Integer Quantum Hall effect - Semantic Scholar...Integer Quantum Hall effect • basics • theories...
Transcript of Integer Quantum Hall effect - Semantic Scholar...Integer Quantum Hall effect • basics • theories...
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Integer Quantum Hall effect• basics
• theories for the quantization
• disorder in QHS
• Berry phase in QHS
• topology in QHS
• effect of lattice
• effect of spin and electron interaction
M.C. Chang
Dept of Phys
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Hall effect (1879), a classical analysis
* *
ˆ; / 0 at steady state
dv v vm eE e B mdt c
B Bz dv dtτ
= − − × −
= =
*
*
/ // /
x x
y y
v Em eB ce
v EeB c mτ
τ⎛ ⎞ ⎛ ⎞⎛ ⎞
= −⎜ ⎟ ⎜ ⎟⎜ ⎟−⎝ ⎠⎝ ⎠ ⎝ ⎠
*
2
0*
2
11
x x xc
y y yc
m BE j jne necE j jB m
nec ne
ω ττ ρω τ
τ
⎛ ⎞⎜ ⎟⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞⎜ ⎟= =⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟−⎜ ⎟ ⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠−⎜ ⎟⎝ ⎠
ρxy
B
j env= −*
0 *2 , cm
neBm ce
ρτ
ω ==
( )1 0
2
10
1
111
1
1
0 // 0
c
c
c
cc
c
c
nec Bnec B
ω τ
ω τ
ω τσω τω τ
ω τσ
ω τ
−
>
−⎛ ⎞= = ⎜ ⎟
+ ⎝ ⎠
−⎛ ⎞⎯⎯⎯→ ⎜ ⎟
⎝ ⎠−⎛ ⎞
⎯⎯⎯→ ⎜ ⎟⎝ ⎠
σ ρ2
0 *
nemτσ =
• Hall conductivity
• Hall resistivity
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Resistance and conductance
,x xx xy x x xx xy xy yx yy y y yx yy y
V R R I I VV R R I I V
Σ Σ⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞= =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟Σ Σ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠
Note:
detyy
xxRΣ
=Σ
So it’s possible to have Rxx and Σxxsimultaneously be zero (provided Rxy and Σxyare nonzero).
0
3 :
2 :
y
y yxx xx yx yx
x xI
yxx xx yx yx
x
V E WL WD R RA I J A A
E WLD R RW J W
ρ ρ
ρ ρ
=
= ≡ = =
= = =
L
Wx
y
Quantum Hall effect 0, .
1 1det det
xx yx
yx yxyx yx
yx yx
const
RR R
ρ ρ
ρσ
ρ ρ
= =
→ Σ = = = = =
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Measurement of Hall resistance
2-dim electron gas (2DEG)
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GaAs/AlGaAs heterojunction
(broadened) Landau levels in a magnetic field
Ener
gy
μ
subband
Dynamics along z-direction is frozen in the ground state
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Ando, Matsumoto, and Uemura JPSJ 1975
Effect of disorder on σxy (theoretical prediction before 1980)
Kawaji et al, Supp PTP 1975
Si(100) MOS inversion layer
9.8 T, 1.6 K
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1985
ρxy deviates from (h/e2)/n by less than 3 ppm on the very first report.• This result is independent of the shape/size of sample. • Different materials lead to the same effect (Si MOSFET, GaAsheterojunction…)
→ a very accurate way to measure α-1 = h/e2c = 137.036 (no unit)→ a very convenient resistance standard.
Quantum Hall effect (von Klitzing, 1980)
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An accurate and stable resistance standard (1990)
Kinoshita, Phys. Rev. Lett. 1995• theory
• experiment
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Condensed matter physics is physics of dirt - Pauli
dirty clean
• Flux quantization
0 2he
φ =
• Quantum Hall effect
• …
Often protected by topology, but not vice versa.
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The triangle of quantum metrology
QCP
e
(to be realized)
I
QHEV h / e 2
Josephsoneffect
f
e / h
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Quantum Hall effect requires • Two-dimensional electron gas• strong magnetic field • low temperature Note: Room Temp QHE in graphene (Novoselov et al, Science 2007)
Plateau and the importance of disorder
Broadened LL due to disorder
Why RH has to be exactly (h/e2)/n ?
