Imaging Coherent Electron Flow
Brian LeRoy
Delft University of Technology
Harvard University
June 10, 2005
www.mb.tn.tudelft.nl
[email protected]
Outline
• Introduction
• Measurement technique
• Imaging a quantum point contact
• Interference fringes
• Measuring electron density
• Electron optics
• Electron-electron interactions
• Conclusions
Electrical transport measurements
v. Klitzing et al., PRL 45 494
(1980)
van Wees et al., PRL 60 848 (1988)
Meirav et al., PRL 65 771 (1990)
Fundamental physics but no spatial information
Imaging
Quantum Hall effect
Quantum point contact
Quantum dot
Yacoby et al., Solid State
Comm. 111 1 (1999)
Crook et al., PRL 91 246803 (2003)
Now possible to image electrons in all these regimes
Two-dimensional electron gas
Quantum Point Contact Gate
Ohmic Contact
GaAs/AlGaAs Heterostructure
Two-Dimensional Electron Gas (2DEG)
2DEG located 57 nm below surface
2DEG properties
Free electrons in 2D with a reduced mass
2
k F2 1 * 2
EF =
= m vF
*
2m
2
m* = 0.067 me
Low (tunable) electron density and long Fermi wavelength
2π
λF =
= 39nm
kF
n=
2π
λF2
= 4.2 × 1011 / cm 2
High mobility and long elastic mean free path
µ = 1.0 x 106 cm2/V s
= vF m* µ e = 11µ m
Good review article for basics about 2DEG and quantum point contacts
Beenakker and van Houten, Solid State Physics 44 1 (1991) (cond-mat/0412664)
Fabrication
Clean Chip
Spin PMMA
Evaporate Metal
Expose with
E-beam, Develop
Lift-off PMMA
Quantum point contacts
Electrons flowing through a narrow constriction
Electrostatic gate
Ohmic contact
300nm
Classical
Quantum
Width
Current in 1-D
Calculate the current carried by a single mode
Jn = e
EF +δµ
∫
velocity
ρ n ( E )vn dE
EF
dN 1 1
ρn ( E ) =
=
dE π vF
Density of states
Density of states and velocity cancel
2e 2
δµ =
∂V
Jn =
π
h
e
In 1-D each mode carries the same amount of current
Quantum point contacts-quantization
What is the differential conductance?
∂I 2e 2
G≡
=
h
∂V
∑T
i
i
What are the values of Tn?
⎧0 En > EF
Tn = ⎨
⎩1 En < EF
Assume 1-D particle-in-a-box of width L
En = EF
nλF
when L =
2
Finite temperature blurs the conductance plateaus
∞
df
2e 2
G ( E , T ) = ∫ G ( E , 0)
dE =
dEF
h
0
∞
∑ f (E
n =1
n
− EF )
Quantum point contacts-modes
Conductance vs. Gate Voltage
14
12
10
8
6
4
2
0
Tunneling Regime
-1.2
-1.0
-0.8
Gate Voltage (Volts)
1st Mode
-0.6
T = 1.7 K
2nd Mode
Quantum point contact-energy levels
Conductance
Gate Voltage
6
6
6
5
4
5
4
3
4
1
g = e2/h
3
2
2
g = 2e2/h
0
Vsd
2
1
g = 2e2/h
Quantum point contact-energy levels
Plot of dG/dVg
Red areas are plateaus,
yellow and blue are steep
14
12
10
8
6
4
2
0
-1.2
-1.0
-0.8
Gate Voltage (Volts)
Horizontal distance between edges of
diamonds gives subband spacing
-0.6
Outline
• Introduction
• Measurement technique
• Imaging a quantum point contact
• Interference fringes
• Measuring electron density
• Electron optics
• Electron-electron interactions
• Conclusions
Liquid helium temperature scanning
probe microscope - cantilever
Tip
AFM Cantilever
∆Rc
= 4 x 10-7/Å
Rc
Rc ≈ 2000 Ω
VBias
25µm
R
R
RC
R
Vout
∆R
c
∆R
c
∝
V
Vbias
Vout
µ
V
out
RR Bias
Liquid helium temperature scanning probe
microscope – scan tube
