Imaging transmission of nanostructures in
a high-mobility heterostructure
Clemens Rössler
Thomas Ihn
Klaus Ensslin
Aleksey Kozikov
C. Reichl
W. Wegscheider
Local electron transport
• Classical/quantum
phenomena
• Diffusive/ballistic
transport
Motivation
Ultra high-mobility:
• lp >> L  Ballistic transport:
electron trajectories are straight lines
• Modulation doping technique  Small-angle
scattering:
electron trajectories are wavy lines
How does small-angle scattering affect transport?
Motivation
Conductance, G
2DEG
QPC
y
x
M. Topinka et al. Nature 410, 183-186 (2001)
Motivation
Scannell et al. PRB 85, 195319 (2012)
300 K
115 K
0.24 K
Local relocation of charge between donor sites
Motivation
Conductance through
a tunneling diode
Wilkinson et al. Nature 380,
608 (1996)
Motivation
Experimental data
Filtered data
Crook et al. PRL 91,
246803 (2003)
Motivation
Experimental data
Filtered data
Theory
Aoki
et al. PRL 108,
136804 (2012)
No
one-to-one
correspondence
Sample
n = 1.2 ×
EF = 4 meV
λF = 72 nm
µ = 850 m2/Vs
lp = 49 µm
DStadium = 3 µm
1015
Golden top gates
m-2
Excellent wafers:
C. Reichl
W. Wegscheider
ETH Zurich
1 µm
2DEG
QPC
Ballistic
stadium
Quantum point contact
Top gates
Electron flow
2
Conductance, (2e /h)
D. A. Wharam et al., 1988
B. J. van Wees et al., 1988
6
4
2
0
-0.8
-0.6
-0.4
Gate voltage, (V)
2DEG
SGM technique
Energy
Top gates
Tip
EF
d
2
Conductance, (2e /h)
D. A. Wharam et al., 1988
B. J. van Wees et al., 1988
6
4
2
0
-0.8
-0.6
-0.4
Gate voltage, (V)
Landauer-Büttiker theory
of transport
2DEG
2e2
G
Tn

h n
Backscattering effect
Electron backscattering through the QPC
x
Differential conductance, dG/dx
y
3rd plateau
Vtip= -6.0 V
d = 70 nm
arXiv:1206.1371
1 µm
•
•
•
•
•
•
•
y (µm)
Scanning gate microscopy on a QPC
0.5 µm
Gate voltage dependence
Tip voltage dependence
Tip-surface distance dependence
Temperature dependence
Source-drain bias dependence
QPC asymmetry dependence
X (µm)
Magnetic field dependence: backscattering is
essential
o Strongly varying interference fringe spacing (50%)
Small-angle scattering
arXiv:1206.1371
Scanning gate microscopy on a stadium
dG/dx
y (µm)
1 µm
Vtip= -8.0 V
Vstadium= -0.5 V
X (µm)
Scanning gate microscopy on a stadium
dG/dx
y (µm)
1 µm
Vtip= -8.0 V
Vstadium= -0.8 V
X (µm)
Scanning gate microscopy on a stadium
dG/dx
y (µm)
1 µm
Vtip= -8.0 V
Vstadium= -2.0 V
X (µm)
Scanning gate microscopy on a stadium
G (2e2/h)
dG/dx
1 µm
1 µm
Vtip= -8.0 V
Vstadium= -0.8 V
Scanning gate microscopy on a stadium
dG/dx
500 nm
Scanning gate microscopy on a stadium
dG/dx
dG/dx
G (2e2/h)
Qualitative model
d
a
c
b
Qualitative model
𝑅 𝑇𝑜𝑡𝑎𝑙 = 𝑅 𝑎 ||𝑅 𝑏 + 𝑅 𝑐 + 𝑅 𝑑 + 𝑅 𝑐𝑟
𝑅 𝑇𝑜𝑡𝑎𝑙
d
𝑒2
𝑒2
=
𝑎+ 𝑏
ℎ
ℎ
a
c Rcr
b
contact
resistance
−1
𝑒2
+
𝑑
ℎ
𝑒2
+
𝑐
ℎ
−1
−1
+ 𝑅𝑐𝑟
𝐺𝑇𝑜𝑡𝑎𝑙 = 1/𝑅𝑇𝑜𝑡𝑎𝑙
+
Qualitative model
Assumptions: Rcr= 0, d = ∞
c = 25, W = 0.9 µm, RTip=0.5 µm
𝐺𝑇𝑜𝑡𝑎𝑙
G (2e2/h)
2𝑒 2 (𝑎 + 𝑏)𝑐
=
ℎ 𝑎+𝑏+𝑐
Model vs. experiment
Model
G (2e2/h)
Experiment G (2e2/h)
µ
Dashed lines are guides to the eye
Model vs. experiment
1D profiles along red lines shown in the previous slide
Magnetic field dependence
dG/dx
y (µm)
1 µm
Vtip= -8.0 V
Vcgate= -1.0 V
B = 0 mT
X (µm)
Magnetic field dependence
dG/dx
y (µm)
1 µm
Vtip= -8.0 V
Vcgate= -1.0 V
B = 50 mT
X (µm)
Magnetic field dependence
