require
magnetic fields stronger than Eq.(1.3) [thus imposing a constraint on the electron concentration, or, if
the latter is given the inequality may require stronger magnetic fields than is required by the Eq.(1.3)].
For instance, in a magnetic field of the order of B 10T the electron concentration N must be smaller
∼
than 2.7 1017 cm−3.
·
Consideringthecaseofthealignedquantumwells,sothatU =U [otherwisetheeffectisexponentially
1 2
small, cf. Eq.(3.12)] and putting N =N we obtain
B
2 4
eV T 3h¯ω
B
J =J , (3.20)
drag 0
4T ε 2ε
(cid:18) F(cid:19) (cid:18) F (cid:19)
J = e5mLyLzkF2 3h¯ωB 4e12(2εF/3h¯ωB)3Φ 4Wk 2εF . (3.21)
0 − 9κ2(4π¯h)3 2ε F3h¯ω
(cid:18) F (cid:19) (cid:18) B(cid:19)
The dragcurrentis a rapidlyincreasingfunction ofthe appliedmagnetic field, as the latterincreasesthe
density of states and decreases the transferred Fermi momentum.
J,nA
0.03
0.02 1 2
0.01
b
0.8 1.2 1.6 2
FIG.2. Drag current versusdimensionless magnetic field b=¯hωB/εF for two values of theinterwell distances
W =40nm (1) and W =50nm (2). Otherparameters are given in thetext.
Tomakeanestimateofthecurrentweputm=0.07m ,h¯ω ε =14meV,κ=13,L L =1µm,
e B F z y
∼ ∼
W =40nm.
J 10−11 A
drag
∼
In the linear response regime we can introduce a drag resistance, i.e. we can introduce the coefficient
that depends only onthe quantum wells parametersandrelates the drive currentJ in the quantum
drive
well 2 to the induced voltage in the drag quantum well J R = V . Here the drive current in the
drive D d
quantum well 2 is
e2 L L
x y
J = V (3.22)
drive − 2π¯h πa2
B
(cf. Eq.(2.18)) and V is determined by the condition of zero total current J = J +J = 0 in
d drag Ohm
the drag quantum well in Eq.(2.15).
6
RD = πe2¯hEεB εT LLz (k 1L )2 34h¯εωB e12(2εF/3h¯ωB)3Φ 4WkF32¯hεωF , (3.23)
F F y F x (cid:18) F (cid:19) (cid:20) B(cid:21)
8
where we introduced effective Bohr energy E =me4/κ2¯h2 and
B
2N ε
pB = B F p , p =√2mε . (3.24)
F 3 N ¯hω F F F
B
With the given above parameters we have the following estimate for the transresistance
R 0.4mΩ.
D
∼
Now let us discuss when one can neglect the screening. Since the transferred momenta are q 2pB/¯h
z ∼ F
we may not take into account the screening of the Coulomb potential if the inverse screening length is
much smaller than the transferredmomentum. We estimate the screeninglength at a transferredenergy
T as
∼
1 πe2N εB
B ln F. (3.25)
rs ∼s κεBF T
The required inequality can be written as (we put N =N)
B
N1/3e2 ε ε 2 ε 4
F F F
ln ε . (3.26)
F
κ " T (cid:18)¯hωB(cid:19) #≪ (cid:18)¯hωB(cid:19)
We will assume this inequality to be satisfied.
