Download Local density of states in quantum Hall systems with a smooth

Local density of states in quantum
Hall systems with a smooth
disordered potential landscape
Thierry Champel (LPMMC – CNRS/Université Joseph Fourier)
Grenoble - France
SUMMARY:
• The quantum Hall effect
• Vortex Green’s function theory
• Applications: Local spectroscopy
• Conclusion/Perspectives
ACKNOWLEDGMENTS
Theory part
Serge Florens
@ Néel institute – Grenoble/France
Rudolf Römer
@ University of Warwick – Coventry/UK
Mikhail Raikh
@ University of Utah – Salt Lake City/USA
Jascha Ulrich (Diplomarbeit) @ Aachen University – Aachen/Germany
Experimental part
Katsushi Hashimoto
@ Tohoku University – Sendai/Japan
Markus Morgenstern
@ Aachen University – Aachen/Germany
WHY STUDY 2D ELECTRON GASES UNDER MAGNETIC FIELDS NOW?
IQHE :Von Klitzing et al. (1980)
T < 1 Kelvin
B > 1 Tesla
8
6
Edge states
explanation:
En(X)
4
µ
2
0
-40
-20
0
x/lB
20
40
Experiments since 2000 (far from being exhaustive):
New effects: microwave induced zero-resistance states
New probes: local sensing techniques in the IQHE regime
New systems: graphene, topological insulators, 2DEG surface states
WHY STUDY 2D ELECTRON GASES UNDER MAGNETIC FIELDS NOW?
STM Experiment: Local DOS in the IQHE regime (InSb surface states)
B=12 T
(LOWEST LANDAU LEVEL)
• Percolation features
• Broad structures close to saddle
points of the potential landscape
Disorder plays an important role!
Theory:
Many fundamental aspects (e.g. for the IQHE) well understood
But: how do we calculate stuff? (quantitative microscopic theory to develop!)
This talk
Goal:
Find an (approximate) analytical solution to the problem
arbitrary potential energy
H = H0 + V (r)
H0 =
1
2m∗
2
e
−i∇r − c A(r)
∇ × A(r) = Bẑ
Theoretical difficulties:
Disorder averaging is questioned (at microscopic scale)
question of origin of irreversibility and dissipation (crucial for transport)
We are in a nonperturbative regime at high magnetic fields
(kinetic energy frozen + degeneracy of Landau levels)
Smooth disorder (finite correlation length)
Complexity of diagrammatrics at high magnetic fields (unsolved problem)
Raikh & Shabhazyan, PRB (1993)
We are at the border between classical and quantum mechanics
The wave function as a basic dynamical object is questioned
Need to develop a new approach/method to tackle the problem
Standard theoretical
approaches (I):
Semiclassical limit
CLASSICAL MOTION IN HIGH PERPENDICULAR MAGNETIC FIELD
Two degrees of freedom with very different timescales
• fast cyclotron motion: θ̇ = ωc = |e|B/(m∗ c)
• slow drift:
vd =
1
BE ×
ẑ
Decoupling in the limit B → ∞
Edges: delocalized
skipping orbits
Disordered bulk: localization on closed equipotential lines
Remark: motion regular and integrable in the limit B → ∞ !
Averaging over disordered potential configurations questionable here!
