Ballistic electron spectroscopy
Frank Hohls
I. Introduction – low dimensional electron
systems and ballistic electrons
II. The idea – ballistic electron spectroscopy
III. Charge readout of detector
IV. Some data – proof of concept
2d electrons – where do I find them?
Confining electrons to an interface
Electrons in metal: thin metal films or surface
- high electron density, short length scales
(Fermi length, mean free path)
- ultra high vacuum
standing
waves on
metal surface
Electrons in semiconductor:
Field effect transistor (Silicon-MOSFET)
+ easy tunable electron density
Quantum Hall Effect (QHE)
Nobel prize 1985
Heterostructure grown atomic layer by layer
(Nobel prize 2000)
+ Very clean system
Fractional QHE, Nobel prize 1998
ballistic electron effects
2d-electrons
2d electrons in a heterostructure
Engineering of band edge to
create a quantum well
Quantized energy in
growth direction (z)
Free motion in x-y plane
conduction band
E
Electron density n ~ 1015 m-2
Electron distance ~ 30 nm
Fermi wave length ~ 80 nm
High interface quality and
remote doping
Very high mobilty µ
(measured from σ = enµ)
Mean free path of up to
100 µm
band gap
∆E ~ 100 meV
10-20 nm
z (growth)
Getting smaller – gating the 2d electrons
metal gate
Metal on GaAs forms Shottky Diode
Current I ~ exp(eVG/kT) negligible
for VG<0 and low T
Gate and 2d electron sheet form plate
capacitor
C = εε0·A/d, Δq = C·VG = A·e·Δn
Change of electron density n by
applied voltage
1 gate + 2 contacts = MESFET
Depletion of electrons beyond
threshold voltage Vdep
d
ε = 13
VG
2d electrons
GaAs
n
Vdep
-VG
gate
Transfer gate pattern into 2DES
depletion
2DES
Ballistic electrons in 2d – focusing
focusing signal
Electrons travel large distance
without scattering
ballistic transport
Revealed in magnetic focusing
experiments
Lorentz force ~ B·vF in
perpendicular magnetic field B
Electrons forced on cyclotron
orbits in with rC ~ 1/B
Spector et al '90

B

F
van Houten et al '88
v
−e
Ballistic electrons – optics and interference
Fermi wave vector kF ~ √n
can be changed by gate
Electron optics
Wave nature of electrons
interference effects
Double slit
interferometer with
magnetic field
Aharonov-Bohm effect
Δφ~Φ/Φ0 (Φ flux)
Spector et al '90
Electron lens
Analogues to light optics
lens
double slit
Schuster et al '97
electron interferometer
Ballistic 1d wire – quantized conductance
Ballistic wire: width w and
length L much smaller than
scattering length l
no backscattering within
channel
quantized conductance
(Wharam et al '88,
van Wees et al '88)
G = N·(2e2/h)
Thomas et al
1d wire – why quantized in 2e2/h ?
Quantized energy perpendicular to wire (y)
Free motion in 1d along wire (x)
y
For one 1d sub-band:
v
Density of states 1d (no spin):
D  E =
Current

