Slides.

ORNÝ, SMRČKA, AND DEUTSCHMANN
Magnetic breakdown
Magnetic breakdown in a 1D superlattice
1D superlattice
MAGNETORESISTANCE CALCULATIONS FOR A TWO- . . .
PHYSICAL
REVIEW
B 66, 205318
!2002"
2DEG
with
strong
1D
modulation
PHYSICAL
VOLUME 86, NUMBER 9
PHYSICAL REVIEW B 66
REVIEW LETTERS
1011 cm22 . Each curve is
n+ GaAs
respect to the inverse field
(110)≡z B
y
21
SL
ponents
D
of the magne
ohmic contacts
results are summarized in
n+ GaAs
frequency components are
50 nm GaAs
substrate
50 nm AlGaAs
scale, where large and smal
blue, respectively.
FIG. 1 (color). Sample cross section. The 2DES is field inTo analyze the transport
at Alternatively,
the GaAs!AlAs
in the (110) direction
G. 1. Right: 2DES with lateral periodical modulation and the effectively two-point-contactduced
geometry.
this mayheterointerface
be
means
of ‘‘barriers’’
a positive
voltage
applied to the n1 GaAs gate.
ived as an array of coupled quantum wires !parallel to the x axis". The ‘‘wires’’ !layers ofby
GaAs"
and the
!layers
of
energy EF of the modulate
a1$x As, x"0.32) were 11.9 and 3.1 nm thick in the experiments !Ref. 13" we refer to. Owing
to
the
height
of
the
‘‘barriers’’
the
The (001)-oriented SL provides an atomically precise potential
information acquired from a
ing between wires is small (E F "4 ! t ! !V 0 ) and a tight-binding model is thus appropriate. Left:
Zero field band
!along k y )
modulation
to structure
the 2DES.
11 keeping
while
i
ated by Kronig-Penney model. We keep only the lowest band !thick
line"
our calculations and we setCalculations
4 ! t ! equal to the and
width experiment.
of this
FIG.
5.in Magnetoresistance.
Electron concentrations #from calculation,
the top$ 5.1%10
cm-2, 3.8%
along the y direction. Her
%1011 cm-2. The lowest concentration corresponds to E F )2 ! t ! .
that depends on the GaAs layer thickness. For our sample
an index n that runs from
2 2
t relevant for these experiments. Other experiments
k
$
this xvariation of the density, integrated over the z direclyingisminiband
to NFor
forinte
the
E ! k x ,k y " R!cϱ
$2 !"ϱ
"
t ! cos k y d.
!1"
presented by Chowdhury et al.15 who fabricated bilatthe quadratic behavior
reconciled.
2m exceeds
yy
xy10%, as determined by a self-consistent 2Dtion,
(001)≡x
200 nm n+ GaAs
100 nm AlAs
30 nm GaAs
with 2DES
PRB 66, 205318 (2002)
...ato
to
1D
superlattices
superlattices
Model of...
1D1D
superlattice
(strong modulation)
Tight-bindingmodel
modelfor
for strongly
strongly modulated
modulatedtwo-dimensional
two-dimensionalsuperlattices
superlattices
Tight-binding
zz
22
BB Current
Current
xx
yy
V(y)
VbVb V(y)
yy
Current
Current
contact
contact
Voltage
Voltage
contact
contact
Figure 1.
1. Two–dimensional
Two–dimensional electron
electron gas
gas with
with periodic
periodic unilateral
unilateral modulation.
modulation.
Figure
Alternatively,the
thesystem
systemmay
maybe
beconceived
conceivedas
asan
anarray
arrayofofwires
wirescoupled
coupledby
bytunnelling.
tunnelling.
Alternatively,
33
1.5
1.5
11
0.5
0.5
-0.5
-0.5
E/2|t|
E/2|t|
55
44
33
22
11
00
-1-1
-4-4
-2-2 00
kxkx
22
44
-1-1
11
0.5
0.5
00 k kd/π
d/π
yy
-0.5
-0.5
Figure2.
2. Band
Bandstructure
structureininzero
zeromagnetic
magneticfield:
field:all
allmodulation
modulationbands
bands(in
(iny–direction)
y–direction)
Figure
exceptfor
forthe
thelowest
lowestare
arediscarded.
discarded. Fermi
Fermicontours
contours(related
(relatedtotoreal
realspace
spacetrajectories)
trajectories)
except
areclosed
closedfor
for−2|t|
−2|t|<<EEFF <<2|t|
2|t|and
andopen
openfor
forEEFF >>2|t|.