SEMICLASSICAL MOTION : THE GUIDING CENTER PICTURE
x = X + ζ = X + vy /ωc
y = Y + η = Y − vx /ωc
v
change in variables:
(x, px ), (y, py ) → (X, Y ), (ζ, η)
H = 12 m∗ v2 + V (X + ζ, Y + η)
then quantization
Semiclassical high field picture (V smooth):
(X and Y treated as classical variables)
Rc
(x, y)
(X, Y )
[v̂x , v̂y ] = −iωc /m∗
2
[X̂, Ŷ ] = ilB
2 = c/(|e|B) → 0
lB
[X̂ , Ŷ ] → 0
Effective energy: En,R = ωc (n + 1/2) + V (R)
Limitations:
• No quantization of energies (e.g. in quantum dot)
• No transverse spread + no tunneling effects (e.g. in QPC)
• Problems to formulate a consistent transport theory
• Captures only the high temperature regime
LDoS in the IQHE regime follows potential landscape
Hashimoto et al., (2008)
but quantum percolation features
MOTIVATION FOR A HIGH MAGNETIC FIELD EXPANSION
At large magnetic field:
Magnetic length
lB = 8 nm at 10 T
Correlation length of the disordered potential in heterostructures:
ξ ≥ 100 nm
The random potential is smooth on the scale lB
The idea of using lB /ξ as a small parameter is not new. The real challenge is to go
beyond the strict limit lB /ξ = 0 !
Some attempts:
• Effective Hamiltonian theory
Haldane & Yang, PRL (1997)
Apenko & Lozovik, J. of Phys. C (1984)
- limited to energy
- includes only virtual transitions = no Landau-level mixing taken into account
Standard theoretical
approaches (II):
Wave functions
TRANSLATION INVARIANT LANDAU STATES
H0 =
1
2m∗
2
e
−i∇r − c A(r)
2
Ψn,k (x, y)
∇ × A(r) = Bẑ
with
Landau states:
take A(r) = xBŷ
En,k = ωc (n + 12 )
iky
Ψn,k (x, y) = e
2 2
(x−klB
)
x−kl2B
Hn
exp − 2l2
lB
B
Landau (1930)
Translationally invariant along y
0
y
x
Localized on the scale l B along x
Remarks:
- Huge degeneracy of Landau levels
- Magnetic field enters in wave functions only via lB = c/|e|B
- Landau states problematic for quantum/classical correspondence
CIRCULARLY INVARIANT STATES
Other possible eigenstates of
H0
Circular states:
2
Ψ
(r,
θ)
l,m
El,m = ωc (l +
take A(r) = B × r/2
|m|+m+1
)
2
Ψl,m (r, θ) = r|m| exp
2
−r
4l2B
|m|
Ll
2
r
2l2B
Rotationally invariant around the origin
0
y
Localized on a scale l B along r
x
Remark:
- States still problematic for quantum/classical correspondence at lB
→0
eimθ
DIGRESSION ON THE LANDAU LEVEL INDEX
What is the physical meaning of the Landau level index n?
2
Translation-invariant states: n comes from |Ψ| → 0 (degree of Hermite polynomial)
Rotation-invariant states:
n=l+
|m|+m
2
angular momentum (single-valuedness)
degree of Laguerre polynomial
v
A semi-classical answer:
1 ∗ 2 1 ∗
E =
m v = m (Rc ωc )2
2
2
Rc2 m∗ ωc
Φc
= ωc
= ωc
2
Φ0
1
To compare with E = ωc n + 2
Rc
(x, y)
(X, Y )
Φc = πRc2 B
Φ0 = hc/|e|
Φc = n Φ0
ωc = |e|B/(m∗ c)
Cyclotron Flux
is quantized
Can we build the right basis of eigenstates where n is only related to the accumulated
phase?
