VOLUML60,
NUMBLR9
PHYSICAL
REVIEW
LETTERS
29
FEBRUARY
1988
Quantized Conductance
of
Point
Contacts
in
a
Two-Dimensional
Electron
Gas
B
J van
Wees
Department
of
Applied
Physics
Delft
Unwersity
of
Technology
2628
CJ
Delft
The
Netherlands
H van
Houten,
C W J
Beenakker,
and J G
Wilhamson,
Philips
Research Laboratories 5600
JA
Eindhoven,
The
Netherlands
L P
Kouwenhoven
and D van der
Marel
Department
of
Applied
Physics
Delft
Unnersity
of
Technology
2628
CJ
Delft,
The
Netherlands
and
C T
Foxon
Philips
Research Laboratories
Redhill,
Surrey
RH1
5HA
United
Kingdom
(Received
31
December
1987)
Ballistic
pomt
contacts,
defined
in
the
two-dimensional
electron
gas of a
GaAs-AlGaAs
heterostruc-
ture,
have
been
studied
in
zero
magnetic
field The
conductance
changes
in
quantized
Steps
of
e
2
/nh
when
the
width,
controlled
by a
gate
on top of the
heterojunction,
is
vaned
Up to
sixteen
Steps
are ob-
served
when
the
pomt
contact
is
widened
from
0 to 360 nm An
explanation
is
proposed,
which
assumes
quantized
transverse
momentum
in
the
pomt-contact
region
PACS
numbers
72 20 Jv 73 40 Cg 73 40 Lq
As a
result
of the
high
mobihty
attamable
in
the
two-
dimensional
electron
gas
(2DEG)
in
GaAs-AlGaAs
het-
erostructures
it
is
now
becoming
feasible
to
study
ballis-
tic
transport
in
small
devices
'"
6
In
metals ideal
tools
for
such
studies
are
constnctions
havng
a
width
W and
length
L
much
smaller
than
the
mean
free
path
l
e
These
are
known
äs
Sharvin
pomt contacts
7
Because
of
the
ballistic
transport through these constnctions,
the
resistance
is
determmed
by the
pomt-contact
geometry
only
Point contacts have been
used
extensively
for the
study
of
elastic
and
melastic
electron
scattermg
With
use
of
biased
pomt contacts,
electrons
can be
mjected
mto
metals
at
energies
above
the
Fermi
level
This
al-
lows
the
study
of the
energy
dependence
of the
scattermg
mechamsms
8
With
the use of
a
geometry
containmg
two
pomt contacts,
with
Separation smaller than
l
e
,
elec-
trons
mjected
by a
pomt contact
can be
focused
mto the
other contact,
by the
application
of a
magnetic
field
This
technique
(transverse electron
focusmg)
has
been
applied
to the
detailed
study
of
Fermi surfaces
9
In
this
Letter
we
report
the first
expenmental
study
of
the
resistance
of
ballistic pomt contacts
m the
2DEG
of
high-mobihty
GaAs-AlGaAs
heterostructures
The
smgle-pomt
contacts
discussed
m
this
paper
are
part
of a
double-pomt-contact
device
The
results
of
transverse
electron focusmg
m
these devices
will
be
published
else-
where
'°
The
pomt contacts
are
dehned
by
electrostatic
depletion
of the
2DEG underneath
a
gate
This
method,
which
has
been used
by
several authors
for the
study
of
l
D
conduction,'
1
offers
the
possibility
to
control
the
width
of the
pomt contact
by the
gate voltage Control
of
the
width
is
not
feasible
in
metal
pomt contacts
The
classical
expression
for the
conductance
of a
pomt
contact
m two
dimensions
(see
below)
is
G=(e
2
/nh)k
Y
W/n
(1)
in
which
kf
is
the
Fermi
wave vector
and W
is
the
width
of
the
contact
This
expression
is
vahd
if
l
e
»
W and the
Fermi wavelength
λρ<ίί
W The first
condition
is
satisfied
in
our
devices,
which
have
a
maximum
width
W
mm
«=
250 nm and
l
e
=8
5
μηι
The
second
condition
should
also
hold
when
the
devices have
the
maximum
width
We
expect
quantum
effects
to
become
important
when
the
width
becomes comparable
to
λρ,
which
is
42 nm m
our
devices
In
this
way we are
able
to
study
the
transi-
tion
from
classical
to
quantum
ballistic transport
through
the
pomt contact
The
pomt contacts
are
made
on
high-mobility
molecular-beam-epitaxy-grown
GaAs-AlGaAs hetero-
structures
The
electron
density
of the
matenal
is
3
56xl0
15
/m
2
and the
mobihty
85
m
2
/V
s (at 0 6 K)
These
values
are
obtamed
from
the
devices containmg
the
studied
pomt
contacts
A
Standard Hall
bar
geome-
try
is
defined
by wet
etchmg
Usmg
electron-beam
lithography,
a
metal gate
is
made
on top of the
hetero-
structure,
with
an
openmg
250 nm
wide
(mset
m
Fig
1)
The
pomt
contacts
are
defined
by the
application
of a
negative
voltage
to the
gate
At
V
g
=
— 0 6 V the
elec-
tron
gas
underneath
the
gate
is
depleted,
the
conduction
takmg
place
through
the
pomt contact
only
At
this
volt-
age the
pomt
contacts have
their
maximum
width
W
max
,
about
equal
to the
openmg between
the
gates
By a
fur-
ther
decrease
of the
gate voltage,
the
width
of the
pomt
contacts
can
gradually
be
reduced,
until
they
are
fully
848
©
1988
The
American
Physical
Society
VOLUME
60,
NUMBER9
PHYSICAL
REVIEW
LEITERS
29
FEBRUARY
1988
-2
-1.8
-l B
-1.4 -1.2
-l
-O.B
-0 6
GATE
VOLTAGE
(V)
FIG.