• see Laughlin’s argument below
( )B ck T ω<
Filling factor
Aoki, CMST 2011
The importance of localized states
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Width of extended states?
256 states in the LLL. ε(Φ) periodic in Φ0Aoki 1983
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• Finite-Size Scaling
~ , 1/ 2xE N x ν−Δ =
ΔE ΔE
Huo
and Bhatt P
RL 1992
Exponent for correlation length
Li et al PR
L 2005
• experiment Ensemble average over 100-2000 disorder configurations
States that can carry Hall current (with non-zero Chern number)
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Quantization of Hall conductance, Laughlin’s gauge argument (1981)21 ( ) ( )
2 i i eiei
eH p A r V Vrm c⎛ ⎞= + +⎜ ⎟⎝
+⎠
∑
1 ( )
x x iix y
x y x y
e ej A rm L L i x c
c H cL L A L
HΦ
− ∂⎡ ⎤= +⎢ ⎥∂⎣ ⎦
∂∂= − = −
∂ ∂Φ
∑
x xA LΦ =
x
y
solve | |H Eψ ψΦ Φ Φ Φ>= >By the Hellman-Feynman theorem, one has
| | | |
xy
H EH
EcjL
ψ ψ ψ ψΦ ΦΦ Φ Φ Φ Φ
Φ
∂ ∂∂< >= < >=
∂Φ ∂Φ ∂Φ∂
∴ = −∂Φ
• EF at localized states, no charge transfer whatever Φ is.
• EF at extended states, only integer charges may transfer along y when Φ is changed by one Φ0. 0
2( ) yx y
y
Vn ej c e EL
nh
−= − =
Φ
• Due to gauge symmetry, the system needs to be invariant under Φ→ Φ+ Φ0,
• Simulate a longitudinal EMF by a fictitious time-dependent flux Φ
1
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Edge state in quantum Hall system
• Bending of LLsGapless excitations at the edges
• Robust against disorder (no back-scattering)
• Classical pictureChiral edge state (skipping orbit)
• number of edge modes = n
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Inclusion of lattice (more details later)
• Bulk states: En(kx,ky) (projected to ky); Edge states: En(ky)
• when the flux is changed by 1 Φ0, the states should come back.
→ Only integer charges can be transported.
Figs from Hatsugai’s ppt
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2Streda formula (1982)
Giuliani and Vignale, Sec 10.3.3
• If ν bands are filled, then the number of electrons per unit area is n=νeB/hc
∴ σH=νe2/h
L
R
Nonzero along edgeˆc M z= ∇ ×
Degeneracy of a LL: D=BA/Φ0
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Current response: conductivity
• Vector potential of an uniform electric field
22
00 0
0
1 ( )2
'
latt
i t
e eH p A V H A p O Am c m ceH p A e
m cω
ω−
⎛ ⎞= + + = + ⋅ +⎜ ⎟⎝ ⎠
= ⋅
1 ( )( )
( ) , then ( ) ;i t i t
A tE tc t
iE t E e A t A e E Ac
ω ωω ω ω ω
ω− −
∂= −
∂
= = =
2
,
( ) m m m
m m m
m m m
m
f f v veiV
v v
α
α
β
α
β
ασ ω ω ω
ω ψ
ω
ω ω ψ
−=
+
≡ − ≡
∑
• 1st order perturbation in E →
Kubo-Greenwood formula
j Eα αβ βσ=
-
00
1=
=
m
m m
m
m
pm
u
u k um
uk k
i
α
α
α
α
α
ε δ ω∂ ∂+∂ ∂
∂ +
Quantization of Hall conductanceThouless et al’s argument (1982)
3
2
2 20
2
1
2 nk nk nk nk
DC m m m m
m m
nknk
u u u ui
p p p pe fim V
e fV k k k k
α β
α β
α
α β
β
β
α
σω≠−
=
⎛ ⎞= ⎜ ⎟⎜ ⎟
∂ ∂ ∂ ∂−
∂ ∂ ∂ ∂⎝ ⎠
∑
∑
ℓ, m = (n, k)
• Berry curvature
( ) ( )2
2
( , , are
(
( )
)
cyclic)
1
2
nk nk nk nk
n zH nBZ
n
e
u u u ukk k
ik
d k k
k
h
γα β β α
α β γ
σπ
