Spin splitting of surface states in HgTe quantum wells
We report on beating appearance in Shubnikov–de Haas oscillations in conduction band of 18–22 nm HgTe quantum wells under applied top-gate voltage. Analysis of the beatings reveals two electron concentrations at the Fermi level arising due to Rashba-like spin splitting of the first conduction subb...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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| Cite this: | Spin splitting of surface states in HgTe quantum wells / A.A. Dobretsova, Z.D. Kvon, S.S. Krishtopenko, N.N. Mikhailov, S.A. Dvoretsky // Физика низких температур. — 2019. — Т. 45, № 2. — С. 185-191. — Бібліогр.: 34 назв. — рос. |
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Dobretsova, A.A. Kvon, Z.D. Krishtopenko, S.S. Mikhailov, N.N. Dvoretsky, S.A. 2021-02-02T19:32:27Z 2021-02-02T19:32:27Z 2019 Spin splitting of surface states in HgTe quantum wells / A.A. Dobretsova, Z.D. Kvon, S.S. Krishtopenko, N.N. Mikhailov, S.A. Dvoretsky // Физика низких температур. — 2019. — Т. 45, № 2. — С. 185-191. — Бібліогр.: 34 назв. — рос. 0132-6414 https://nasplib.isofts.kiev.ua/handle/123456789/175794 We report on beating appearance in Shubnikov–de Haas oscillations in conduction band of 18–22 nm HgTe quantum wells under applied top-gate voltage. Analysis of the beatings reveals two electron concentrations at the Fermi level arising due to Rashba-like spin splitting of the first conduction subband H₁. The difference ΔNs in two concentrations as a function of the gate voltage is qualitatively explained by a proposed toy electrostatic model involving the surface states localized at quantum well interfaces. Experimental values of ΔNs are also in a good quantitative agreement with self-consistent calculations of Poisson and Schrödinger equations with eightband k p⋅ Hamiltonian. Our results clearly demonstrate that the large spin splitting of the first conduction subband is caused by surface nature of H₁ states hybridized with the heavy-hole band. Виявлено появу биття в осциляціях Шубнікова–де Гааза в зоні провідності HgTe квантової ями завтовшки 18–22 нм при прикладенні верхньої затворної напруги. Аналіз биття вказує на два типи електронів з різними концентраціями на рівні Фермі, що виникають внаслідок рашба-подібного спінового розщеплення першої підзони провідності H₁. Різниця двох концентрацій ΔNs як функція затворної напруги якісно пояснюється запропонованою спрощеною електростатичною моделлю поверхневих станів, локалізованих на гетерограниці квантових ям. Експериментальні значення ΔNs також знаходяться в хорошій кількісній згоді з самоузгодженими розрахунками рівнянь Пуассона та Шредінгера для восьмизонного k · p гамільтоніана. Отримані результати наочно демонструють, що велике спінове розщеплення першої підзони провідності обумовлено поверхневою природою станів H₁, гібридизованих із зоною важких дірок. Обнаружено появление биений в осцилляциях Шубникова–де Гааза в зоне проводимости HgTe квантовой ямы толщиной 18–22 нм при приложении верхнего затворного напряжения. Анализ биений указывает на два типа электронов с различными концентрациями на уровне Ферми, возникающих вследствие рашба-подобного спинового расщепления первой подзоны проводимости H₁. Разность двух концентраций ΔNs как функция затворного напряжения качественно объясняется предложенной упрощенной электростатической моделью поверхностных состояний, локализованных на гетерограницах квантовых ям. Экспериментальные значения ΔNs также находятся в хорошем количественном согласии с самосогласованными расчетами уравнений Пуассона и Шредингера для восьмизонного k · p гамильтониана. Полученные результаты наглядно демонстрируют, что большое спиновое расщепление первой подзоны проводимости обусловлено поверхностной природой