Desbois.Les.Houches. Impurities and quantum Hall effect
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SEMINAR 5
RANDOM MAGNETIC IMPURITIES AND QUANTUM HALL EFFECT
J. DESBOIS
Laboratoire de Physique Theorique et Modeles Statistiques, 91406 Orsay Cedex, France
Contents
1 |
Average density of states (D.O.S.) |
897 |
2 |
Hall conductivity |
901 |
3 |
Magnetization and persistent currents |
904 |
RANDOM MAGNETIC IMPURITIES
AND QUANTUM HALL EFFECT
J. Desbois
Abstract
In this talk, we present works done in collaboration with Ouvry et al. [1{3]. Random magnetic elds have already been studied in several papers [4]. Here, we will consider a model where the disorder is contained in the de nition of the magnetic eld itself. By magnetic impurities, we will mean in nitely thin vortices carrying a flux ( e =2 ), perpendicular to a plane. Those vortices are randomly dropped according to a Poisson's law with average density . In arst part, we will consider the average density of states (D.O.S.) of a charged particle coupled to the impurities. In particular, we will show that this system exhibits broadened Landau Levels for small values. This fact has motivated us to study the Hall Conductivity (part II) and, nally, persistent currents and magnetization.
1Average density of states (D.O.S.) [1]
We consider Hamiltonians for an electron minimally coupled to a vector
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potential A(~r) with the additional coupling of the electron spin up or downz = 1 to the local magnetic eld B(~r) (we set the electron mass me =h = 1)
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p~ − eA~(~r) |
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eB(~r) |
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H = |
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z: |
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It rewrites |
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z = +1 |
Hu = |
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z = −1 |
Hd = |
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+ − |
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where = (px − eAx) |
i(py − eAy) vx |
ivy are the covariant |
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momentum operators. In the homogeneous eld case, the spin coupling is a trivial constant shift, but, in general, it has important e ects. In the one vortex or magnetic impurity cases, it is a sum of (~r−~ri) functions, which is
c EDP Sciences, Springer-Verlag 1999
898 |
Topological Aspects of Low Dimensional Systems |
needed to de ne in a non ambiguous way [5,6] the short distance behavior of the wavefunctions at the location of the impurities ~ri. It can be attractive or repulsive and in the sequel we will only be concerned with the repulsive case ( z = −1, H = Hd).
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2 |
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For one vortex located in O (eA(~r) = k ~r=r |
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, eB(~r) = 2 (~r), k |
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is the unit vector perpendicular to the plane), it is easy to realize that the partition function reads:
Z (t) = Z0(t)hei 2 nifCg |
(4) |
where 2 n is the angle wound around O by the closed brownian curve C of length t. h ifCg stands for averaging over the set of all such curves and Z0(t) is the free partition function.
Z (t) is unchanged when ! + 1 and ! − ; so, we can restrictto the interval [0;1=2] when there is no additionnal magnetic eld. The D.O.S. exhibits a depletion at the bottom of the spectrum:
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(E) − 0(E) = |
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(E); 0(E) = |
V |
(5) |
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(V is the (in nite) area of the system). |
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~ |
Turning now to magnetic impurities located in ~ri, i = 1;2; :::;N, (eA(~r) = |
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P |
N ~ |
2 ~ |
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N |
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i=1 k (~r−~ri)=j~r−~rij ), eB(~r) = 2 i=1 (~r−~ri), we get for a given con guration of the N vortices:
Z (t) = Z0(t) |
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jN=1 2 nj |
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P |
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fCg
2 nj is the angle wound around vortex j by C. Averaging over disorder, we are left with:
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Z (t) = Z0(t)he n Sn(e2i n−1)ifCg
(6)
(7)
Sn is the arithmetic area of the n-winding sector (Sn 0; −1 < n < +1). Remarking that the random variables Sn scale like t, we rewrite Z (t) as:
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Z (t) = Z0(t)he− t(S−iA)ifCg |
(8) |
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Sn sin2( n); hSi = (1 − ) |
(9) |
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S = |
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A = |
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(10) |
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J. Desbois: RandomMagneticImpuritiesandQuantumHallE ect 899
In this formalism, the rescaled algebraic area enclosed by C should be written:
A = |
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nSn: |
(11) |
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From (8), it is easy to deduce that the average D.O.S. (E) is a function of E= and .