⎛ ⎞∂ ∂ ∂ ∂Ω ≡ −⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂⎝
Ω⎦
⎠
⎡ ⎤= ⎢ ⎥
⎣∫
• Hall conductivity for the n-th band
cell-periodic function um
an integer for a filled band
( )
( )
n n nk k
nk
k i u u
A k
Ω = ∇ × ∇
= ∇ ×
( )n n nkA k i u u≡ ∇
• Berry curvature (for n-th band)
• Berry connection
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2
( ,0) ( , ) ( , ) (0, )
BZb c d a
a b c d
x x x x x y y y x y y y
d k
dk A dk A dk A dk A
dk A k A k g dk A g k A
A
k→ ↑
= ⋅ + ⋅ + ⋅ + ⋅
⎡ ⎤ ⎡ ⎤= − + −⎣ ⎦
∇
⎦
×
⎣
∫
∫ ∫ ∫ ∫∫ ∫
1 2( ) ( )ˆ ˆ
2 2
,
( ,0) ( , ) ( ) ( )
etc
y x
x y
i k i kk k g x k k g y
x x x x x y
u e u u e u
dk A k A k g a b
θ θ
θ θ
+ +
→
= =
⎡ ⎤− = −⎣ ⎦∫
2
2 2 1 1( ) ( ) ( ) ( )2
BZ
d k
a b d a
A
nθ θ θ θπ
= −
∇×
+ −=
∫
Pf:a b
cd
Brillouinzone
• Niu-Thouless-Wu generalization to system with disorder and electron interaction (PRB 1985).
BZ
Zeros and vortices
total vorticity in the BZ
Czerwinski and Brown, PRS (London) 1991
gx
gy
21 intege( r )2 B
zZ
k kd nπ
Ω =∫
[ ]
1
2
1
2
1 2 1 2
( )
( )
( )
( )
( ) ( ) ( ) ( )
i aa b
i bb c
i dc d
i ad a
i a b d aa a
u e u
u e u
u e u
u e u
u e u
θ
θ
θ
θ
θ θ θ θ
−
−
+ − −
=
=
=
=
∴ =
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Connection with localization in disordered system (Anderson, 1958)
• one-parameter scaling hypothesis(Abrahams et al, 1979 < Thouless, Landauer…): assumeβ(g) depends only on g
Localized
extended
Quasi-extended
20( ) , ( ) 2
dg L L g dσ β−= = −
• For large g (good conductor)
• For small g (insulator)/( ) , ( ) lnLc
c
gg L g e gg
ξ β−= =
Lagendijk et al, Phys Today 2009
• All wave functions of disordered systems in 1D and 2D are localized.
• QHE belongs to a new class of disordered systems.
This analysis does not apply to the QHS, since the extended states are crucial there.
Flow follows the increase of L
MIT
conductance
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Fig from Altshuler’s ppt
Spectral distribution of random matrix (rank N>>1)• eigenvalues Ei• mean level spacing d1= (taking ensemble average)
• spacing between NN s=(Ei+1-Ei)/d1• P(s): distribution function of s
• spectral rigidity: P(0)=0
• level repulsion: P(s
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Fig from Altshuler’s ppt
Wigner-Dyson classes
GOE
GUE
GSE
AI
A
AII
Altland-Zirnbauerclasses
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Quantization ofmagnetic monopole (see Sakurai Sec 2.6)
• Vector potential (use 2 “atlas” to avoid Dirac string)
• gauge transformation between 2 atlas
→ monopole charge is quantized
YM
Shn
ir, M
agne
tic m
onop
oles
Note:
2 2
2 /
N S ig ig
N S ieg c
A A ie ee
ϕ ϕ
ϕψ ψ
−
⋅
− = − ∇
=
2eg nc=
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Analogy in QH system
• Gauge transformation
Kohmoto, Ann. Phys, 1985
• Two atlases
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, , ,( , ; ) ( ) ( )n n nH r p x E xλ λ λλ ψ ψ=
• Fast variable and slow variable
• “Slow variables Ri” are treated as parameters λ(t)
(Kinetic energies from Pi are neglected)
• solve time-independent Schroedinger eq.