YES : the vortex states basis
Champel & Florens, PRB (2007)
VORTEX (SEMI-COHERENT) EIGENSTATES
Other possible eigenstates of
Vortex states: Ψm,R (r) = r|m, R
H0
2
Ψm,R (x, y)
Ψm,R (r) = |r − R|m eim arg(r−R)
(r − R)2 − 2iẑ · (r × R)
× exp −
2
4lB
0
y
Em,R = ωc (m + 12 )
x
R1 |R2 Overcomplete semicoherent states basis:
m1 , R1 |m2 , R2 = δm1 ,m2
2
(R1 − R2 ) − 2iẑ · (R1 × R2 )
exp −
2
4lB
+∞ 2
d R
2 |m, Rm, R| = 1
2πl
B
m=0
Remarks:
- States OK for quantum/classical correspondence
- States with no preferred symmetry: can adapt to arbitrary V(r)
SemiSemi-coherent vortex states
Green’
Green’s function formalism
Champel, Florens & Canet, PRB (2008)
Champel & Florens, PRB (2009)
Champel & Florens, PRB (2010)
THEORY: VORTEX GREEN’S FUNCTIONS
Our approach: electron dynamics projected in the vortex representation
Vortex states:
Ψm,R (r) = r|m, R
+∞ 2
d R
2 |m, Rm, R| = 1
2πl
B
m=0
Exact result:
′
G(r, r , ω) =
d2 R ′
K
(R,
r,
r
) g̃m1 ;m2 (R, ω)
m
;m
1
2
2
2πlB m m
1
−
=e
known
l2
B
4
« Vortex Dyson’s » equation:
+
to determine
2
∆R
∗
′
Ψm2 ,R (r )Ψm1 ,R (r) (cyclotron motion)
(ω − Em1 + i0 )g̃m1 ;m2 (R, ω) = δm1 ,m2 +
ṽm1 ;m3 (R) ⋆ g̃m3 ;m2 (R, ω)
m3
2 ←
− −
→
←
− −
→
l
with the star-product ⋆ = exp i B ∂ X ∂ Y − ∂ Y ∂ X
2
connection to the deformation (Weyl) quantization theory
THEORY: VORTEX GREEN’S FUNCTIONS
High magnetic field regime: (ωc → ∞ while keeping lB finite)
LL mixing negligible
Effective potential
2 ←
−
−
→
←
−
−
→
l
⋆ = exp i B ∂ X ∂ Y − ∂ Y ∂ X
(ω − Em + i0+ )g̃m (R) = 1 + ṽm (R) ⋆ g̃m (R)
2
+ −1
Trivial for 1D potentials: g̃m (R) = [ω − Em − ṽm (R) + i0 ] Trugman, PRB (1983)
Raikh & Shahbazayan, PRB (1995)
Some non trivial questions: - How to get quantized energies for a closed system?
- How to get tunneling effects in QPC?
V (R) = V (R0 ) + [R − R0 ] · ∇V (R0 ) + 12 [(R − R0 ) · ∇]2 V (R0 )
everything is encoded
in quadratic (curvature)
terms!
Dyson’s equation up to second-order derivatives of V:
2m + 1 2
1 = ω − Em − V (R) −
lB ∆R V + i0+ g̃m (R)
4
4 lB
2
2
2
2
+
∂ V ∂X + ∂X V ∂Y − 2∂X ∂Y V ∂X ∂Y g̃m (R)
8 Y
This ugly equation can be
exactly solved!
Champel & Florens, PRB (2009)
EXACT SOLUTION FOR ANY QUADRATIC POTENTIAL
Solution (m=0):
g̃m (R) = −i
where
γ
=
η(R) =
4
lB
4
4
lB
8
0
+∞
dt
√
√
i η(R)
γ [t−tan( γt)/ γ]
e
e
√
cos( γt)
2
∂XX V ∂Y Y V − (∂XY V )
2
it[ω−V (R)−lB
∆V (R)/4+i0+ ]
Related to the Gaussian curvature of V
2
2
∂XX V (∂Y V ) + ∂Y Y V (∂X V ) − 2∂XY V ∂X V ∂Y V
Solution embraces all
possible cases of quadratic
potentials
Stability of vortex
quantum numbers
γ>0
Solution periodical in time
energy quantization
Open and closed quantum mechanics unified!
