Physics and technology in quantum point contacts (QPCs ...
35
ISSP Int. Summer School (2003/8/18) Physics and technology in quantum point contacts (QPCs) Yoshiro Hirayama NTT Basic Research Laboratories and CREST-JST 1. Fabrication and quantized conductance 2. Effect of confinement potential 3. 0.7 structure 4. Magnetic field dependence 5. Series and multi-parallel QPCs 6. Surface effects 7. Nanoscale understanding of QPCs
Transcript of Physics and technology in quantum point contacts (QPCs ...
ISSP
Int.
Sum
mer
Sch
ool (
2003
/8/1
8)
Phys
ics
and
tech
nolo
gy in
qua
ntum
poi
nt c
onta
cts
(QPC
s)
Yosh
iro
Hir
ayam
aN
TT B
asic
Res
earc
h La
bora
tori
es a
nd C
REST
-JST
1. F
abri
catio
n an
d qu
antiz
ed c
ondu
ctan
ce2.
Effe
ct o
f con
finem
ent p
oten
tial
3. 0
.7 s
truc
ture
4. M
agne
tic fi
eld
depe
nden
ce5.
Ser
ies
and
mul
ti-pa
ralle
l QPC
s6.
Sur
face
effe
cts
7. N
anos
cale
unde
rsta
ndin
g of
QPC
s
QP
Cs:
Qua
ntum
poi
nt c
onta
cts （on
e-di
men
sion
al b
allis
tic c
hann
el）
)(2
)(
*
n
sn
EEm
hgE
D−
=
)2(
/2
/
/
/)
()
(
)(2
21
22
1
2
*
2*
==
=
==
==
−=
=− ∑ =
s
si n
n
sF
nn
nF
nF
gh
ie
VI
G
hVi
eg
II
hV
eg
eVev
ED
I
mE
Ev
vm
EE
One
-dim
ensi
onal
D
OS
of e
ach
subb
and
Ele
ctro
ns m
ove
on fe
rmis
urfa
ce a
t T=0
App
licat
ion
of s
mal
l V
Tot
al c
urre
nt =
Sum
of c
urre
nt o
f eac
h su
bban
d（
i;num
ber
of su
bban
dsun
der
ferm
ilev
el）
chan
ge to
con
duct
ance
Ene
rgy
Density of states
Bal
listic
mea
n fre
e pa
th o
f hig
h m
obili
ty 2
DE
G
nen
hle
,2
µπ
µ∝
=
l e ~
10 µ
m
( n ~
3x1
011 cm
-2,
µ ~
106 cm
2 /Vs)
I
V
1.5
K
l e ~
100
µm (
n ~
3x1
011 cm
-2,
µ ~
107 cm
2 /Vs)
Hir
ayam
a et
al.,
App
l. Ph
ys. L
ett.
56, 2
672
(199
1)
QP
Cs:
Qua
ntum
poi
nt c
onta
cts （on
e-di
men
sion
al b
allis
tic c
hann
el）
(ｂ)
split
-gat
e or
in
-pla
ne g
ate
wid
th: v
aria
ble
dens
ity: ~
cons
tant
Ene
rgy
DOS
Ene
rgy
Ene
rgy
DOS
DOS
elec
tron
elec
tron
depl
etio
n
(a)
insu
latio
n +
gate
wid
th: ~
cons
tant
dens
ity: v
aria
ble
Spl
it-S
chot
tky
gate
QP
Cs
and
quan
tized
con
duct
ance
gate
vol
tage
[ V
]
gate
vol
tage
[ V
]
resistance [ kΩ] conductance [ 2e2/h]
B. J
. van
Wee
set a
l., P
hys.
Rev
. Let
t. 60
, 848
(198
8)D
. A. W
hara
met
al.,
J. P
hys.
C21
, L20
9 (1
988)
and
oth
ers
Oth
er ty
pes
of Q
PC
s(in
-pla
ne g
ate
QP
Cs)
2DE
G g
ate
2DE
G/1
DE
G
chan
nel
A. D
. Wie
ckan
d K
. Plo
og, A
ppl.
Phys
. Let
t. 56
, 928
(199
0)J.