2|t|.
are
2 22
2
1. Model
Model for
for Zero
Zero Magnetic
Magnetic Field:
Field: Semiclassics
Semiclassics
~~ kkxx
1.
cond-mat/0208224
E(kxx,,kkyy)) =
=
+ 2t
2tcos
coskkyydd
cond-mat/0208224
E(k
+
⇤
Sincethe
thepotential
potentialin
inour
oursystem
systemisisseparable,
separable,zero
zeromagnetic
magneticfield
fieldband
bandstructure
structure
2m⇤ isisanan
Since
2m
Experimental determination of Fermi surface
198
Ph. Hofmann / Progress in Surface Science 81 (2006) 191–245
the bulk band structure on different Bi surfaces. The result of the calculation is show
Fig. 2, the bulk Brillouin zone is shown in Fig. 3.
An inspection of Fig. 2 immediately reveals the semimetallic character of Bi: The b
structure can very nearly be described by two filled s bands and three filled p ba
Fermi surfaces
FIRST-PRINCIPLES SCATTERING MATRICES FOR¼
Reminder:
Shubnikov-de Haas
Shubnikov - oscillations
de Haas oscillations in graphene
Potassium
binding energy (eV)
LETTERS
2.0
1.0
10
8
6
4
2
0
0
35
0.04 0.08 0.12
1/B (T¬1)
0.16
0.5
10
Ig (nA)
30
C (nF cm¬2)
1.5
Landau index m
Longitudinal resistance Rxx (B) ¬ Rxx (B = 0 T) (kΩ)
Copper
0
2
4
6
10
0
¬20
25
¬30
¬5
20
0
5
10
Vg (V)
15
20
X
Dark
Ultraviolet illuminated
Simulation
0
¬0.5
T = 300 K
Aluminium
10
100
8
B (T)
150
valley/spin degeneracy
200
T (K)
250
L U T
12
14
16
de Haas-van Alphen
Organic 2D metal
effect
Nat. Mat.
10, 357 (2011)
39
PRB 73, 064420
Γ
Fig. 2. Bulk band structure of Bi from the tight-binding calculation of Liu and Allan (Ref. [15], green lines)
principles calculation by Gonze et al. (Ref. [18], red lines), only in the C–T direction. (For interpretati
colours in this figure legend, the reader is referred to the web version of this article.)
Bismuth
300
T
Figure 4 | Measurement and simulation of the temperature dependent
capacitance of sample HD2. The capacitance is calculated using the
difference of the charge carrier density at Vg = 0 V and Vg = 3 V. The
xx
strong increase of the capacitance for T > 260 K indicates the transition
from the IPC to the SC regime. The solid line shows the simulated
g
The inset shows a fan diagram of the SdH oscillations that yields a Berry
temperature dependence using an extended Schottky model with
phase of . Straight lines are best fits to the data.
self-consistent field distribution. By illuminating the sample with ultraviolet
SC regime741
can –
be744
extended to lower temperatures. The error bars
742
A. Usher et al. / Physica light
E 22the(2004)
(equivalent
to 100INVESTIGATIONS
nm of SiO2 dielectric) compared
with
C(T = are
estimated using the errors of the measurements of the Hall resistance
DE HAAS-VAN
ALPHEN
OF
THE
ORGANIC
ETC.
200 K) = 8.5 nF cm 2 .
at Vg = 0 V and Vg = 3 V. The inset shows the asymmetric (Schottky-like)
1=2
The SC regime is reached for all samples at elevated gate current at T = 300 K.
U
X
L
L
U
Γ
2DEG
B [5], with the addition of a constant background inverse mag. flux quantum
temperatures. The higher the implantation dose, the lower is the
DOS between
LLs
[4,5]. to the4SC regime. In addition
α-(ET)
Θ = 1.8
transition
temperature
dDLo depends case. Interestingly, as soon as the ultraviolet light is switched off, the
2TlHg(SeCN)
1.5
on
the
applied
gate
voltage.
As
a
consequence,
the
device undergoes a transition back to the IPC regime. This confirms
Potts et al. [8] were the !rst to analyse the differential
dHvA
capacitance is no longer independent of the gate voltage as for the our notion that the two regimes are not kinetically distinguished,
oscillations
usingbuta itrigorous
procedure.
They
IPC regime,
decreases for!tting
higher gate
voltages. Furthermore
but each is stable in time.