Wheatstone
Bridge
Feedback
Circuit
Cross Section
+Vy
1”
-Vx
Vz
+Vx
Piezoelectric Tube
Scan Range 20µm
-Vy
Liquid helium temperature scanning probe
microscope
Height
50 nm
1µ m
0 nm
Experimental technique
AFM Cantilever
GaAs/AlGaAs
heterostructure
Quantum Point
Contact (QPC) Gate
Two-dimensional
Electron Gas (2DEG)
n-type GaAs
substrate
Perturbation from tip causes scattering
which changes the conductance of the QPC
Image obtained by measuring conductance
through the QPC as a function of tip position
High flow ÆHigh scattering Æ High signal
No flow Æ No scattering Æ No signal
Imaging mechanism
Weak Scatterer
No Backscattering
=
Conductance Unchanged
Strong Scatterer
Backscattering
=
Conductance Reduced
Effect of tip height
10 nm
15 nm
20 nm
150nm
150nm
150nm
25 nm
150nm
30 nm
150nm
35 nm
150nm
1.6 e2/h
∆G: 0 e2/h
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0.00
0
Tip voltage -3.5 Volts
20
40
60
Height (nm)
80 100
Signal only when voltage on
tip depletes the 2DEG
Electrostatic simulation
⎛V
∆n ( ρ ) =
2 ⎜
1 + ( ρ / d ) ⎝ Vd
ns
⎞
εε 0
V
+
⎟
⎠ d + εh
Induced charge
n
Quantum mechanical simulation
Gates
Tip
0.04
0.03
0.02
0.01
0.00
20
40
60
Height (nm)
Signal only when tip depletes 2DEG
80
100
Importance of depletion region
Vtip -3.0V
Vtip -2.6V
Vtip -2.2V
Vtip -1.8V
Vtip 0.0V
50 nm
0 .0 6
0 .0 4
0 .0 2
-3
-2
-1
0
1
Tip Volt age (V )
2
3
Vtip +3.0V
Imaging requirements
Perturbation from tip must backscatter
electrons through the QPC.
This requires that there is an area depleted of
electrons beneath the tip.
Can be accomplished by either bringing tip close
to surface or increasing the tip voltage
Imaging only works with a negative tip voltage
Size of perturbation is set by distance
between the tip and the 2DEG
Approximately Lorentzian in shape
Outline
• Introduction
• Measurement technique
• Imaging a quantum point contact
• Measuring electron density
• Interference fringes
• Electron optics
• Electron-electron interactions
• Conclusions
Imaging a quantum point contact
1st Plateau
∆G: 0 e2/h
2nd Plateau
3rd Plateau
-1.7e2/h
Topinka, LeRoy et al., Science 289, 2323 (2000)
Coherent fringes
Constructive & Destructive Backscattering
Fringes spaced by λF/2
Constructive Interference: 2kFL = 2π n
Destructive Interference: 2kFL = 2π (n+1/2)
Opening QPC
New Modes of current only appear on plateaus
Pattern remains constant between plateaus
Heating electrons
Small Bias Voltage Vbias = 0.2mV
G = 2e2/h
G = 4e2/h
G = 6e2/h
Large Bias Voltage Vbias = 3.0 mV
G = 2e2/h
G = 4e2/h
∆G: 0 e2/h
Large bias voltage heats electrons
G = 6e2/h
1.7 e2/h
Blurred interference fringes
Extracting a modal pattern
Experiment
Theory
Experiment
Theory
Derivation
2nd Mode Flow
2nd Plateau Flow
1st Plateau Flow
Modal summary
Experiment
Theory
1st Mode
2nd Mode
3rd Mode
∆G
|Ψ |2
(percent) (arb unit)
Selective mode suppression
No Tip
Tip Blocking Center
Tip Blocking Side
Tip
Tip
500 nm
500 nm
500 nm
6
4
2
1.7e /h
2
2
2.0e /h
0
Vg (Volts)
Tip selectively modifies transmission coefficient of individual modes
Electron-wave flow through 2DEG
“Idealized” 2DEG Potential
& Electron Wave
Actual 2DEG Potential &
Electron Wave?
