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Laurens W. Molenkamp
Physikalisches Institut, EP3Universität Würzburg
Overview
- HgTe/CdTe bandstructure, quantum spin Hall effect- HgTe as a Dirac system- Dirac surface states of strained bulk HgTe
band structure
D.J. Chadi et al. PRB, 3058 (1972)
fundamental energy gap
meV 30086 EE meV 30086 EE
semi-metal or semiconductor
HgTe
-1.0 -0.5 0.0 0.5 1.0k (0.01 )
-1500
-1000
-500
0
500
1000
E(m
eV) 8
6
7
-1.0 -0.5 0.0 0.5 1.0k (0.01 )
-1500
-1000
-500
0
500
1000
E(m
eV) 8
6
7
Eg
Type-III QW
VBO = 570 meV
HgCdTeHgCdTeHgTe
HgCdTe
HH1E1
QW < 63 Å
HgTe
inverted normal
band structure
conduction band
valence band
HgTe-Quantum Wells
Layer Structure
gate
insulator
cap layer
doping layer
barrier
barrierquantum well
doping layer
buffer
substrate
Au
100 nm Si N /SiO
3 4 2
25 nm CdTe
CdZnTe(001)
25 nm CdTe10 nm HgCdTe x = 0.79 nm HgCdTe with I10 nm HgCdTe x = 0.74 - 12 nm HgTe10 nm HgCdTe x = 0.7 9 nm HgCdTe with I10 nm HgCdTe x = 0.7
symmetric or asymmetricdoping
Carrier densities: ns = 1x1011 ... 2x1012 cm-2
Carrier mobilities: = 1x105 ... 1.5x106 cm2/Vs
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 80
100
200
300
400
500
µ=1.06*106cm2(Vs)-1
nHall=4.01*1011cm-2
Q2134a_Gate
B[T]
Rxx
[]
-15000
-10000
-5000
0
5000
10000
15000
Graph2
Rxy
[]
123456
k (0.01 -1)
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.20
Ener
gyE(
k)(e
V)
k || (1,1)k || (1,0)k = (kx,ky)
k || (1,1)k || (1,0)k = (kx,ky)
4 nm QW 15 nm QW
normal
semiconductor
inverted
semiconductor
1 2 3 4 5 6
k (0.01 -1)
-0.20
-0.15
-0.10
-0.05
0.00
0.50
0.10
0.15
0.20
E2
H1H2
E1L1
0.6 0.8 1.0 1.2 1.4
dHgTe (100 )
E2E2
E1E1H1H1
H2H2H3H3
H4H4 H5H5
H6H6L1L1
Band Gap Engineering
Bandstructure HgTe
E
k
E1
H1
invertedgap
4.0nm 6.2 nm 7.0 nm
normalgap
H1
E1
B.A Bernevig, T.L. Hughes, S.C. Zhang, Science 314, 1757 (2006)
QSHE, Simplified Picture
normalinsulator
bulk
bulkinsulating
entire sampleinsulating
m > 0 m < 0
QSHE
-1.0 -0.5 0.0 0.5 1.0 1.5 2.0103
104
105
106
G = 2 e2/h
Rxx
/
(VGate- Vthr) / V
Observation of QSH Effect
(1 x 0.5) m2
(1 x 1) m2(2 x 1) m2
(1 x 1) m2
non-inverted
M. König et al., Science 318, 766 (2007).
1 m 2 m
1 2 3
6 5 4
1 m
1 m
5 m
1 2
34
(a) (b)
Verify helical edge state transport
Multiterminal /Non-local transport samples
Multi-Terminal Probe
210001121000012100001210000121100012
T
heIG
heIG
t
t
2
23
144
2
14
142
232
generally
22 2)1(
ehnR t
3exp4
2 t
t
RR
heG t
2
exp,4 2
Landauer-Büttiker Formalism normal conducting contacts no QSHE
Configurations would be equivalent in quantum adiabatic regime
-1 0 1 2 30
5
10
15
20
25
30
35
40
R (k
)
V* (V)
I: 1-4V: 2-3
R14,23=1/2 h/e2
R14,14=3/2 h/e2
I: 1-3V: 5-6
R13,13=4/3 h/e2
R13,54=1/3 h/e2
-1 0 1 2 3 4
V* (V)
Multi-Terminal Measurements
A. Roth et al., Science 325, 294 (2009).
0.0 0.5 1.0 1.5 2.00
5
10
15
20
25
R
(k
)
V* (V)
I: 1-4V: 2-3
1
3
2
4
R14,23=1/4 h/e2
R14,14=3/4 h/e2
Non-Local data on H-bar
A. Roth et al., Science 325, 294 (2009).
H-bar for detection of Spin-Hall-Effect
(electrical detection through inverse SHE)
E.M. Hankiewicz et al ., PRB 70, R241301 (2004)
200 nm 200 nm
-2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.00
100
200
300
400
500
0
2
4
6
8
10
12
R12
,36
()
Vg (V)
I (n
A)
-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.00.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
T = 2 K
Rno
nloc
al /
kVGate / V
I / n
A
n-conductingp-conducting
insu
latin
g
– Suppress non-local QSHE using long leads or narrow wires
– Intrinsic metallic SHE only shows up for holes: larger spin-orbit
– Amplitude in agreement with modeling (E. Hankiewicz, J. Sinova)
H-bar experiments
C. Brüne et al., Nature Physics 6, 448 (2010).
-1,5 -1,0 -0,5 0,0 0,5 1,0 1,5 2,00
100
200
300
400
500
R21
,36
()
Vg* (V)
-1,5 -1,0 -0,5 0,0 0,5 1,0 1,5 2,00
100
200
300
400
500
R36
,21
()
Vg* (V)-1,5 -1,0 -0,5 0,0 0,5 1,0 1,5 2,0
0,0
0,5
1,0
1,5
2,0
R45
,21
(k
)
Vg* (V)
-1,5 -1,0 -0,5 0,0 0,5 1,0 1,5 2,00,0
0,2
0,4
0,6
0,8
1,0
1,2
1,4
1,6
R21
,45
(k
)
Vg* (V)
Q2197(a)
p-cond. insul. n-cond.
p-cond. insul. n-cond.
p-cond. insul. n-cond.
