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The energy spectrum of the carriers in HgTe
quantum wells:
"Myth and Realities"
Grigory Minkov1,2
Andrew Sherstobitov 1,2
Alexander Germanenko 2
Оlga Rut 2
Sergey Dvoretskii 3
Nikolai Mikhailov 3
(decoration of Eugenius Rumyantsev)2
1Institute
of Metal Physics, RAS,
2Ural Federal University
3ИInstitute of Semiconductor Physics, RAS,
What I meant under "Myth and Realities"?
Myths..
Reality.
.
Mythlets
theorist
…
Cooking..
Art!!
experimenter
technologist
experimenter's commandments
Do not believe a word of it ....
not any!,
unless stated what is neglected..
Do not believe a word of it ....
not any!
Additional commandments
Do not trust yourself too…
What I meant under "Myth and Realities"?
Myths..
Experiment
shows that…
Reality.
.
Mythlets
…
experiment never shows nothing ..
Only the interpretation of experiment shows…sometime..
experimenter shows only those results that, as it seems, he understands…
“normal” and gapless semiconductors A2B6
Hg1-xCdxTe
gapless
“normal”
Gap versus width of CdTe/HgTe/CdTe
quantum well
250
200
150
E
Independent quantization of
conduction and valence
bands
Г8
100
Quantization in framework of
the Kane Hamiltonian
50
0
-50
-15 -10 -5
0
5
10 15
k
200
200
E
e3
energy (meV)
e3
e2
e2
e1
100
E
k
100
dc
e1
k
c1
gap
0
0
h1
h1
h2
-100
5
10
h3
15
20
QW width (nm)
25
-100
h2
5
10
15
20
QW width (nm)
25
5
Size quantization and the surface states in semiconductors
V. A. Volkov and T. N. Pinsker
Some history….
Institute of Radio Engineering and Electronics, USSR Academy of
Sciences
(Submitted December 25, 1975)
Не вырожденное
состояние
Zh. Eksp. Teor. Fiz. 70, 2268-2278 (June 1976)
1.
2.
3.
M.I. Dyakonov, A.V. Khaetskii, Size quantization of
holes in semiconductors with complex valence band
and a carriers in gapless semiconductor .
Zh. Eksp. Teor. Fiz. 82, 1584 (1982); Sov. Phys.
JETP 55,. 917 (1982)
Y.R. Lin-Liu and L.J. Sham, Interface states and
subbands in HgTe-CdTe heterostructures. Phys. Rev. B
32, 5561–5563 (1985).
M.V. Kisin and V.I. Petrosyan, Size quantization in
narrow gap and gapless semiconductors,
Fiz.Tekn.Poluprovodn. 22, 829 (1988)
[Sov.Phys.Semicond 22, 523 (1988)].
200
E
e3
e2
E
k
100
dc
e1
k
c1
0
h1
-100
h2
5
10
15
20
QW width (nm)
25
Value of QW width is very important!!
How do you know?: d<dc; d=dc; d>dc?
200
E
e3
e2
E
k
100
dc
e1
k
c1
0
h1
-100
h2
5
10
15
20
QW width (nm)
25
Value of QW width is very important!!
How do you know?: d<dc; d=dc; d>dc?
Very elegant way
Single valley Dirac fermions in
zero-gap HgTe quantum wells
Nature Phys.7:418-422,2011
inverted
normal
IV. QUANTUM HALL EFFECT AND THE IDENTIFICATION OF ZERO-GAP SAMPLES
zero mode
Landau levels
En
En
B
V а не Е!!
gap
B
Let us inspect the result in more detail..
Vg, but not Е!!
1. why does one extrapolate this point-xy=0 ?..
… zero-mode Landau levels!
IV. QUANTUM HALL EFFECT AND THE IDENTIFICATION OF ZERO-GAP SAMPLES
2. H =n*(h/e) ,
n=CV/e C=const
H =CVh/e2
Does not depend on gap Δ !!
(E)
n
EF
0
Vg
3. In principle C=(1/Cg+1/Cq(B,Vg))-1 ..
but with real (Cq) it does not change
the answer!! (печалька…..)
Сильно быстро спешили 1.
greatly hastened quickly??
Peculiarities of the energy spectrum results in unique transport,
optics and other properties……
The energy E(k) for the conduction band is simple enough both
for the wide (d > dc) quantum wells with the inverse subband
ordering and for the narrow (d < dc) wells with the normal
spectrum.
what about valence band spectrum?
What is known about valence band
spectrum?
Сильно быстро спешили 2.
Experimentally:
Theoretically
Is it OK?
What does mean such a coincidence?
4*1012
k2
EF
2
0
0
0
0
0
0
Xx
Xx
Xx
Xx
Xx
2
Yy
0
Yy
0
Yy
Yy
0
Yy
4*1012
What does mean such a coincidence?