2µm
2µm
Operating Point
Operating Point
Coherent electron flow through a 2DEG
1µm
∆G: 0 e2/h
0.4 e2/h
Modal patterns far away from QPC
1st Mode
3rd Mode
∆G: 0 e2/h
2nd Mode
4th Mode
0.4 e2/h
Simulated image
Does the presence of the tip influence the image?
Simulated electron flow
Simulated image
Image the unperturbed electron flow
Unexpectedly high resolution
100nm
Tip Perturbation ~ 120nm FWHM
but
Fringe Resolution ~ better than 10nm
Lateral Resolution ~ better than 20nm
High spatial resolution
Tip Moving Through Current
Unperturbed Current
and Tip Perturbation
Quantum and classical simulations
0.2 EF
Potential, U(x,y)
1µm
-0.2 EF
Classical
1µm
Quantum Mechanical
1µm
Topinka, LeRoy et al., Nature 410, 183 (2001)
Formation of Caustics
Classical Trajectories
Py-y Phase Space
Potential Dip
Moving QPC location
QPC Location shifts by 20 nm
500 nm
Shifting QPC
Unequal voltage on QPC gates
∆G
e2/h
0.0
1
500 nm
2
3
0.4
Difference of the two scans
Gdiff
1
-0.1e2/h
0.0e2/h
500 nm
500 nm
0.1e2/h
Mapping 2DEG potential
Tip Locations
0.5
∆G (e2/h)
Potential?
0.4
0.3
200 nm
0.2
0.1
-4.0
-3.5
-3.0
-2.5
Tip Voltage (Volts)
-2.0
Imaging a QPC
Imaged electron flow near a quantum point contact
Modal pattern associated with
wavefunctions in QPC
Interference fringes spaced by λF/2
Imaged electron flow far from a quantum point contact
Electron flow forms narrow branches
Branches are due to caustics caused by dips
in the potential
Interference fringes spaced by λF/2
Outline
• Introduction
• Measurement technique
• Imaging a quantum point contact
• Interference fringes
• Measuring electron density
• Electron optics
• Electron-electron interactions
• Conclusions
Interference fringes
Presence of interference fringes shows
that electrons are coherent
Use interference to create new types of devices
that rely not only on amplitude of signal but also
the phase information
Want to understand what causes
the interference fringes
Can this interference also be controlled?
Coherent fringes
Constructive & Destructive Backscattering
Fringes spaced by λF/2
Constructive Interference: 2kFL = 2π n
Destructive Interference: 2kFL = 2π (n+1/2)
Finite temperature- thermal smearing
Fermi energy for free electrons in a 2DEG
2
k F2 1 * 2
EF =
= m vF
*
2m
2
m* = 0.067 me
Electrons are not monoenergetic, they have a range of
energy from the finite temperature
Consider two electrons that differ in energy by 2kT
They will have slightly different wavevectors, k
2
E+ ,− = EF ± kT =
k+2,−
2m
*
k + , − = k F ± ∆k
Assume kT<<EF (ignore ∆k2)
m*kT kT
∆k =
=
2
kF
vF
Thermal smearing (continued)
Now suppose these electrons interfere after a distance L
Each of the electrons will have accumulated a phase, kL
So, their phase difference is
2kT
2∆kL =
L
vF
Define the thermal length as when this phase difference is 1
vF
Lt =
2kT
Lt ~ 200 nm @ 4.2 K
The thermal length can be better defined by using the spread in energy of the
Fermi distribution
Thermal smearing (continued)
As the length is increased, the range of accumulated phases
will start to wash out the interference fringes
Each electron accumulates a different phase on the
roundtrip from the tip to the QPC.