i) When ! 0, ! 1 with 2 ( ehBi) xed, we get after a careful analysis [1]:
Z (t) ! !0 e−tehBi=2ZhBi(t) |
(12) |
ZhBi(t) is the Landau partition function for the average magnetic eld hBi
(h!ci ehBi=2). (12) shows that (E) = P1 (E − 2nh!ci) i.e. we get
n=1
the Landau spectrum shifted by h!ci.
ii) = 1=2. (10) shows that A 0 and (8) leads to:
Z E=
(E) = 0(E) P(S)dS (13)
0
where P(S) is the probability distribution of S. (E) is a monotonically growing function of E with a depletion of states at the bottom of the spectrum.
iii) 0 < < 1=2. Using an argument based on the speci c heat c ( kt2d2 lnZ=dt2, k is the Boltzmann constant), we can show that the D.O.S. surely oscillates when is smaller than some value 0. The argument runs as follows. With the expression (8), we get:
c − c0 t!0 kt2(hS2ifCg − hSif2Cg − hA2ifCg) |
(14) |
(c0 is the free speci c heat). Numerically, one obtains that (c − c0) < 0 for
0 < < num :28. On another hand, Z (t) is the Laplace Transform of |
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(E). Integrating by parts, we show that: |
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c − c0 t!0 kt22 2 |
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1 Z0 |
1 dEdE0 d (dEE)=V d (dEE0)0=V (E − E0)2: (15) |
Thus, we deduce that (E)=V is a non-monotonic function of E for 0 < <
num0 .
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Fig. 1. Average density of states of the random magnetic impurity model.
To precise the shape of (E), we remark that variables jAj and jAj are
strongly correlated, especially for small values. |
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Assuming the linear relation: |
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( |
e ! !0 h i |
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when p |
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5), and |
p |
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jAj |
= (eBe = )jAj |
with |
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B |
B ; |
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introducing the new |
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variable S0 = S − jAj such that S0 and jAj are uncorrelated, we can write:
Z (t) he− tS0 ifCghe− t jAj cos(eBe tA)ifCg:Z0(t): |
(16) |
Performing the inverse Laplace Transform, we get (E) as shown in Figure 1.
Now, let us show briefly how the critical 0 value can be recovered analytically. With the concentration expansion:
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− V |
N |
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e ( V ) |
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(17) |
Z = |
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N=0 |
N! |
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ZN = hTr e−tHN i=V |
(18) |
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J. Desbois: RandomMagneticImpuritiesandQuantumHallE ect 901
(ZN is the average N impurities partition function per unit volume), we see that 0 is given by the equation:
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+ 1 = |
Z0 |
− 1 |
2 |
(19) |
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Z2 |
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Z1 |
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Computing Z2 diagrammatically to fourth order in , we nally get that0 is solution of:
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(3) ( ( − 1))4 : |
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(20) |
In the interval ]0;1=2], one obtains: 0 :29.
To end up with this part, let us mention what happens when we consider correlated impurities that are spatially distributed like fermions at T = 0 [1]. Diagrammatic computations lead to:
Z (t) = |
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(1 + ( − 1) t + 0( t)2 + :::( t)3 + :::) |
(21) |
2 t |
and for the speci c heat:
c − c0 t!0 −k(t (1 − ))2 < 0 |
(22) |
when 0 < 1=2. From this, we conclude that the D.O.S. has always oscillations.
2Hall conductivity [2]
For a review on the Integer Quantum Hall E ect, see, for instance, reference [7].
In this part, we develop a Kubo inspired formalism and compute the linear response of the system to a small uniform electric eld applied in the
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~x direction, E = (t)Eo. The local current |
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f~vj~rih~rj + j~rih~rj~vg |
(23) |
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− ~
(~v is the velocity operator ~v = p~ eA) is proportional to the conductivity
ij(~r;t) = i (t)eTrf^[ji(~r;t);rj ]g |
(24) |
where (t) is the Heaviside function. Tr^ is the thermal Boltzmann or Fermi-Dirac average. ji(~r;t) is the current density operator in the Heisenberg representation
~j(~r;t) = eiHt~j(~r)e−iHt: |
(25) |
902 |
Topological Aspects of Low Dimensional Systems |
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Considering the combination |
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−(~r;t) = xx(~r;t) − i yx(~r;t) |
(26) |
and ^ = e− H=Z (Boltzmann statistics), the global conductivity averaged over volume reads:
−(t) V |
Z |
d~r −(~r;t) |
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−Git(r;~0)x0G −it(~r 0;~r) − (it |
! it + ) |
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(G is the thermal propagator).