1
the leakage
the insulating
area of
at T multi-layer
= 300 K shows the
For the sample HD3, ⇥(Vg ) and 1/eRH (Vg ) and the associated
found that
data current
takenacross
from
a range
typical asymmetric shape of a Schottky diode (inset Fig. 4), which gate current were measured in the IPC (T = 29 K) and the SC regime
2DEGs !t
best
to
a
model
DOS
which
was
Lorentzian
underlines that for high temperatures the gate has to be described (T = 295 K; Fig. 5a,b). In the SC regime a much higher capacitance,
in shapein0.5
with
a width
a Schottky
as in the IPC regime, is observed, resulting
in a steeper slope of
0model.that
30 was
60 independent of B. The
!the charge
To
model
the
transition
from
the
IPC
to
the
SC
regime
we
1/eR
(V
).
Notice
that
in
the
SC
regime
neutrality point
H
g
Θ
(deg)
best !t was
obtained
with no background.
A Gaussian
consider
a Schottky contact with graphene
0.5
1=2 as the metal and SiC as is already reached at Vg ⇤ 14 V.
form with
width
proportional
gave
the15
semiconductor.
thatan
theequally
electrostatic
The minimal measured charge carrier density nmin =
−1 In addition
−2toweBincluded
gate
potential
drops
partially
along
the
DL,
partially
along
the
highly
(1/eR
H )min is a measure of potential fluctuations in the graphene
good !t at low B.
resistive semiconductor (Fig. 1c,d). This unusual situation is due layer (electron–hole puddles). Typically, nmin is temperatureImprovements
in sample
quality
and magnetometer
to the high activation
energy
of the vanadium
dopand compared independent or increases with temperature if thermally generated
with
the
low
injection
barrier;
for
typical
contacts
design have enabled measurements toSchottky
be made
onthe charge carriers become important. Here, nmin is much smaller in
0
situation
is vice versa. As a consequence, the extension of the DL the high temperature SC regime, indicating a more homogeneous
2 effect is presumably
high-mobility
single-layer
2DEGs
[9–11].
and the associated
voltage drop
has to be
calculatedSaw-tooth
self-consistently lateral potential distribution. This unexpected
' locallyBZ
10
15
20
25
30
for different
temperatures, were
taking into
account theatcourse
deform the
magnetization
oscillations
observed
lowof !the caused by defects in the gate stack, which may
(T) the vanadium donor level electrostatic profile in the IPC regime. Improved wafer quality
quasi-Fermi level20 , which is well B
above
from which ˝!c could be determined without !tting.
F (103 T)
Γ
de Haas - van Alphen effect
10
✓ ◆
Figure 3 | Shubnikov–de Haas oscillations for different gate voltages.
1
2⇡e 1
e f
Longitudinal magnetoresistance R of sample HD1 at 6 K. A clear change in
= is visible with changing=gate voltages
the oscillation frequency
(V = 210
V triangles/dotted).
BV squares/solid, 0~V circles/dashed,
Ae +2102⇡~
p
L
L
electron
pocket
T
hole
pocket
FIG. 5. %Color& Top row, left-h
Prog Surf Sci 81, 191 (2006)
Cu;
middle panel: majority-spin F
viewed along the %111& direction
Brillouin zone %BZ& onto a plane p
of the two-dimensional BZ. Botto
els: projections onto a eplane perp
Fig. 3. Bulk Brillouin zone of Bi and a schematic sketch of the Fermi surface (not to scale). The C–
corresponds to the C3 axis and the [1 1 1] direction in real space.
FIG. 4. Interface conductance G %111& %in units of
10 & m & for an fcc Cu/ Co%111& interface for majority and
minority spins plotted as a function of the normalized area element
EPL 35, 37
(1996)
used in the Brillouin zone summation,
)k
/ A = 1 / Q2. Q, the
M (arb. units)
5
¬10
15
p = 4.7 × 1012 cm¬2
p = 6.1 × 1012 cm¬2
p = 7.3 × 1012 cm¬2
holes
spin-orbit
gap
0 electrons
NATURE MATERIALS DOI: 10.1038/NMAT2988
et al.:
PHYSICA
~
Extremal cross section ? B
(1/B) / 1/A
Obrázek 20: Teplotní pr*b'h vzork* E081#1 a D101#8. Jeho charakter
z molekulov!ch svazk%)
Extreme quantum limit
je dobrou indicií správné funk&nosti vzorku a nakontaktování 2DEGu.
∆ρminim
=v ideálních pozicích.