“snapshot” solution
{ }( , ; , )i iH r p R P
electron; {nuclei}
Born-Oppenheimer approximation
e-
H+2 molecule
nuclei move thousands of times slower than the electron
Instead of solving time-dependent Schroedinger eq., one uses
Connection with Berry phase First, a brief review of Berry phase:
-
• After a cyclic evolution
0
, ( ) , (
'
0
(
)
' )T
ni dt E
n
t
n T eλ λψ ψ− ∫=
Dynamical phase
Adiabatic evolution of a quantum system
0 λ(t)
E(λ(t)) ( ) (0)Tλ λ=
xx
n
n+1
n-1
• Phases of the snapshot states at different λ’sare independent and can be arbitrarily assigned
(, ( , ( )
))
nn t n t
ieλ λγ λψ ψ→
• Do we need to worry about this phase?
• Energy spectrum:
( , ; )H r p λ
-
, , 0n n ni λ λγ ψ ψ λλ∂
= ⋅ ≠∂
≣An(λ)
• Fock, Z. Phys 1928• Schiff, Quantum Mechanics (3rd ed.) p.290
No!
Pf :
( ) ( )H t i ttλ λ∂
Ψ = Ψ∂
0' ( ')( )
,( )t
nni dt E ti
nt e eγ λ
λ λψ−
Ψ = ∫
Consider the n-th level,
Stationary, snapshot state
, ,nn nH Eλ λψ ψ=
( ), ,'
nn n
ie φ λλ λψ ψ=
nφλ
∂= −
∂Choose a φ (λ) such that,
Redefine the phase,
Thus removing the extra phase
An’(λ) An(λ)
An’(λ)=0
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• One problem: ( )Aλφ λ∇ =does not always have a well-defined (global) solution.
0C
A d λ⋅ =∫ 0C A d λ⋅ ≠∫
Vector flow A
Contour of φ
C
Vector flow
Contour of φ
A φ is not defined here
C
0' ( ')
( ) (0)C
Ti dt t
Ti Ee eγλ λψ ψ
− ∫=
0C C i dλ λγ ψ ψ λλ∂
= ⋅ ≠∂∫
• Berry phase (path dependent)
M. Berry, 1984 : • Parameter-dependent phase NOT always removable!
Index nneglected
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• Berry connection (or Berry potential)
• Berry curvature (or Berry field)
( )A i λλ λλ ψ ψ≡ ∇
( ) ( )F A iλ λ λλ λλ λ ψ ψ≡ ∇ × = ∇ × ∇
C C SA d A daλγ λ= ⋅ = ∇ × ⋅∫ ∫
• Stokes theorem (3-dim here, can be higher)
• Gauge transformation
( )
( ) ( )
( ) ( )
i
C C
e
A A
F F
φ λλ λ
λ
ψ ψ
λ λ φ
λ λγ γ
→
→ −∇
→→
i
i
ii
Redefine the phases of the snapshot states
Berry curvature and Berry phase are gauge invariant
2λ
3λ
1λ
( )tλ
C
S
Some terminology
λ→ k in QHS
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spin × solid angle
Example: spin-1/2 particle in slowly changing B field
BBH Bλ μ σ= = ⋅
y
z
x( )B t
CS
• Real space • Parameter space
Berry curvature
a monopole at the origin
Berry phase
S
1= ( )2
F da Cγ ± ± ⋅ = Ω∫ ∓
yB
zB
xB( )B t
C
E(B)
B
Level crossing at B=0
−
+
2, ,
ˆ1( )2B BB B
BF B iB
ψ ψ± ± ±= ∇ × ∇ = ∓
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Examples of the Berry phase:
-
Magnetic monopole / Berry phase / fiber bundle
( )A k i λ λ λψ ψ≡ ∇
in parameter space
Berry connection
Berry curvature (in 3D)
( ) ( )F Aλλ λ≡ ∇ ×
S
( )
=
C CA d
F da
γ λ λ= ⋅
⋅
∫∫
Berry phase
Total curvature
1 ( ) integer2
F daλπ
⋅ =∫
in real space
Vector potential
Magnetic field
( ) ( )B r A r≡ ∇×
( )A r
( )
=C
S
A r dr
B da
Φ = ⋅
⋅
∫∫
Magnetic flux
Monopole charge
1 ( ) integer4
B r daπ
⋅ =∫
connection
curvature
1st Chern number
horizontal lift (along a U(1) fiber)
U(1) fiber bundle
A
F
γ
C1
(QHE: λ→ k in BZ)
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Connection with geometry First, a brief review of topology:
K≠0
G≠0K≠0
G=0
• extrinsic curvature K vs• intrinsic (Gaussian) curvature G
G>0
G=0
G
-
Euler characteristic 歐拉特徵數
2 ( ), 2(1 )M
da G M gπχ χ= = −∫
• Gauss-Bonnet theorem (for a 2-dim closed surface)
0g = 1g = 2g =
The most beautiful theorem in differential topology
• Gauss-Bonnet theorem (for a surface with boundary)
( ),2gM M M Mda G ds k π χ∂ ∂+ =∫ ∫
Marder, Phys Today, Feb 2007
• Can be generalized to higher dimension.