dG/dx
y (µm)
1 µm
Vtip= -8.0 V
Vcgate= -1.0 V
B = 100 mT
X (µm)
Magnetic field dependence
dG/dx
y (µm)
1 µm
Vtip= -8.0 V
Vcgate= -1.0 V
B = 200 mT
X (µm)
Magnetic field dependence
dG/dx
y (µm)
1 µm
Vtip= -8.0 V
Vcgate= -1.0 V
B = 300 mT
X (µm)
Magnetic field dependence
dG/dx
y (µm)
1 µm
Vtip= -8.0 V
Vcgate= -1.0 V
B = 500 mT
X (µm)
Magnetic field dependence
dG/dx
y (µm)
1 µm
Vtip= -8.0 V
Vcgate= -1.0 V
B = 0 mT
X (µm)
Magnetic field dependence
dG/dx
dG/dx
Dr. Dietmar Weinmann,
Strasbourg, France
Summary (experimental observations)
QPC:
• Backscattering effect
• Interference effect
1 µm
1 µm
500 nm
Ballistic stadium:
• Two fringe patterns
• Conductance fluctuations
Summary (experimental features not
covered by the model)
• Center of the stadium
• Positions of the lens-shaped
regions
• Magnetic field
dependence
THANK YOU
Numerical simulations (top panel) vs. experiment
(bottom panel)
RTip=0.05 µm
Vtip = - 4 V
RTip=0.5 µm
Vtip = - 6 V
G ≈ 17× 2e2/h without the tip
RTip=1 µm
Vtip = - 8 V
Features not explained by simulations
• A region of reduced conductance in the center of the
stadium at low tip biases (experiment)
• Positions of the lens-shaped regions:
inside the stadium in the experiment
in the centers of the constrictions in the simulations
Numerical simulations (B = 0 mT):
same as in the previous slide, but the color scales are different
RTip=0.05 µm
RTip=0.5 µm
RTip=1 µm
SGM technique
Tip
Tip-induced
potential
Energy
μS
Top gates
μD
2
Conductance, G (2e /h)
D. A. Wharam et al., 1988
B. J. van Wees et al., 1988
7
2DEG
6
5
4
3
2
Gating effect
1
0
-0.8
-0.7 -0.6 -0.5 -0.4
Gate voltage, Vg (V)
Influence of the tip on the conductance
VTip = -6 V, B = 25 mT
3
(a)
(b)
2
1
0
-0.9
-0.8
-0.7
Vg (V)
-0.6
2
2
2
2
2
(c)
G (2e /h)
VTip = -6 V, B = 0 mT
G (2e /h)
3
VTip = 0 V, B = 0 mT
G (2e /h)
3
Side branch
Central branch
Off branch
2
2e /h
1
0
-0.9
-0.8
-0.7
Vg (V)
-0.6
1
0
-0.9
-0.8
-0.7
Vg (V)
-0.6
Scanning inside the stadium
Vtip=-8.0 V
Vcgate=-1.0 V
VQPC=0 V
Scanning inside the stadium
Vtip=-8.0 V
Vcgate=-1.0 V
VQPC=-0.38 V
B=0 mT
Profiles
A
B
Vtip=-8.0 V
Vcgate=-1.0 V
B=0 mT
A
B
Left QPC is biased,
3 modes. This is the
case only in this slide.
Profiles
I (nA)
A
B
Vtip=-8.0 V
Vcgate=-1.0 V
B=300 mT
A
B
Profiles
I (nA)
A
B
Vtip=-8.0 V
Vcgate=-1.0 V
B=500 mT
A
B
Magnetoresistance measurements
30
=1
25
R (kOhm)
20
15
=2
10
=3
=4
=5
5
=6
0
0
1
2
3
B (T)
4
5
Magnetoresistance measurements
3.0
2.5
R (kOhm)
2.0
Stadium
voltage
B (mT)
rc (um)
-2.5 V
120
0.48
-2,0 V
100
0.58
80
0.72
60
0.96
40
1.44
10
5.75
1.5
1.0
-1,5 V
-1,2 V
-1,0 V
-0,8 V
0.5
-0,5 V
0.0
0V
0
40
80
120 160 200
B (mT)
240
280
320
Magnetic focusing
80 mT
100 mT
50 mT
B (mT)
rc (um)
120
0.48
100
0.58
80
0.72
60
0.96
40
1.44
10
5.75
Summary (experimental observations)
Scanning gate microscopy on a quantum point contact:
•
•
•
•
Imaging electron backscattering
Observation of branches and interference fringes
Detailed investigation of the branching behaviour
Strongly varying interference fringe spacing
Scanning gate microscopy on a ballistic stadium:
• Two fringe pattern close to the constrictions
• Measurements at high magnetic fields
• Proposed model explains some of the observed
features, but not all of them