L /ρav le<í
polohách ( n1 25.813 k , n = 4, 6, 8, . . .), SdHO pom4ry
⌫⇡
1
⌫ koncentrace
1 nosi39 nSdHO
11
2
Z jejich polohy byla ur3ena
⇥
2.5
10
cm
. Zúdery
Hallovy
C
cos
2φ
+
C
cos(2φ
4θ)
I
I ,C ionty
povrch vzorku. Napra:ování
(Leybold), kde argonové
do +
slitiny
AuGe +
vy11
2
atomy a ty
:í9í ke vzorku.
P9i této metod6
je rychlost atom; v6t:í,
sm4rnice potom koncentrace nHall ⇥ 2.4 10 rá>ejí
cm
a sesv:esm6rov6
ohledem
na rozm4ry
hallbaru
pronikají tedy hloub6ji do materiálu a pokr=vají i mírn6 zakryté prostory (nevytvá9ejí
+ údaj9m
4θ)
+ CzUdoby
cos(2φ + 2θ
2 +C
Cse cos(4φ
„stíny@).
Titanová
vrtsva
i vodpovídá
tomto p9ípad6 pouze
pa9í.
(l=1000 µm, d=100 µm) mobilita nosi39 µ ⇥ 1.1
cm
/Vs.
V8e
P9esto>e sou5et v=:ek jednotliv=ch pater sandwiche by m6l b=t 100 nm, na Dektaku
r9stu waferu.
se zm69ila v=:ka reliéfu 73 nm (pa9ení) a 160 nm (prá:ení). P9esto>e se metody sna>í
Shubnikov - de Haas
SdH
oscillations
Plain 2DEG
b=t obdobné, srovnatelné se nezdají b=t. Pro to hovo9í i srovnání morfologie na obrázku
ní>e. Povrch legendárních kontakt; od pana Melichara vykazovaly charakter podobn=
spí:e sou5asnému Leyboldu. D101#8
de Ranieri et al., NJP ’08
Hall bar geometry
E081#1
3
10
Rxx
Rxy
2.5
Integer Quantum
IQHE
Hall Effect (IQHE)
6
Rxx
Rxy
8
[110]
3
out-of-plane
B
M
5
4
1
2
0.5
⌫n=4= 4
⌫ n=6
=6
0
0
0
0.5
1
1.5
2
B [T]
2.5
3
3.5
4
2
Rxy [kΩ]
1.5
Rxx [kΩ]
6
Rxy [kΩ]
Rxx [kΩ]
2
Obrázek 3: Srovnání morfologie ohmick+ch kontakt*. Zleva: napra)ované,
napa(ované, pan Melichar, pan Melichar. Rozdílnost metod je zjevná.
Autor obrázk*: Z. V+born+.
#
!
3
Za>íhání je provád6no v pícce a typické hodnoty jsou 450 C1po 2 minuty.
Na2 tyto ohmické kontakty (tedy kontaktu kov-polovi5, kde nedochází k vytvo9ení
n=6 materiálu do druhého, tzv. Shottkyho bariéry)
bariéry p9i pr;chodu nosi5; z jednoho
je posléze nanesena dostate5n6 velká plocha n=4
:icího kovu (nej5ast6ji zlata), na n6j> lze
1
0 zlata je t9eba plochu
p9ipevnit ultrazvukovou fixací st9íbrn= vodi5. V p9ípad6 vyu>ití
0
0.5
1
1.5
2
2.5
3
3.5
nejprve pokr=t titanem, kter= zvy:uje adhezi Au.
B [T]
Obrázek 21: Magnetotransport pro vzorky E081#1 a D101#8.
Zatímco p7ede8l; vzorek fungoval2skv4le, D101#8 se skute3n4 choval nekulturn4 –
e
signál 2⇡~
byl zna3n4 za8um4n;, pravd4podobn4
vlivem
(ostré
Obrázek 4:neideálního
P(íklad kontaktování naokontaktování
zlaté )icí desky. Autor obrázk*:
Z. V+born+.
⇢xyúzké
= struktury
2nco jsem rychle pro8el kolem vodi39. . . ). Oscila3ní chaxy =poté,
se objevily
2
e
⌫
2⇡~
Courtesy
of L. Nádvorník
(MFF/FZU
rakter sice dob7e viditeln; je, nicmén4 Hall9v odpor
má p7ibli<n4
t7ikrát men8í
sm4rnici,AV)
10
ne< bylo o3ekáváno, zcela neprochází nulou a nevykazuje <ádná plata. Z SdH oscilací
"