Expectation:
Fringes decay with distance from QPC
Note: The electrons have not lost their coherence it
is only being “hidden” by the thermal smearing
Fringe persistence
1D model of fringe persistence, calculate transmission
through two delta functions as a function of position
Fringe amplitude decays with distance
Fringe Size (%)
Fringe persistence
Distance from QPC (nm)
No distance dependence observed
Backscattering from Tip and Impurities...
Calculate wave backscattered to the QPC
Ψ=
Atip e
2 ikRtip
Rtip
Ai e 2ikRi
+∑
Ri
i
Look at the amplitude of terms that vary with tip position
Ψ = 2 Re ∑
2
Ai Atip e
2 ik ( Rtip − Ri )
Ri Rtip
i
Finite temperature Æ Range of wavevectors
signal = ∫ Ψ
2
∂f
dE
∂E
Fringe persistence far from the QPC
Integrand oscillates rapidly with energy except for terms with Ri ≈ Rtip
s ( Rtip ) = 2∑
i
Ai Atip
Ri Rtip
T
cos ⎡⎣ 2k F ( Rtip − Ri ) ⎤⎦e
=
2
− ( Rtip − Ri ) 2 /
2
T
k F π 1/ 2 / 4mkT
Impurity
Tip
Annular band, width =
T
Combined backscattering off the tip and the impurities in the annular
band around rtip produces interference fringes spaced at λF/2*
*S.E.J. Shaw et al, cond-mat/0105354 (2001)
Test this theory with an artificial scatterer
Fringe persistence far from the QPC
Impurity
Tip
Annular band, width = thermal length
Combined backscattering off the tip and the impurities in the annular
band around rtip produces interference fringes spaced at λF/2*
*S.E.J. Shaw et al, cond-mat/0105354 (2001)
Test this theory with an artificial scatterer
Scattering induced fringe enhancement
QPC Gates
Reflector
Impurities
Scan Area
∆G
(e2/h)
0.0
0.05
Vrefl = 0 V
Vrefl = -0.4 V
Vrefl = -0.8 V
Fringe strength analysis
Reflector
Voltage
0.0 Volts
-0.4 Volts
-0.8 Volts
0.00
∆G (e2/h)
0.05
0
FFT (arb. units)
300
Reflector enhances interference fringes
Single Bounce
Fringe movement
100 nm
Double
Reflecting Arc
Single
Double Bounce
100 nm
Movies range from Vrefl = -0.5 V to -1.0
V in steps of 0.02 V
Fringes move twice as fast at
twice the distance from the QPC
LeRoy et al., PRL 94 126801 (2005)
Fringe movement
Single Bounce
Average Peak Movement
0.10 µm/Volt
Double Bounce
Average Peak Movement
0.23 µm/Volt
Interference fringes at twice the radius of the reflecting arc
move twice the speed of the arc
Fringe amplitude
Fringes enhanced
Fringes suppressed
Reflector backscattering suppressed
Reflector backscattering enhanced
Fringe amplitude anti-correlated with reflector conductance
Heating electrons
QPC Gates
Reflector
1.7K
“4.2K”
Scan Area
“8.4K”
Temperature dependence
Increased temperature destroys interference away from reflector
Interference fringes
Spaced by half the Fermi wavelength, λF/2
Due to interference between electrons backscattered by
the tip an ones backscattered from impurities
Persist throughout image because of backscattering from
impurities
Can control their location by introducing artificial impurities
Demonstrated their movement
with a reflecting gate
They become more localized with
increasing temperature
Outline
• Introduction
• Measurement technique
• Imaging a quantum point contact
• Interference fringes
• Measuring electron density
• Electron optics
• Electron-electron interactions
• Conclusions
Measuring density
Electron density measured by Shubnikov-de Haas oscillations
gives the average value over the entire sample.