Yy
EF
EF
0
0
0
0
k1
k2
k1
Yy
Yy
Сильно быстро спешили 2.
Theoretically:
Experimentally:
0
0
0
Xx
Xx
Xx
2
/2 2
Q/e=(2,4)*(k1-k2)
/2
1-k2)
/2 Q/e=(2,4)*(k1-k2)
2
2
2
2
2
-k2 )/2-k2 )/2
Q/e=(k1
-k2Q/e=(k1
)/2 Q/e=(k1
2
2
2
/2
nH/2
=(2,4)*(k1-k2)
/2
-k2)
 nH=(2,4)*(k1-k2)
2
2
2
n
=(2,4)*(k1-k2)
/2
nH=(2,4)*(k1-k2)
/2
n
=(2,4)*(k1-k2)
/2
2
2
2
H
H
nShdH=(k1-k2)
/2
/2
) /2nShdH=(k1-k2)
2
2
2
nShdH=(k1-k2)
/2
nShdH=(k1-k2)
/2
nShdH
=(k1-k2)
/2
=/ nH
nShdH=/ n
nShdH
H
nShdH=/ nH nShdH=/ n
=/ nH
nShdH
H
2
2
2
Q/e=k
/2
Q/e=k
/2

Q/e=k /2

2
2
2
nH=k /2nH=k /2
nH=k /2
2
2
2
nShdH=k n/2
 =kn/2
 =k /2
ShdH
ShdH
nShdH= nnHShdH= nnHShdH= nH
Our data
0.0
11
Vg= -0.5V
2x10
11
10
2x10
-0.2
1x10
5.5e10*(x-1.8)
n_e
-0.4
10
1x10
0
1.4e9*(x+6)
6e9*(x+0.15)
0
11
0
-1x10
2
Vg
p, n
d/dH(xx/xx(0)
5.8e10*(x-1.8)
1/eRh(0.5kOe)
1/eRh(20)
p_ShdH
11
-2x10
-0.6
11
-3x10
-0.8
Что следует из
совпадения?
11
-4x10
Vg= -9.5V
-1.0
10
20
H, kOe
30
40
11
-5x10
-6
-4
-2
0
Vg
2
4
6
To study spectrum we have found the hole effective mass
from temperature dependence of the amplitude of SdH oscillations
p=1.05*1011см-2
Calculated mass is negative!!
Electron-like states
The energy spectrum is close
to parabolic one at k>8*105 .
Electron-like states, if there
is, occures for small k.
А как при малых k?
ОБЛАСТЬ ПЕРЕКРЫТИЯ.
ПРОВОДИМОСТЬ ДВУМЯ ТИПАМИ НОСИТЕЛЕЙ.
Описывается неплохо; с ростом поля n+p=const,
но n, p могут зависеть от поля..
Проводимость двумя типами носителей
inverted
zero mode
Landau levels
En
En
B
The dispersion E(k) reconstructed from the data
analysis and calculated one within framework of
isotropic six-band kP-model.
This is qualitatively similar but quantitatively ……
What are the possible reasons?......
kP?
single-particle approximation?
What about valence band spectrum at d<dc ? (“normal” spectrum)
Сильно быстро спешили 3.
!!
happy coincidence
Unfortunately there was ungated structure….
Our data for analogous structure:
The only model that describes our results is as follows:
the subband of spatial quantization H1 is strongly split by spin-orbit
interaction into two subbands, H1+ and H1−, due to asymmetry of the
quantum well.
Effective mass, spectrum
At first sight the large spin-orbit splitting of the energy
spectrum in the nominally almost symmetrical structure
seems very surprising….
Peculiarity of QHE
И что мы знаем (понимаем) про HgTe quantum well?
What does we know (understand) about HgTe quantum wells?
Почти ничего….
Спектр валентной зоны….
Транспорт при пересечении zero-mod LL..
Роль краевых состояний…почему они столь слабо protected…
Роль интерференции….
Роль взаимодействия….
we know almost nothing :
Valence band energy spectrum…
Conductivity, Hall effect at zero-mod LL crossing..
Role of edge states…. Why they are so poorly protected…
Role of the interference (weak localization)….
Role of e-e, h-h, e-h interaction….
И только совместные усилия теоретиков, экспериментаторов, технологов
могут как-то прояснить картину….
Only joint efforts of theorists, experimentalists, technologists
can somehow clarify the picture ....
Post Scriptum:
(Conclusion)
Myths..
Reality.
.
Mythlets
…
Cooking..
Art!!
Do not shoot the pianist - he plays as he can…
«Черт прячется в деталях»…..
Готов обсудить их с любой (доступной) степенью подробностей..
"The devil is in the details" .....
Ready to discuss them with any (available) level of details ..
к. №
Thanks…