γ=0
γ<0
Solution with lifetime
tunneling and
dissipation
Champel & Florens, PRB (2009)
Applications of vortex
formalism (I):
Local density of states
Champel & Florens, PRB Rapid Com (2009)
Champel & Florens, PRB (2009)
Champel & Florens, PRB (2010)
Hashimoto, Champel, Florens, et al., PRL (2012)
LOCAL DENSITY OF STATES
Vortex view of LDoS at high field:
1
ρ(r, E) = −
π
g̃m,m = g̃m δm,m
+∞
d2 R Fm (R − r) Im g̃m (R, E)
2
2πlB
m=0
cyclotron orbit
(−1)m
with structure factor: Fm (R) =
2 Lm
πlB
m=0
m=1
2R2
2
lB
guiding center drifting
−R2 /l2B
e
can be negative
(Wigner’s distribution)
m=2
Lowest order result for vortex Green’s function (local 1D drift):
m=3
for smooth potential
and lowest LL
− π1 Im g̃m (R, E) = δ[E − Em − ṽm (R)]
2
= d rFm (R − r)V (r) ≈ V (R)
HIERARCHY OF LOCAL ENERGY SCALES IN VORTEX REPRESENTATION
Tip
LDoS for the lowest Landau level at the center (r = 0)
within different approximation schemes
Example:
U0 2
V (r) =
r
2
ρSTS (r, µ, T )
1000
Semiclassical
Gradient
Curvature
100
ρ(r, µ, T ) = −
10
1
n′F [V (r)]
2
2πlB
γ = η = 0 (lowest order,
curvature neglected)
1
0.001
exact expression
0.01
0.1
1
T /h̄ωc
Existence of a hierarchy of local energy scales
Champel & Florens, PRB (2009)
Champel & Florens, PRB (2010)
controlled theory for a
smooth arbitrary potential !
APPLICATION: LDOS IN GRAPHENE
graphene
Theory: LDoS for different temperatures
(with lowest order vortex Green’s function)
Experiment (at fixed Temp)
ρST S (r, ε, T )
7.5
5
2.5
0
−2
−1
Miller et al., Science (2009)
Experimental features captured by theory:
0
ε/h̄Ωc
1
2
Champel & Florens, PRB (2010)
the width of the LDoS peaks at fixed tip position grows roughly as
the heights of the LDoS peaks decrease with m
Origin: wave function broadening
m
at fixed tip position
What about spatial dependence of LDOS?
REAL SPACE LDOS DATA
InSb surface states
B=6 T
Hashimoto, Champel, Florens et al., PRL (2012)
4 successive LLs are observed (spin resolved)
The drift trajectories are blurred in the high LLs
… but no obvious signature of the nodal structure associated to cyclotron motion
MOMENTUM-SPACE LDOS DATA
InSb surface states
B=6 T
Structures appear at scale
Hashimoto, Champel, Florens et al., PRL (2012)
1/lB ≈ 0.1 nm−1
LLn shows n kinks in the momentum-dependence
Good comparison experiment/simulations
REVEALING THE NODAL STRUCTURE OF CYCLOTRON MOTION
Guiding-center spectral density (disorder-dependent, length scale ~ 50 nm)
LDOS (theory):
ρ(r, E) =
+∞
d2 R Fm (R − r) Am (R; E)
2
2πlB m=0
for B=6 T
Structure factor for cyclotron motion
(length scale lB ~ 10 nm)
Deconvolution in Fourier space:
ρ̃(q, E) =
+∞
F̃m (q) Ãm (q; E)
n=0
where
InSb surface states
F̃n (q) = Lm
2 2
q
lB
2
e
m=3
2 2
−lB
q /4
m=2
Kinks of ρ̃(q, E) follow the nodes of F̃m (q)
m=1
The nodal structure of LLs is robust to disorder
m=0
Key property of quantum Hall states!