Usually a Hall bar of
~ 100 microns
2e 1
ns =
h ∆ (1 B )
For nanoscale devices, important to be able to measure the
density in the vicinity of the device
Therefore want a local probe of density
Electron density
Free electrons in two dimensions
2
k2
EF =
2 m*
m* = 0.067 me
Fermi surface is a circle
2
k2
⎛ k ⎞
n = 2π ⎜
⎟ =
2π
⎝ 2π ⎠
Wavelength of interference pattern is related to k
Use interference of electron waves
Changing Fermi energy
Effect of reducing Fermi Energy from 15 meV to 5 meV
500 nm
Reducing Fermi Energy increases relative strength of
bumps and dips in potential
Effect of reducing Fermi energy
Back Gate 0 V
Back Gate -1 V
∆G: 0 e2/h
Back Gate -3 V
Back Gate -5 V
0.15 e2/h
Reducing Fermi energy, increases fringe spacing and
causes flow to be more diffuse
Measuring local electron density
Density Decreases, Wavelength Increases
Back gate -1V
Wafer Profile
Back gate -3V
Back gate -5V
Measured Density
2DEG
1.12 µm GaAs
0.82 µm Al0.3 Ga0.7As
n-type GaAs
Backgate
Parallel plate
capacitor model
∆n
= 0.36 × 1011 cm -2 V
∆V
LeRoy et al., APL 80 4431 (2002)
Is it really local?
Calculate spacing between two adjacent fringes
Phase accumulated, φ
Need phase to change by 2π, must move tip by L
L=
π
k
k is wavevector at this location
Yes it is local, with resolution ~ fringe spacing
Mapping electron density
∆G
(e2/h)
0.00
200 nm
0.15
Average Density, n = 2.95 x 1011 cm-2
Standard Deviation σ = 0.48 x 1011 cm-2
Electron density
Interference fringes provide a way to measure
the local electron density
spacing =
π
2n
Density can be controlled by back gate voltage
Pattern of flow become more diffuse as
density is decreased
Potential felt by the electrons becomes a large
fraction of their energy at low density
Outline
• Introduction
• Measurement technique
• Imaging a quantum point contact
• Interference fringes
• Measuring electron density
• Electron optics
• Electron-electron interactions
• Conclusions
Electron optics
Create elements to electrostatically control electrons
Electrostatic lens
Sivan et al., PRB 41 7937 (1990)
Image how electrons travel through these types of devices
Electron optics theory
Analog to Optical Index of Refraction…
p1Sin(θ1 ) = p2 Sin(θ 2 )
n1 Sin(θ1 ) = n2 Sin(θ 2 )
Electron optics element - prism
1µm
“Refractive switch for two-dimensional electrons”, J. Spector, H. L. Stormer, K. W.
Baldwin, L. N. Pfeiffer, and K. W. West , App Phys Let 56, 2433-2435 (1990)
Electron optics element - ball
Ball -0.5 V
Ball 0.0 V
∆G
e2/h
0.0
0.5
Addition of the ball creates elliptical interference fringes
Electron optics element - channel
∆G
e2/h
0.0
0.75
Simulated Trajectory
Electron trajectories are bent by the potential from the gate
Electron optics
Imaged electrons traveling through the following devices
Electrostatic prism
Reflecting ball
Channel
Can learn about the potential from electrostatic gates
The images show why it is difficult to make efficient
devices with these type of gates
Branches of current limit the ability
to control the flow
Outline
• Introduction
• Measurement technique
• Imaging a quantum point contact
• Interference fringes
• Measuring electron density
• Electron optics
• Electron-electron interactions
• Conclusions
Interactions
To design devices relying on quantum information must
have a way to measure time (distance) over which the
electrons decohere.