Reg
ulet
al.,
App
l. Ph
ys. L
ett.,
81,
202
3 (2
002)
Oth
er ty
pes
of Q
PC
s(fo
cuse
d-io
n-be
am w
ritte
n Q
PC
s)
Y. H
iray
ama
and
T. S
aku,
App
l. Ph
ys. L
ett.,
54,
255
6 (1
989)
Gat
e vo
ltage
[ V
]
Resistance [ kΩ]
rD
eff
i1
2λ
=l
FF
Emh *
2=
λ
)(
2*
1n
FD
EE
mh
−=
λ
lis
repr
esen
ted
by a
uni
t of λ
F
E. T
ekm
anan
d S.
Cir
aci,
Phys
. Rev
. B39
, 877
2 (1
989)
B40
, 855
9 (1
989)
theo
ry
Con
finem
ent p
oten
tial a
nd q
uant
ized
con
duct
ance
cha
ract
eris
tics
----
-ref
lect
ion
of e
lect
ron
wav
e at
bot
h en
ds --
----
Abr
upt w
idth
cha
nge
in a
wav
egui
de
resu
lts in
a w
ave
refle
ctio
n.
Ref
lect
ed e
lect
ron
wav
e m
akes
in
terf
eren
ce a
nd c
ondu
ctan
ce
osci
llatio
n ap
pear
s.
Osc
illat
ion
is d
eter
min
ed b
y λ
1Dan
d l ef
f.
(i r:
inte
ger）
Con
finem
ent p
oten
tial a
nd q
uant
ized
con
duct
ance
cha
ract
eris
tics
----
-ref
lect
ion
of e
lect
ron
wav
e at
bot
h en
ds --
----
smal
l ele
ctro
n de
nsity
larg
e de
plet
ion
spre
adin
g ro
unde
d co
rner
larg
e el
ectr
on d
ensi
ty
smal
l dep
letio
n sp
read
ing
shar
p co
rner
Exp
erim
ents
:Y. H
iray
ama
et a
l., J
pn. J
. App
l. Ph
ys. 2
8, L
701
(198
9)
Qua
ntiz
ed c
ondu
ctan
ce c
hara
cter
istic
s fo
r mod
el p
oten
tials
()
()
1si
nco
s2
2
2
2
=−
αα
Cx
Cy
2
2
2hc
mq
FF
ε=
vu
cy
vu
cx
cos
cosh
sin
sinh
==
)(
3 an
d4,
6,8,
16,0
fa
→
=π
ππ
ππ
α
hype
rbol
a bo
unda
ry
ellip
tic c
oord
inat
e
A. K
awab
ata,
J. P
hys.
Soc
. Jap
an, 5
8, 3
72 (1
989)
Qua
ntiz
ed c
ondu
ctan
ce c
hara
cter
istic
s fo
r mod
el p
oten
tials
Sadd
le p
oten
tial c
onfig
urat
ion:
M
. Büt
tiker
, Phy
s. R
ev. B
41, 7
906
(199
0). 2
2*
22
*0
2121
),
(y
mx
mV
yx
Vy
xω
ω+
−=
xy
nn
nV
nE
eT
ωω
επε
hh
/)) 21
((2
11
0−
+−
=+
=−
∑=
nTT
Sadd
le p
oten
tial
Tra
nsm
issi
on r
ate
of
each
cha
nnel
Tot
al tr
ansm
issi
on
y
x
y
x
Con
finem
ent p
oten
tial a
nd q
uant
ized
con
duct
ance
cha
ract
eris
tics
----
-Bac
kgat
edQ
PC
s--
---
n-G
aAsb
ackg
ate
GaA
s
AlA
s/G
aAsb
arri
er
1DE
G
split
gat
es
d 1W
smal
l d1
(sm
all W
)
smal
l ωx
and
larg
e ω
y
larg
e d 1
(larg
e W
)
larg
e ω
xan
d sm
all ω
y
Con
finem
ent p
oten
tial a
nd q
uant
ized
con
duct
ance
cha
ract
eris
tics
W=
300n
m, d
1=
250n
m
•Q
uant
ized
step
s bec
ome
obsc
ure
in
low
-den
sity
regi
ons d
ue to
a th
erm
al
broa
deni
ng e
ffec
t.