IV. NON-OHMIC CASE
The product of distribution functions Eq.(2.22) can be recast into the form
=2sinh(eV/2T)exp (ε(1) µ )/T exp (ε(2) µ )/T (4.1)
P { pz − B } { pz′− B }
f(ε(1) µ )f(ε(2) µ eV/2)f(ε(1) µ )f(ε(2) µ +eV/2)
× pz − B p′z − B− p′z − B pz − B
As is a sharp function of p and p′ one can take out of the integral all the slowly varying functions
P z z
and get
∞dp dp′ P h¯Lx/2a2B 2Lydpy dp′yg (p +p′)/¯h,(p p′)/¯h
Z0 z z(pz+p′z)Z−h¯Lx/2a2B 2π¯h 2π¯h 00 z z y− y
(cid:2) (cid:3)
eV U eV U
12 12
= Lym2a2BT2exp(2aBkFB)2Φ(4WkB)sinh eV 4T − 2T 4T + 2T (4.2)
4(4π¯h)3(a kB)6 F 2T eV U · eV U
B F (cid:18) (cid:19)sinh 12 sinh + 12
4T − 2T 4T 2T
(cid:18) (cid:19) (cid:18) (cid:19)
The drag current is
eV U eV U
12 12
2 +
1 T eV 4T − 2T 4T 2T
J =J sinh (4.3)
drag 02 εB 2T eV U · eV U
(cid:18) F(cid:19) (cid:18) (cid:19)sinh 12 sinh + 12
4T − 2T 4T 2T
(cid:18) (cid:19) (cid:18) (cid:19)
For eV T one gets from Eq. (4.3) the result of Eq. (3.20). Let us consider the opposite case eV T.
≪ ≫
InthiscaseonegetsanonvanishingresultforEq.(4.3)onlyif U < eV/2andoneobtainsthefollowing
12
| |
equation for the drag current
2 2 4
eV U 3h¯ω
12 B
J =J (4.4)
drag 0
"(cid:18)4εF(cid:19) −(cid:18)2εF(cid:19) #(cid:18) 2εF (cid:19)
Thus the drag current vanishes unless eV > 2U .
12
| |
9
V. SUMMARY
We have developed a theory of the Coulomb drag between two quantum wells in a strong longitudinal
magnetic field. We have considered a comparatively simple limiting case where only the lowest Landau
level is occupied. The strong magnetic field makes transverse motion of an electron one dimensional.
These one dimensional electron states can be visualized as quantum ”tubes” or ”wires”. Therefore, the
Coulomb drag problem in this situation becomes similar to the Coulomb drag problem between two
parallel nanowires.
It is interesting to compare our results with two different geometries of experiment. First, let us
consider the influence of magnetic field on 1D Coulomb drag for the longitudinal geometry. In this case
the magneticfieldisdirectedalongz axisandisparallelto1Dnanowires. Forsimplicity,weassumethat
the confining potential in the absence of the magnetic field is
mΩ2
U(x,y)= (x2+y2). (5.1)
2
The applied magnetic field shortens the radius of the state a so that it becomes
B
a2 Ωmc ¯h
a2 = 0 , B =2 , a = (5.2)
B 1+(B/Bc)2 c |e| 0 r2mΩ
q
where a is the radius in the absence of the magnetic field. For the lowest Landau level we have
0
1 1 ¯h2 p2
φ= exp ρ2/4a2 , ε = + z . (5.3)
√2πaB − B p 2ma2B 2m
(cid:0) (cid:1)
The wave function of the electron in the second wire can be obtained by a gauge transformation of the
wave function in the first one. Since the interaction term g is not phase sensitive we are left only with a
shift by the distance W between the centers of the wires in the argument of the wave function (5.3). As
a result, one gets for the interaction
g(2pBF)=4e−W2/2a2B ∞dρρe−ρ2I0 aW ρ K0 4pBF¯haBρ 2, (5.4)
(cid:20)Z0 (cid:18) B (cid:19) (cid:18) (cid:19)(cid:21)
where I (x), K (x) are the modified Bessel functions. The quasimomentum
0 0
1
pB = π¯hNB
F 2 L
must satisfy the inequality
¯h2
T (pB)2/2m < , (5.5)
≪ F 2ma2
B
since we have assumed that only the lowest Landau level is occupied. Here NB is the electron density
L
per unit length in magnetic field. The expression (5.4) demonstrates that provided the magnetic field
goes up the localizationradius a of the wave functions suppresses the probability of the backscattering
B
processes. Note that if one assumes NB = N , where N is the electron density per unit length for
L L L
B =0thenthe effectiveinteractiondependsonthemagneticfieldonlyviaa . Thereforeinthiscasethe
B
magnetic field does not change the magnitude of transferred momentum, in contrast with the previous
case where such a change leads to a rapid increase of the drag current in a strong magnetic field. The
drag current is
10