At low temperatures, electron-electron interactions limit
the distance electrons remain coherent in the 2DEG
In order to create devices relying on coherence
must be able to measure the electron-electron
scattering time
1
τ ee
∼ ( p − pF ) ln ( p − pF )
2
Images can give a direct measure of this scattering time
DC bias technique
No Energy Loss
Conductance Reduced
Energy Loss
Conductance Unchanged
Voltage applied across QPC
0.0 meV Across QPC
∆g
e2/h
0.0
2.4 meV Across QPC
0.4
Images of electron flow
2.74 µm from QPC
1.96 µm from QPC
50 nm
∆G (e2/h) 0.00
0.13
1.20 µm from QPC
50 nm
50 nm
0.00
0.00
0.35
0.55
EF
EF
Tip
QPC
Movies range from 0.0 mV to 3.1 mV in steps of 0.15 mV
Measuring electron energy
Fringe spacing indicates energy of electrons
Measuring e-e scattering length
Electron-Electron Scattering Time
⎛ 2qTF
EF ⎛ ∆ ⎞ ⎡ ⎛ EF ⎞
1
=
⎜
⎟ ⎢ln ⎜
⎟ + ln ⎜
τ ee 4π ⎝ EF ⎠ ⎣ ⎝ ∆ ⎠
⎝ pF
2
⎞ 1⎤
⎟+ ⎥
⎠ 2⎦
Giuliani and Quinn, PRB 26, 4421 (1982).
Valid at T=0, ∆ is excess energy
What is probability that an electron can travel distance L without a collision?
⎡ L ⎤
⎡ L ⎛ ∆ ⎞2 ⎤
⎥ ~ Exp ⎢ − ⎜
P ~ Exp ⎢ −
⎟ ⎥
⎢⎣ vτ ee ⎥⎦
⎢⎣ v ⎝ EF ⎠ ⎥⎦
Assuming τee < τel
Signal relies on electrons not colliding should decay like a gaussian…
EF
EF
Tip
QPC
Differential conductance vs. excess energy
Signal decays more quickly at longer distances from the QPC
0 meV
1 meV
2 meV
3 meV
0.6 µm
50 nm
∆g
(e2/h)
0.00
0.55
∆g
(e2/h)
0.00
1.4 µm
50 nm
0.13
Differential conductance vs. excess energy
Signal decays more quickly with increasing back gate voltage
0 meV
1 meV
2 meV
3 meV
∆g
(e2/h)
0.00
Back Gate 0V
50 nm
Back Gate -2V
0.35
50 nm
Signal Vs. excess energy
Signal decays more quickly at longer distances from the QPC
Signal decays more quickly with increasing back gate voltage
Interactions
Images are sensitive to whether or not an electron has
lost energy
This is used to measure the distance over which
electrons remain coherent
The measurements show that the scattering time depends
on the Fermi energy and the excess applied energy.
1
τ ee
EF ⎛ ∆ ⎞ ⎡ ⎛ EF
=
⎜
⎟ ⎢ln ⎜
4π ⎝ EF ⎠ ⎣ ⎝ ∆
2
⎛ 2qTF
⎞
⎟ + ln ⎜
⎠
⎝ pF
⎞ 1⎤
⎟+ ⎥
⎠ 2⎦
Acknowledgements
Harvard University
Experiment
Theory
Mark Topinka
Ania Bleszynski
Kathy Aidala
Robert Westervelt
Scot Shaw
Allison Kalben
Eric Heller
University of California – Santa Barbara
Kevin Maranowski
Art Gossard
Conclusions
Image pattern of electron
flow from a QPC
Control interference fringes
Image local electron density
Image electron flow through
electron optics elements
Image electron-electron scattering
Current work
Combined scanning tunneling spectroscopy and
transport measurements of carbon nanotubes
Phonon generation and absorption
LeRoy et al., Nature 432 374 (2004)
Improved fringe resolution
Wiggling Tip Voltage
Fixed Tip Voltage
100nm
100nm
1.1
dI/dVtip (nA/V)
-0.8
0.0
∆G (e2/h)
0.6
Wiggling tip voltage, images the spatial derivative of the flow
Imaging with AC voltage on tip
Tip Voltage Series
100 nm
Tip Voltage goes from 0 to -3.5 V
Bias Voltage Series
100 nm
Bias Voltage goes from -6.25 to 6.25 mV