To deduce from (27) the conductivity of a gas of electrons at zero temperature and Fermi energy EF, one uses the integral representation of the step function (EF − H)
EF |
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1 dt0 eiEFt0 |
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lim |
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0 (t) |
(28) |
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where 0 and 0 are regulators which have to be set to zero at the end.
In general, it will be more convenient to calculate the derivative (t) of(t) with respect to time, rather than (t) itself. In the case of the thermal Boltzmann conductivity, one gets
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Z |
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V |
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−(it ! it + ) :
d~rd~r0 eB(~r) −Git(r;~0)x0G −it(~r0;~r) (29)
To derive (29), the identity |
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[H; ] = |
eB(~r) + ; eB2(~r) |
V (~r) |
(30) |
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= (p~ − |
has been used, which is valid in general for an Hamiltonian H |
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~ 2 |
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eA) =2 + V (~r). |
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The appearance of the local magnetic eld B(~r) in (29) { in the magnetic impurity case, it is a sum of (~r−~ri) functions { greatly simpli es the space integrals. We now discuss some examples:
i) homogeneous magnetic eld
(29) rewrites as
L−(t) = |
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(t) + 2i!c L−(t) |
(31) |
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J. Desbois: RandomMagneticImpuritiesandQuantumHallE ect 903
leading to: |
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L−(!) = |
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V − i(! + 2!c) |
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For a gas of electrons at T = 0: |
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Re L |
(!)j |
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e2 |
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2!c |
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EF |
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F V 4!c2 − |
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In (33), the limit !c ! 0 is properly de ned only if one keeps ! 6= 0, in which case it vanishes, as it should. The Hall conductivity nally reads
Re L |
(! = 0)jyx = −N(EF) |
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EF |
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This is the classical straight line, showing no plateaus in the Hall conductivity as a function of the number of electrons N(EF), or of the inverse magnetic eld 1=B.
ii) Hall conductivity for one vortex
With the standard Aharonov-Bohm propagator
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(35) |
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G (r;~0) = 2 e−2 |
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eim( − |
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(I (z)'s are the modi ed Bessel functions), one gets: |
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e2 |
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i sin( ) |
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(t |
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(t + i ) |
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(36) |
Its Fourier transform reads |
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−(!) |
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e2 i |
1 − |
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sin( ) |
(1 + )Ψ(1 + ;3; (! + i )) |
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V (! + i ) |
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− (2 − )Ψ(2 − ;3;− (! + i )) |
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(37) |
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where the Ψ(a;b;z)'s are the unregular confluent hypergeometric functions.
In the ! ! 0 limit, the Hall conductivity reads: |
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he2 sin(2 ) |
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Re (!)jyx = |
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(h !)2 ln(h !) + |
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904 |
Topological Aspects of Low Dimensional Systems |
For a gas of electrons coupled to the vortex at zero temperature, one gets, in the limit ! EF,
Re EF(!)jyx = N(EF) |
e2 |
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' !0 N(EF) |
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sin(2 ) |
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(39) |
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consistent with the homogeneous magnetic eld result (Eq. (33) with 2 =V = eB = 2!c).
It is possible to generalize those results when an external uniform B eld is added to the vortex. When is small:
EF (! = 0) = |
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N(EF) + |
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V hBtoti |
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dEF N(EF) + (40) |
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(ehBtoti = eB + 2 =V ).
iii) Perturbative hall conductivity for magnetic impurities
Considering the nonunitary wavefunction rede nitions
(~r) = e−21 h!cir2 |
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(41) |
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j~r −~rij ~0(~r) U0d(~r) ~0(~r) |
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i=1 |
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one obtains the Hamiltonian acting on the new wavefunctions ~0 |
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~0 |
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hLi hLi |
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hLi |
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H |
d = |
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− i (Ω − hΩi) − |
(42) |
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where −hLi = −2i(@z − h!ciz=2), Ω = |
iN=1 1=(z − zi) and hΩi = z. |
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It allows for perturbative |
computations. Skipping details [2], we get the |
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nal simple result: |
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1 e |
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Re EF (! = 0)jyx = −N(EF + h!ci) |
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(43) |
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hBi |
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In Figure 2, xy exhibits small oscillations above the classical straight line. It is worthwhile to notice that Hall plateaus shifted above the classical straight line have already been observed experimentally when the Quantum Hall device contains repulsive impurities [8].
3Magnetization and persistent currents [3]
Since the pioneering work of Bloch [9], several questions concerning persistent currents have been answered. The conducting ring case has been