W=
800n
m, d
1=
500n
m
•Q
uant
ized
step
s are
obs
cure
eve
n in
hi
gh-d
ensi
ty re
gion
s and
at v
ery
low
te
mpe
ratu
re d
ue to
the
conf
inem
ent
pote
ntia
l eff
ect.
•Th
e 0.
7 st
ruct
ure
clea
rly re
mai
ns.
Con
duct
ance
of t
he a
nom
alou
s pla
teau
dr
ops t
o ar
ound
0.5
G0
whe
n th
e el
ectr
on
dens
ity d
ecre
ases
.S.
Nut
tinck
et a
l., J
JAP3
9, L
655
(200
0)
T=
1.4
K
Bac
kgat
edQ
uant
um P
oint
Con
tact
at 1
00 m
K
W=
300n
md 1
= 25
0nm
100m
k
V b: 1
→2.
4V (0
.1V
/ste
p)el
ectro
n de
nsity
:1.
2 x
1011→
3.3
x 10
11cm
-2
2DE
G
QPC
(qua
ntum
poi
nt c
onta
ct)
heG
2
02
=
K. H
ashi
mot
o et
al.,
Jpn
. J. A
ppl.
Phys
., 40
, 300
0 (2
001)
Expa
nded
Vie
w o
f the
Con
duct
ance
bel
ow 1
.2 G
0 at
100
mK
Qua
ntiz
ed st
ep:
appe
ar ~
1.0
G0
•cl
ear s
tep
over
ent
ire
rang
e of
den
sitie
s
0.7
anom
aly :
appe
ar 0
.68
~ 0.
8G0
•ki
nk ra
ther
than
step
at
arou
nd 0
.8G
0fo
r the
in
term
edia
te e
lect
ron
dens
ity (~
2 x
1011
cm-2
)
•m
ore
appa
rent
step
at
arou
nd 0
.7G
0fo
r bot
h th
e hi
gher
and
low
er
dens
ity re
gion
s
3.3x
1011
cm-2
1.2x
1011
cm-2
W=
300n
m, d
1=
250n
mK
. Has
him
oto
et a
l., J
pn. J
. App
l. Ph
ys.,
40, 3
000
(200
1)
0.7
anom
aly
----
sum
mar
y of
exp
erim
ents
0.7
anom
aly
* in
trin
sic
feat
ure
rela
ted
with
ele
ctro
n sp
in
* m
ore
prom
inen
t at h
ighe
r te
mpe
ratu
res
* be
twee
n 0.
5 an
d 0.
8, a
nd p
rom
inen
t ste
p at
aro
und
0.7
* sh
ift to
0.5
und
er lo
w-e
lect
ron-
dens
ity, l
ong-
chan
nel,
and
high
-ele
ctro
n-de
nsity
(str
ong
inte
ract
ion
?)
Ref
eren
cesK
. J. T
hom
as e
t al.,
PR
L, 7
7, 1
35 (1
996)
; K. J
. Tho
mas
et a
l., P
RB
58, 4
846
(199
8), A
. Kris
tens
enet
al.,
Phy
sica
B24
9-25
1, 1
80 (1
998)
, S.N
uttin
cket
al.,
JJA
P39,
L6
55 (2
000)
, K. J
. Tho
mas
et a
l., P
RB
61, R
1336
5 (2
000)
, K. S
. Pys
hkin
et a
l., P
RB
62,
1258
4 (2
000)
, K. H
ashi
mot
o et
al.,
JJA
P40,
300
0 (2
001)
, D. J
. Rei
lly e
t al.,
Phy
s. R
ev.
B63
, 121
311
(200
1), A
. Kris
tens
enan
d H
. Bru
us, P
hysi
caSc
ripta
(200
2), S
. M.