Q2198(b)
p-cond. insul. n-cond.
I
V
I
V
I
V
I
V
C. Brüne et al., Nature Physics 6, 448 (2010).
Bandstructure HgTe
E
k
E1
H1
invertedgap
4.0nm 6.2 nm 7.0 nm
normalgap
H1
E1
B.A Bernevig, T.L. Hughes, S.C. Zhang, Science 314, 1757 (2006)
Dispersion at d=dc is Dirac-like
For well thickness d=6.3 nm, the gap closes,especially the conductionband shows a linear dispersion: single Dirac cone
Zero mode dispersion
Zero mode spin splitting allows to select sample at dc.
B. Büttner et al., Nature Physics doi:10.1038/nphys1914
Large g-factor (g=55) responsible for spin slitting already at low fields.Hall quantization reflects single valley character of the band structure:a HgTe quantum well at d=6.3 nm is half-graphene.
B. Büttner et al., Nature Physics doi:10.1038/nphys1914
Quantum Hall effect shows Berry phase
Landau-fan
Color coded: gate voltage derivative of longitudinal resistivity.Fits: left – 8-band Kane model, right – Dirac Hamiltonian
B. Büttner et al., Nature Physics doi:10.1038/nphys1914
Dirac peak at B=0
Peak width and mobilities comparable with/better than free standing grapheneScattering mechanisms: probably mass fluctuations + Coulomb (fit is Kubo model)
Mobility for finite Dirac mass
B. Büttner et al., Phys. Rev. Lett. 106, 076802 (2011).
Originally increase in mobility from reduced impurity scattering, then changeover to behavior due to well width (Dirac mass) fluctuations.
Mobility for finite Dirac mass
B. Büttner et al., Phys. Rev. Lett. 106, 076802 (2011).
Modeling by Grigory Tkachov and Ewelina Hankiewicz:Mass and disorder induce backscattering of Dirac fermions.
Bulk HgTe as a 3-D Topological ‚Insulator‘
-1.0 -0.5 0.0 0.5 1.0k (0.01 )
-1500
-1000
-500
0
500
1000
E(m
eV) 8
6
7
-1.0 -0.5 0.0 0.5 1.0k (0.01 )
-1500
-1000
-500
0
500
1000
E(m
eV) 8
6
7
Bulk HgTe is semimetal,
topological surface state overlaps w/ valenceband.
k(1/a)
E-E
F(eV
)ARPES:
Yulin Chen, ZX Shen, Stanford
C. Brüne et al., Phys. Rev. Lett. 106, 126803 (2011).
70 nm layer on CdTe substrate:coherent strain opens gap
0 2 4 6 8 10 12 14 160
2000
4000
6000
8000
10000
12000
14000
16000
0
2000
4000
6000
8000
10000
12000
14000
Rxx (SdH)
Rxx
in O
hm
B in Tesla
Rxy (Hall)
Rxy
in O
hm
Bulk HgTe as a 3-D Topological ‚Insulator‘
@ 20 mK: bulk conductivity almost frozen out - Surface state mobility ca. 35000 cm2/Vs
C. Brüne et al., Phys. Rev. Lett. 106, 126803 (2011).
2 4 6 8 10 12 14 160
2
4
6
8
10
12
14
xy [e
2 /h]
B [T]
Bulk HgTe as a 3-D Topological ‚Insulator‘
@ 20 mK: same data, plotted as conductivity
3D HgTe-calculations
2 4 6 8 10 12 14 160
2000
4000
6000
8000
10000
2.73.54.45.67.69.711 33.94.96.78.510.112
experiment
Rxx
in O
hm
B in Tesla
n=3.7*1011 cm-2
n=4.85*1011 cm-2
n=(4.85+3.7)*1011 cm-2
DO
S
Red and blue lines : DOS for each of the Dirac-cones with the corresponding fixed 2D-density,Green line: the sum of the blue and red lines
C. Brüne et al., Phys. Rev. Lett. 106, 126803 (2011).
Conclusions– HgTe quantum wells: normal and inverted gap, linear (Dirac) dispersion
– First observation of Quantum Spin Hall Effect
– At d=dc, a HgTe QW is ideal model system for zero massDirac fermion physics
– Can conveniently study Dirac fermions w/ finite Dirac mass
– Strained 3D layers show QHE of topological surface statesCollaborators:Bastian Büttner, Christoph Brüne, Hartmut Buhmann, Markus König, Matthias Mühlbauer, Andreas Roth, Volkmar Hock
Theory: Alina Novik, Chaoxing Liu, Ewelina Hankiewicz , Grigory Tkachov,Patrick Recher, Björn Trauzettel (all @ Würzburg), Jairo Sinova (TAMU), Shoucheng Zhang, Xiaoliang Qi (Stanford)
Funding: DFG (SPP Spintronics, DFG-JST FG Topotronics), Humboldt Stiftung, EU-ERC AG