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• Nontrivial fiber bundleMöbius band
• Trivial fiber bundle(a product space R1 x R1)
Simplest examples:
R1
R1base
fiber
• Fiber bundle ~ base space × fiber space
Fiber bundle: a generalization of product space
• In physics, a fiber bundle ~ Physical space × Inner space
• In QHS, we have T2 x U(1)
• The topology of a fiber bundle is classified by Chern numbers ~the topology of a closed surface is classified by Euler characteristics
(spin, gauge field…)
base space
fiber space
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Lattice electron in a magnetic field: magnetic translation symmetry
consider a uniform B field
Indep of r• Magnetic translation operator
Commute if this is 1
Xiao et al, RMP 2010
B
a1
a2
-
Simultaneous eigenstates: magnetic Bloch states
• If Φ=(p/q)Φ0 per plaquette, then Magnetic Brillouin zone = BZ/q.
e.g., p/q=1/3
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Hofstadter spectrum
Band structure of a 2DEG subjects to both a periodic potential V(x,y) and a magnetic field B.
Can be studied using the tight-binding model (TBM).
B The tricky part:
q=3 → q=29 upon a small change of B!Also, when B → 0, q can be very large.
1
3 110
1029
13
187−
= = +
Surprisingly complex spectrum!Split of energy band depends on flux/plaquette.If Φplaq/Φ0= p/q, where p, q are co-prime integers, then a Bloch band splits to q subbands (for TBM).
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Hofstadter’s butterfly (Hofstadter, PRB 1976)
• A fractal spectrum with self-similarity structure
Self-similarity (heirarchy)
B → 0 near band button, evenly-spaced LLs
• The total band width for an irrational q is of measure zero (as in a Cantor set).
Wid
th o
f a B
loch
ban
d w
hen
B=0
Landau subband
-
集異璧
著作:Douglas R. Hofstadter翻譯:郭維德
Pulitzer 1980M
IT: h
ttp://
ocw
.mit.
edu/
high
-sch
ool/c
ours
es/g
odel
-esc
her-b
ach/
-
C1 = 1
C2 = −2
C3 = 1
Bloch energy E(k) Berry curvature Ω(k) p/q=1/3
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Distribution of Hall conductance among subbands(Thouless et al PRL 1982)
r rr pt qs= +• Diophantine equation
• for rectangular lattice: sr should be as small as possible
• for triangular lattice: sr and tr cannot both be odd(Thouless, Surf Sci 1984)
e.g., / 2 / 52 5
5 2(0) 5(1)4 2(2) 5(0)3 2( 1) 5(1)2 2(1) 5(0)1 2( 2) 5(1)0 2(0) 5(0)
r r
p qr t s
== += += += − += += − += +
See Xiao et al RMP 2010 for another derivation
1
/
H
tot
tot plaq
plaq
H r
necB
N r rnA q A q
BApq hc e
t
σ
σ
∂=
∂
= =
=
∴ =
• Streda formula
• for weak magnetic field: (σH)r = tr - tr-1• for strong magnetic field: (σH)r = sr - sr-1
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Jump of Hall conductance induced by band-crossingLee, Chang, and Hong, PRB 1998
-
Φ=2/5
Lattice with edges
• Energy dispersion of edge states
Hatusgai, J Phys 1997