Hashimoto, Champel, Florens et al., PRL (2012)
Other Applications of
vortex formalism (II):
Averaged density of states and LDOS correlations
Champel & Florens, PRB (2010)
Champel, Florens & Raikh, PRB (2011)
Ulrich, Florens & Champel, in preparation (2012)
SAMPLE AVERAGED LDOS
Disorder correlator (Gaussian)
2 −
V (R1 )V (R2 ) = v e
|R1 −R2 |2
ξ2
DOS with lowest order vortex result:
2
+∞ −( ω−Em )
Γm
1
e
√
ρ(r, ω) =
2
2πlB m=0
πΓm
with
Γ2m =
π 2 24
v ξ lB
2
2 2
2 2
lB q
2
− ξ 4q
d q F̃m
e
2
The lowest order Green’s function
is exact at ξ ≫ lB
Opposite limit ξ ≪ lB
analytically solved by Wegner
most stringent test!
LDOS CORRELATIONS (I)
Tip 1
r1 , ω1
r
Two-point LDoS correlator:
Tip 2
r2 , ω2
Perform the following sample averaging of LDOS-LDOS signal
χ(r, ω1 , ω2 ) = ρ(r1 , ω1 )ρ(r2 , ω2 ) − ρ(r1 , ω1 )ρ(r2 , ω2 )
Geometrical interpretation: overlap of quantum rings
a)
b)
11
00
1
r2
r1 0
11
00
11
r1 00
r2
Area for c) > Area for b)
lB
c)
d)
0
1
0
r1
1
0
1
0
r1
2
R
11
00
00
11
r1
L
11
00
00
11
r2
Robust way to reveal some nodes in real space?
χ peaks again for r ≈ 2Rc
LDOS CORRELATIONS (II)
Procedure for computation
+∞ +∞
d2 R2 ρ(r1 , ω1 )ρ(r2 , ω2 ) =
Fm1 (R − r1 )Fm2 (R2 − r2 )
2
2πlB m =0 m =0
1
2
1
dt1
dt2 i(ω1 −Em )t1 +i(ω2 −Em )t2 −i[ṽm1 (R1 )t1 +ṽm2 (R2 )t2 ] 1
2
×
e
e
2π
2π
2
d R1
2
2πlB
this can be done analytically!
Spatial dependence confirms previous expectations
1
ξ/lB = 5
ω = E0
ω = E1
ω = E2
0.4
semiclassics
χ(r, En , En )
χ(r, ω, ω)
0.6
0.2
0.75
lB = 0
0.5
0.25
0
0
0
1
2
3
r/lB
4
5
6
0
1
2
3
r/RnL
Champel, Florens, Raikh, PRB (2011)
LDOS CORRELATIONS (III)
Energy dependence: DOS versus LDOS correlations at equal position
has broad peaks with width ∼ V 2 has narrow resonances with width ∼ (lB /ξ)
V 2 Positive correlations if ∆ω = nωc , negative otherwise
Champel, Florens, Raikh, PRB (2011)
Prospects: compare analytic theory with experiments and numerics
PERSPECTIVES: PROBABILITY DISTRIBUTION FOR THE LDOS
Probablity distribution for the LDOS:
On-going work with J. Ulrich and S. Florens
P (ρ) =
∞
dλ iλρ (−iλ)n
[ρ(ω, r = 0)]n e
2π
n!
n=0
Higher moments of the LDOS correlations
can be computed analytically at ξ ≫ lB
Preliminary results:
Assumptions: LL0 + additional external broadening (temperature T)
p = ρ/ρ
T ∗ = (lB /ξ)v
Wide-stretched distribution at T ≪ T ∗
Sharply peaked distribution at high T
(LDOS gets more and more uniform
accross the sample)
CONCLUSION
The rigorous formulation of a quantum guiding center theory was
established in terms of semi-coherent state Green’s functions
The overcompleteness of the vortex representation
makes possible the unification of closed and open
systems (bulk and edge states on the same footing)!
Local equilibrium observables such as the LDOS can be calculated accurately
from systematic gradient expansion using semi-coherent state Green’s functions
STM experiments show percolating states in 2DEG at high magnetic
fields, and revealed the robust nodal structure of Landau levels
Theory works well for capturing disordered averaged quantities
(averaged DOS, LDOS correlations, probability distribution, …)