состояний H₁, гибридизованных с зоной тяжелых дырок This work was supported by the Russian Foundation for Basic Research (projects no. 17-52-14007) and Government of Novosibirsk region (projects no. 17-42-543336). ru Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур Спеціальний випуск. «XXII Уральська міжнародна зимова школа з фізики напівпровідників» (20–23 лютого, 2018) Spin splitting of surface states in HgTe quantum wells Спінове розщеплення поверхневих станів в квантових ямах HgTe Спиновое расщепление поверхностных состояний в квантовых ямах HgTe Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Spin splitting of surface states in HgTe quantum wells |
| spellingShingle |
Spin splitting of surface states in HgTe quantum wells Dobretsova, A.A. Kvon, Z.D. Krishtopenko, S.S. Mikhailov, N.N. Dvoretsky, S.A. Спеціальний випуск. «XXII Уральська міжнародна зимова школа з фізики напівпровідників» (20–23 лютого, 2018) |
| title_short |
Spin splitting of surface states in HgTe quantum wells |
| title_full |
Spin splitting of surface states in HgTe quantum wells |
| title_fullStr |
Spin splitting of surface states in HgTe quantum wells |
| title_full_unstemmed |
Spin splitting of surface states in HgTe quantum wells |
| title_sort |
spin splitting of surface states in hgte quantum wells |
| author |
Dobretsova, A.A. Kvon, Z.D. Krishtopenko, S.S. Mikhailov, N.N. Dvoretsky, S.A. |
| author_facet |
Dobretsova, A.A. Kvon, Z.D. Krishtopenko, S.S. Mikhailov, N.N. Dvoretsky, S.A. |
| topic |
Спеціальний випуск. «XXII Уральська міжнародна зимова школа з фізики напівпровідників» (20–23 лютого, 2018) |
| topic_facet |
Спеціальний випуск. «XXII Уральська міжнародна зимова школа з фізики напівпровідників» (20–23 лютого, 2018) |
| publishDate |
2019 |
| language |
Russian |
| container_title |
Физика низких температур |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| format |
Article |
| title_alt |
Спінове розщеплення поверхневих станів в квантових ямах HgTe Спиновое расщепление поверхностных состояний в квантовых ямах HgTe |
| description |
We report on beating appearance in Shubnikov–de Haas oscillations in conduction band of 18–22 nm HgTe
quantum wells under applied top-gate voltage. Analysis of the beatings reveals two electron concentrations at the
Fermi level arising due to Rashba-like spin splitting of the first conduction subband H₁. The difference ΔNs in
two concentrations as a function of the gate voltage is qualitatively explained by a proposed toy electrostatic
model involving the surface states localized at quantum well interfaces. Experimental values of ΔNs are also in a
good quantitative agreement with self-consistent calculations of Poisson and Schrödinger equations with eightband k p⋅ Hamiltonian. Our results clearly demonstrate that the large spin splitting of the first conduction
subband is caused by surface nature of H₁ states hybridized with the heavy-hole band.
Виявлено появу биття в осциляціях Шубнікова–де Гааза в
зоні провідності HgTe квантової ями завтовшки 18–22 нм
при прикладенні верхньої затворної напруги. Аналіз биття
вказує на два типи електронів з різними концентраціями на
рівні Фермі, що виникають внаслідок рашба-подібного спінового розщеплення першої підзони провідності H₁. Різниця
двох концентрацій ΔNs як функція затворної напруги якісно
пояснюється запропонованою спрощеною електростатичною
моделлю поверхневих станів, локалізованих на гетерограниці
квантових ям. Експериментальні значення ΔNs також знаходяться в хорошій кількісній згоді з самоузгодженими розрахунками рівнянь Пуассона та Шредінгера для восьмизонного
k · p гамільтоніана. Отримані результати наочно демонструють, що велике спінове розщеплення першої підзони провідності обумовлено поверхневою природою станів H₁, гібридизованих із зоною важких дірок.