m
1
=
2 h2 E h v
y
2
e
I =e⋅v⋅[ D E ⋅ E ] = V
h
E
Add spin degeneracy g = 2
2
I 2e
G= =
V
h
Add contributions of N occupied 1d sub-bands
G = N (2e2/h)
Conductance steps smoothed for short wire
Quantum point contact (QPC)
2d
1d
2d
ΔE = eV
E0
x
Open questions: Electron interaction in 1d
Additional conductance plateau
at G = 0.7·(2e2/h)
Effect of electron-electron
interaction
Different models proposed
Kondo model
Spontaneous spin
polarization
Jumping of subbands
For all models we expect change
of density of states D(E)
How to measure without
changing the system (no backaction) ?
Thomas et al
PRL ‘96
Proposal: ballistic electron spectroscopy
V
µD
µS = µD + eV
T(E)
2d
object of
interest
ballistic
electrons
2d
spectrometer
and detector
E
µS
eV
T(E)
2d
µD
2d
Source non-equilibrium
ballistic electrons by an
object of interest
Energy distribution maps
Transmission T(E)
Inner structure mirrored
by transmission
Energy resolved detection of
ballistic electrons
No back-action of
measurement on object of
interest
Use quantum dot as
spectrometer
APL 89, 212103 (06)
Quantum dot – electrons in 0d
Confinement in all dimensions
Quantum dot (QD)
N electrons in QD
1st approximation: disk with
capacitance C
Coulomb blockade energy:
EC = q2/C = e2/C
Spacing of dot chemical
potential: µN+1-µN = EC
2d
2d
Vg
N+2
µS
x
x
µD
N+1
N
N-1
conductance
otherwise Coulomb blockade (CB)
Conductance G displays peak for
each change in electron number
N+2
N
Shift potential of quantum
dot with gate Vg
Resonant single electron tunneling
only for µS > µN+1 > µD
N+1
N-2
µdot
N-1
N-1
N
N+1
N+2
gate voltage Vg
N+3
Meirav
et al '90
Analogue: Fabry-Perot spectrometer
Fabry Perot spectrometer
Quantum dot spectrometer
Transmission for selected
energies (frequencies)
Periodic transmission
Frequency comb tuned by
Energy comb shifted by
distance d
applied gate voltage
ballistic
electrons
incident
light
N+2
mirrors
PM
N+1
N
N-1
T(E)
d
I
T(E)
VG
d
VG
E
hf
Quantum dot charge detection
conductance
operation
point
Detect charge state of quantum dot
Use 1d wire near dot (Field et al '93)
Potential in wire changes due
to charge on quantum dot
At step edge conductance
very sensitive to local
potential
Step in wire conductance for
each change of quantum
dot charge
C. Fricke et al., PRB '05
M. Rogge et al., PRB '05
QPC potential
IQPC
IQD
N
QPC
dot
(QD)
N-3
N-1
N-2
Spectrometer with charge readout
Occupation of the dot depends on occupation of impinging electrons Ω→(E)
and accessible empty outgoing states 1 – Ω←(E) (0 ≤ Ω ≤ 1)
 01 =2  ⇒  E 
 10 =  1−⇐  E 
Dot occupancy p (0 ≤ p ≤ 1):
1− p  01 = p  1 0
Measure charge Q = -pe
For dilute ballistic electrons:
E
 0 1
Ω(E)
⇒ ≪1
1−⇐ ≈1
p
 1 0
quantum dot
}
⇒ p≈2⇒
Q(E)
Iwire
charge
detector
Γ=0
Γ>0
VG
Spectrometer calibration
4
QD prop. p
0
-2
-2
Fermi
sea
-4
 0 1
2
0
p
10
-4
-6
-8
0.0
-8
1.0 0.0
0.5
occupation
spectrometer
0.1
0.2
dp/dE ~ dI/dV
6
5
0.57 K
0.39 K
0.25 K
4
3
0.8
∆ (mV)
dI/dVg (a.u.)
ΔE/ΔVg = 0.0719 ± 0.0007
4
-6
Measure dI/dVg ~ dp/dE
Use temperature
dependence of peak width Δ
for spectrometer calibration
6
2
energy E/(kBT)
Spectrometer signal near
Fermi energy maps Fermi
distribution f(E)
2
p=
1
1
f  E
6
2
1
0
0.6
0.4
0.2
-1
-2
-822
-820
-818
Vg (mV)
-816
-814
0.4
T (K)
0.6
Ballistic signal
Iwire
2µm
Source ballistic electrons with
QPC set to first plateau for well
defined injection spectrum
Inject Ω(E) = const
Magnetic field B bends ballistic
electrons into the spectrometer
Vary maximum energy Emax =
-eVsd of ballistic electrons
V
B = 30 mT
QPC
2.0
AC-modulation: Mark electrons
injected at Emax
-Vsd (mV)
1.5
1.0
Linear shift of ballistic peak
position: -eVsd = 1.01·E
6
1
0.5
0.1
0.0
0.0
0.5
1.0
E (meV)
1.5
2.0
No energy scaling factor
E= -α·eVsd with α<1 as
claimed previously
It works!
Ballistic peak – amplitude
Signal decays for rising energy
2.0
Scattering of ballistic electrons
-Vsd (mV)
1.5
1.0
Compare to theory for e-e scattering
with equilibrium electrons
(Giuliani and Quinn, 1982)
6
1
0.5
Deviation due to large population
of non-equilibrium electrons?
0.1
6
0.0
Signal amplitude
5
0.0
0.5
1.0
1.5
2.0
E (meV)
4
theory
3
Superimposed oscillations due to
interference of different paths
2
1
Agreement observed in
interference experiment (Yacoby
et al '91) – difference between
phase and energy relaxation?
data
0
0.0
0.5
1.0
1.5
2.0
2µm
excess energy -eVsd (meV)
B
Energy + angle distribution
Maximum signal shifts to
larger field
Velocity increases with
increasing energy
Energy spread ΔE wider for
larger excess energy -eV
Small energy scattering
No noticeable change in angle
spread ΔB ∝ Δφ
Small energy scattering
results in little momentum
change
44
B (mT)
-800 µV
20
44
-500 µV
ΔE
20
44
6
ΔB
Vsd = -200 µV
0
20
0
200
400
600
E (µeV)
800
1000
Δφ
B
Full DOS
Again QPC on 1st plateau
Measures full energy distribution
Ω(E) of ballistic electrons
50
dot occupation p (%)
40
30
20
10
0.1
0.2 0.3 0.4 0.5 0.6 0.7 0.8
0.9 mV
0
-10
-0.2
p=2 ⇒  1−2 ⇒ ⇐ O 3 
-Vsd = 0 mV
0.0
0.2
0.4
E (meV)
0.6
0.8
1.0
Summary
Ballistic electron spectroscopy
APL 89, 212103 (2006)
44
QPC charge detector for
measurement with empty dot
Measurement of energy and angle
distribution of non-equilibrium ballistic
electrons
Apply now: QPC (0.7 anomaly), QD
(Kondo effect), energy relaxation, ....
B (mT)
-800 µV
20
44
-500 µV
20
44
6
Vsd = -200 µV
0
20
0
E
T(E)
 0 1
 1 0
QPC
Q(E)
charge
detector
QD
B
200
400
600
E (µeV)
800
1000
Thanks ...
This work was done at the Cavendish Laboratory in
Cambridge
Thanks to
Sir Michael Pepper, head of SP group
Jonathan Griffiths and Geb A.C. Jones for the
electron-beam write jobs
Dave Ritchie for growing the wafer (himself!)
Funding was provided by the EU (RTN COLLECT)
and the UK (EPSRC)