1.
Point-contact resistance
äs
a
function
of
gate
volt-
age at 0.6 K.
Inset:
Point-contact
layout.
GATE VOLTAGE
(V)
FIG.
2.
Point-contact conductance
äs
a
function
of
gate
voltage,
obtained
from
the
data
of
Fig.
l
after
subtraction
of
the
lead
resistance.
The
conductance shows plateaus
at
multi-
ples
of
e
^
/πh.
pinched
off at
V
g
= — 2.2 V.
We
measured
the
resistance
of
several point contacts
äs
a
function
of
gate
voltage.
The
measurements
were
performed
in
zero
magnetic
field, at 0.6 K. An ac
lockin
technique
was
used,
with
voltages
across
the
sample
kept
below
kT/e,
to
prevent
electron
heating.
In
Fig.
l the
measured
resistance
of a
point
contact
äs
a
function
of
gate
voltage
is
shown.
Unexpectedly,
plateaus
are
found
in the
resistance.
In
total,
sixteen
plateaus
are
observed
when
the
gate
voltage
is
varied
from
—0.6
to
—2.2
V.
The
measured
resistance
consists
of the
resistance
of the
point
contact,
which
changes
with
gate
voltage,
and a
constant
series
resistance
from
the
2DEG
leads
to the
point
contact.
As
demonstrated
in
Fig.
2, a
plot
of the
conductance,
calculated
from
the
measured
resistance
after
subtraction
of a
lead
resistance
of 400
Ω,
shows
clear
plateaus
at
integer
multiples
of
ε
2
/π}ϊ.
The
above
value
for the
lead resistance
is
consistent with
an es-
timated value
based
on the
lead
geometry
and the
resis-
tivity
of the
2DEG.
We do not
know
how
accurate
the
quantization
is. In
this experiment
the
deviations from
integer multiples
of e
2
/nh
might
be
caused
by the
uncer-
tainty
in the
resistance
of the
2DEG
leads.
Inserting
the
point-contact
resistance
at
V
g
= — 0.6 V
(750
Ω)
into
Eq.
(1)
we find
for
the
width
W
/
ma
x
==
360
nm,
in
reason-
able
agreement with
the
lithographically
defined
width
between
the
gate
electrodes.
The
average conductance increases almost
linearly
with
gate
voltage. This indicates that
the
relation
be-
tween
the
width
and the
gate
voltage
is
also almost
linear.
From
the
maximum width
W
mm
(360
nm) and
the
total number
of
observed Steps
(16)
we
estimate
the
increase
in
width between
two
consecutive steps
to be 22
nm.
We
propose
an
explanation
of the
observed quantiza-
tion
of the
conductance, based
on the
assumption
of
quantized
transverse momentum
in the
contact constric-
tion.
In
principle this assumption requires
a
constriction
much
longer than wide,
but
presumably
the
quantization
is
conserved
in the
short
and
narrow
constriction
of the
experiment.
The
point-contact conductance
G
for
ballis-
tic
transport
is
given
by
7
'"
G=e
2
N
Q
W(h/2m)(\k
x
(2)
The
brackets denote
an
average
of the
longitudinal wave
vector
k
x
over directions
on the
Fermi
circle,
TVo
=ηι/πίϊ
2
is the
density
of
states
in the
two-dimensional
electron
gas,
and W is the
width
of the
constriction.
The
Fermi-circle
average
is
taken over
discrete
transverse
wave vectors
k
y
=
±nn/W
(n
=1,2,..
.), so
that
we can
write
ηπ
W
W
n
=
l
Carrying
out the
Integration
and
substituting into
Eq.
(2),
one
obtains
the
result
N,
i
(3)
(4)
where
the
number
of
channels
(or
one-dimensional
subbands)
N
c
is the
largest integer smaller than kfW/π.
For
849
VOLUME
60,
NUMBER
9
PHYSICAL REVIEW LETTERS
29FEBRUARY
1988
l
this expression reduces
to the
classical formula
[Eq.
(1)].
Equation
(4)
teils
us
that
G is
quantized
in
units
of
β
2
/π}ί
in
agreement with
the
experimental
ob-
servation.