Cro
nenw
ette
t al.,
Phy
s. R
ev. L
ett.,
88,
226
805
(200
2)
QP
Cs
mad
e on
diff
eren
t mat
eria
ls
high
-mob
ility
SiM
OSF
ET
µ : 2
.2x1
04 cm
2 /Vs (
n: 5
.6x1
011cm
-2)
Hig
her
tem
pera
ture
ope
ratio
n
Diff
eren
t deg
ener
acy
Spin
-orb
it in
tera
ctio
n
Si: v
alle
y de
gene
racy
2 e2 /h
----
-->
4 e
2 /h
S. L
. Wan
g et
al.,
Phy
s. R
ev. B
46, 1
2873
(199
2)
In0.
53G
a 0.47
As/
InA
lAs
in-p
lane
gat
es (F
IB)
T. B
ever
et a
l., J
pn. J
. App
l. Ph
ys.,
33, L
800
(199
4)
QP
Cs
mad
e on
diff
eren
t mat
eria
ls
4.2
K
In0.
53G
a 0.47
As/
InP
in-p
lane
gat
esJ.
J. W
estr
oem
et a
l., A
ppl.
Phys
. Let
t. 70
, 130
2 (1
997)
InA
s/A
lSb
split
-gat
esS.
J. K
oest
er e
t al.,
App
l. Ph
ys. L
ett.
62, 1
373
(199
3)
Tran
spor
t cha
ract
eris
tics
of Q
PC
s: p
erpe
ndic
ular
mag
enet
icfie
ld d
epen
denc
e
B. J
. van
Wee
set
al.,
Ph
ys. R
ev. B
38, 3
625
(198
8).
Dep
opul
atio
n of
1D
subb
and:
( 1
D su
bban
den
ergy
sepa
ratio
n)
0
from
QPC
sto
inte
ger
quan
tum
Hal
l eff
ects
ωh
2
2 0c
ωω
+h
Four
-term
inal
and
two-
term
inal
mea
sure
men
ts
R2t
R4t
R’ 4t
−=
wid
et
ii
ehR
11
22
4
wid
et
cFw
ide
cF
oc
F
ieh
ieh
R
Ei
EE
i
12
12
22
2
22
→=≈
→+
≈
ω
ωω
ω h
hh
ieh
Rt
12
22
=
high
mag
netic
fiel
d lim
it
H. v
an H
oute
net
al.,
Ph
ys. R
ev. B
37, 8
534
(198
8).
Ser
ies
QP
Cs
QPC
dto
tal
QPC R
Ti
ehRR
2
22
2
<+
=<
iehR
RQ
PCto
tal
222
2==
inte
ger)
:(
12
2i
iehR
RQ
PCto
tal ==
ypr
obab
ilit
nsm
issi
on
dire
ct tr
a:
dT
iT d
<<
0
Ser
ies
QP
Cs
W
depl
etio
n re
gion
L sp
WW
depl
etio
n re
gion
L sp
Y. H
iray
ama
and
T. S
aku
, Phy
s. R
ev. B
41, 2
927
(199
0)
Ser
ies
QP
Cs
A. A
. M. S
tari
ng e
t al.,
Phy
s. R
ev. B
41, 8
461
(199
0)
Mul
tiple
par
alle
l QP
Cs
Larg
e m
agne
to-d
epop
ulat
ion
Y. H
iraya
ma
and
T. S
aku,
Phy
s.
Rev
. B42
, 114
08 (1
990)
K. N
akam
ura
et a
l., A
ppl.
Phys
. Let
t. 56
, 385
(199
0)
Mul
tiple
par
alle
l QP
Cs
AB
-typ
e in
terf
eren
ce e
ffect
Y. H
iray
ama
and
T. S
aku,
Phy
s. R
ev.
B42
, 114
08 (1
990)
Bac
kgat
ed h
eter
ostr
uctu
re
n-G
aAs (
Bac
k-ga
te)
AlA
s(2n
m)/G
aAs(
2nm
)Su
per l
attic
e ba
rrie
r
20nm
Al 0.