With
the
increase
of W by an
amount
of
λρ/2,
an
extra channel
is
added
to the
conductance. This
com-
pares well with
the
increase
in
width between
two
con-
secutive
steps,
determined
from
the
experiment. Equa-
tion
(4)
may
also
be
viewed
äs
a
special
case
of the
mul-
tichannel
Landauer
formula,
12
"
14
-
Σ
η,m
™
1
t
n
n
(5)
for
transmission
coefficients
|
t
nm
\
=ö„
m
corresponding
to
ballistic
transport
with
no
channel mixing.
It
is
interesting
to
note
that
this
multichannel
Lan-
dauer formula
has
been
developed
to
describe
the
ideal-
ized
case
of the
resistance
of a
quantum
wire,
connected
to
massive
reservoirs,
in
which
the
inelastic-scattering
events
are
thought
to
take
place
exclusively.
As
dis-
cussed
by
Imry,
13
|
t
nm
\
2=
δ
ηηι
corresponds
to the
case
that
elastic scattering
is
absent
in the
wire also.
The
fact
that
the
conductance
G
=
N
c
e
2
/nh
of
such
an
ideal
wire
is
finite
15
is a
consequence
of the
inevitable contact resis-
tances associated with
the
connection
to the
thermalizing
reservoirs.
The
Undings
described
in
this Letter
may
im-
ply
that
we
have realized
an
experimental System which
closely approximates
the
behavior
of
idealized mesocopic
Systems.
In
summary
we
have reported
the
first
measurements
of
the
conductance
of
single ballistic point contacts
in a
two-dimensional
electron gas.
A
novel
quantum
effect
is
found:
The
conductance
is
quantized
in
units
of
e
2
/nh.
We
would
like
to
thank
J. M.
Lagemaat,
C. E.
Tim-
mering,
and L. W.
Lander
for
technological support
and
L. J.
Geerligs
for
assistance with
the
experiments.
We
thank
the
Delft
Center
for
Submicron Technology
for
the
facilities
offered
and the
Stichting voor Fundamen-
teel
Onderzoek
der
Materie (FOM)
for financial
sup-
port.
'T.
J.
Thornton,
M.
Pepper,
H.
Ahmed,
D.
Andrews,
and
G.
J.
Davies, Phys. Rev. Lett.
56,
1198
(1986).
2
H.
Z.
Zheng,
H. P.
Wei,
D. C.
Tsui,
and G.
Weimann,
Phys. Rev.
B 34,
5635
(1986).
3
K.
K.
Choi,
D. C.
Tsui,
and S. C.
Palmateer,
Phys. Rev.
B
32,
5540
(1985).
4
H.
van
Houten,
C. W. J.
Beenakker,
B. J. van
Wees,
and
J. E.
Mooij,
in
Proceedings
of the
Seventh International
Conference
on the
Physics
of
Two-Dimensional
Systems,
Santa
Fe,
1987,
Surf.
Sei.
(to be
published).
5
G.
Timp,
A. M.
Chang,
J. E.
Cunningham,
T. Y.
Chang,
P.
Mankiewich,
R.
Behringer,
and R. E.
Howard, Phys. Rev.
Lett.
58,
2814
(1987).
6
G.
Timp,
A. M.
Chang,
P.
Mankiewich,
R.
Behringer,
J. E.
Cunningham,
T. Y.
Chang,
and R. E.
Howard, Phys. Rev.
Lett.
59,
732(1987).
7
Yu.V.
Sharvin,
Zh.
Eksp.
Teor.
Fiz.
48, 984
(1965)
[Sov.
Phys.
JETP
21, 655
(1965)].
8
For
a
review,
see I. K.
Yanson
and O. I.
Shklyarevskii,
Fiz.
Nizk.
Temp.
12, 899
(1986)
[Sov.
J. Low
Temp.
Phys.
12, 509
(1986)].
9
P.
C. van
Son,
H. van
Kempen,
and P.
Wyder, Phys. Rev.
Lett.
58,
1567
(1987).
10
H.
van
Houten,
B. J. van
Wees,
J. E.
Mooij,
C. W. J.
Beenakker,
J. G.
Williamson,
and C. T.
Foxon
(to be
pub-
lished).
"l.
B.
Levinson,
E. V.
Sukhorukov,
and A. V.
Khaetskii,
Pis'ma
Zh.
Eksp.
Teor.
Fiz.
45, 384
(1987)
[JETP
Lett.
45,
488
(1987)].
12
R.
Landauer,
IBM J.
Res. Dev.
l, 223
(1957);
R.
Lan-
dauer, Phys. Lett. 85A,
91
(1981).
13
M.
Buttiker,
Y.
Imry,
R.
Landauer,
and S.
Pinhas,
Phys.
Rev.
B 31,
6207
(1985).
For a
survey,
see Y.
Imry,
in
Direc-
tions
in
Condensed
Matter
Physics,
edited
by G.
Grinstein
and
G.
Mazenko (World
Scientific,
Singapore,
1986),
Vol.
l, p.
102.
14
D.
S.
Fisher
and P. A.
Lee, Phys. Rev.
B 23,
6851
(1981).
15
The
original Landauer formula (Ref.
12)
containing
the ra-
tio
of
transmission
and
reflection
coefficients does give
an
infinite
conductance
for a
perfect System.
However,
this for-
mula
excludes contributions from
the
contact resistances.
850