33G
a 0.67
As
2DEG
GaA
s
Ohm
ic c
onta
cts
d 1 d 2
d 1(c
hann
el d
epth
) : 5
4-50
0 nm
d 2(b
arri
er th
ickn
ess)
: 37
5-82
0 nm
grow
n by
MB
E
106234567
mobility (cm2/Vs)
56
78
9 1011
23
45
6
elec
tron
den
sity
(cm
-2)
Dar
k
Aft
er il
umin
atio
n
・D
ensi
ty tu
nabi
lity・
Hig
h m
obili
ty
・A
ppro
pria
te s
yste
m to
stu
dy th
e su
rfac
e ch
arac
teri
stic
sY
. Hir
ayam
a et
al.,
App
l. Ph
ys. L
ett.
72,
1745
(199
8)
Cha
ract
eris
tics
of fr
ee G
aAs
surfa
ce--
---m
idga
ppi
nnin
g m
odel
(MP
M) -
----
-
()(
)bt
hb
bth
b
s
sbt
h
VV
VV
ed
n
eVga
pm
iddd
V
>−
=
≈−
≈
= 22
21
12
9.07.0
εφ
φεε
～
Hig
h qu
ality
--->
neg
lect
impu
rity
cha
rge
in G
aAs
Vbt
his
det
erm
ined
by
a ba
lanc
e of
su
rfac
e ch
arge
, 2D
EG
and
ba
ckga
te.
Cha
ract
eris
tics
of fr
ee G
aAs
surfa
ce--
---f
roze
n su
rface
mod
el (F
SM
) ---
---
non-
equi
libri
um su
rfac
e
()
()(
)bt
hb
bth
b
sbt
hs
ss
s
sbt
h
VV
VV
ed
n
dd
dV
gap
mid
Qd
d
dens
itye
chsu
rfac
efr
ozen
sQ
Qd
V
>−
=
=∴
−
== 22
11
22
2
22
1122
/
:
arg
:
ε
φε
ε
φε
εφ
ε
＋
冷却前の平衡状態
＋eq
uilib
rium
bef
ore
cool
ing
Thr
esho
ld b
ackg
ate
bias
of b
ackg
ated
undo
ped
hete
rost
ruct
ures
The
obt
aine
d re
sults
are
wel
l exp
lain
ed b
y th
e fr
ozen
-sur
face
-mod
el.
A. K
awah
araz
uka
et a
l., P
hys.
Rev
. B63
, 245
309
(200
1)
Shift
of t
he th
resh
old
back
gate
bias
Cha
rge
tran
sfer
(e
lect
ron
tunn
elin
g to
th
e su
rfac
e) is
ob
serv
ed a
t hig
her
tem
pera
ture
s.
d 1=
220
nm
A. K
awah
araz
uka
et a
l.,
Phys
. Rev
. B63
, 245
309
(200
1)
Era
sabl
e el
ectr
on li
thog
raph
y
by u
sing
low
-tem
pera
ture
scan
ning
nan
opro
be
R. C
rook
et a
l., p
rese
nted
EP2
DS-
15 (
Nar
a, 2
003)
H3
Ele
ctro
n flo
w th
roug
h Q
PCs
M. A
. Top
inka
et a
l., S
cien
ce, 2
89, 2
323
(200
0)
Art
ifici
al p
oint
def
ect
and
devi
atio
n of
the
quat
ized
cond
ucta
nce
M. A
. Top
inka
et a
l., S
cien
ce, 2
89, 2
323
(200
0)
Ele
ctro
n flo
w p
athe
sin
real
2D
EG
(hig
h-qu
ality
)
M. A
. Top
inka
et a
l., N
atur
e, 4
10, 1
83 (2
001)
Qua
ntum
Poi
nt C
onta
cts (
QPC
s)
I dis
cuss
ed p
rinc
iple
cha
ract
eris
tics o
f QPC
sand
rel
ated
ph
enom
ena.
Qua
ntiz
ed c
ondu
ctan
ce a
nd o
ther
feat
ures
are
wel
l exp
lain
ed b
y si
ngle
par
ticle
pic
ture
(qua
ntum
eff
ects
and
bal
listic
eff
ects
)
Fut
ure
Com
bina
tion
of Q
PCsa
nd o
ther
nan
ostr
uctu
res
Spin
-spl
ittin
g at
zero
mag
netic
fiel
d?
Car
rier
inte
ract
ion
----
0.7
feat
ure
en
tang
led
elec
tron
s?