Обнаружено появление биений в осцилляциях Шубникова–де Гааза в зоне проводимости HgTe квантовой ямы толщиной 18–22 нм при приложении верхнего затворного напряжения. Анализ биений указывает на два типа электронов
с различными концентрациями на уровне Ферми, возникающих вследствие рашба-подобного спинового расщепления
первой подзоны проводимости H₁. Разность двух концентраций ΔNs как функция затворного напряжения качественно
объясняется предложенной упрощенной электростатической
моделью поверхностных состояний, локализованных на гетерограницах квантовых ям. Экспериментальные значения
ΔNs также находятся в хорошем количественном согласии с
самосогласованными расчетами уравнений Пуассона и Шредингера для восьмизонного k · p гамильтониана. Полученные
результаты наглядно демонстрируют, что большое спиновое
расщепление первой подзоны проводимости обусловлено
поверхностной природой состояний H₁, гибридизованных с
зоной тяжелых дырок
|
| issn |
0132-6414 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/175794 |
| citation_txt |
Spin splitting of surface states in HgTe quantum wells / A.A. Dobretsova, Z.D. Kvon, S.S. Krishtopenko, N.N. Mikhailov, S.A. Dvoretsky // Физика низких температур. — 2019. — Т. 45, № 2. — С. 185-191. — Бібліогр.: 34 назв. — рос. |
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| fulltext |
Low Temperature Physics/Fizika Nizkikh Temperatur, 2019, v. 45, No. 2, pp. 185–191
Spin splitting of surface states in HgTe quantum wells
A.A. Dobretsova1,2, Z.D. Kvon1,2, S.S. Krishtopenko3,4,
N.N. Mikhailov1, and S.A. Dvoretsky1
1Rzhanov Institute of Semiconductor Physics, Novosibirsk 630090, Russia
2Novosibirsk State University, Novosibirsk 630090, Russia
3Institute for Physics of Microstructures RAS, GSP-105, Nizhni Novgorod 603950, Russia
4Laboratoire Charles Coulomb, UMR CNRS 5221, University of Montpellier, Montpellier 34095, France
E-mail: kvon@thermo.isp.nsc.ru
dobretsovaaa@gmail.com
Received October 11, 2018, published online December 20, 2018
We report on beating appearance in Shubnikov–de Haas oscillations in conduction band of 18–22 nm HgTe
quantum wells under applied top-gate voltage. Analysis of the beatings reveals two electron concentrations at the
Fermi level arising due to Rashba-like spin splitting of the first conduction subband H1. The difference ΔNs in
two concentrations as a function of the gate voltage is qualitatively explained by a proposed toy electrostatic
model involving the surface states localized at quantum well interfaces. Experimental values of ΔNs are also in a
good quantitative agreement with self-consistent calculations of Poisson and Schrödinger equations with eight-
band ⋅k p Hamiltonian. Our results clearly demonstrate that the large spin splitting of the first conduction
subband is caused by surface nature of H1 states hybridized with the heavy-hole band.
Keywords: spin splitting, Rashba effect, surface states, Shubnikov–de Haas oscillations, quantum wells.
Introduction
Thin films based on HgTe are known by a number of its
unusual properties originating from inverted band structure
of HgTe [1–4]. The latter particularly results in existence
of topologically protected gapless states, arising at HgTe
boundaries with vacuum or materials with conventional
band structure. Although these states were theoretically pre-
dicted more than 30 years ago [5–7], clear experimental
confirmation was not possible at that time due to lack of
growth technology of high quality HgTe-based films. Exper-
imental investigations of wide (the width 70d ≥ nm)
strained HgTe quantum wells (QWs), which started only in
2011, confirmed existence of the predicted surface states and
revealed their two-dimensional (2D) nature [4,8,9].
In comparison with other materials with the inverted
band structure, in which the surface states are known being
Dirac-like [10–12], HgTe spectrum involves heavy-hole
band 8| , 3 / 2Γ ± 〉 modifying the surface state dispersion. Alt-
hough strain opens a bulk band-gap and results thus in three
dimensional (3D) topological insulator state of wide HgTe
quantum wells [4,8,9], it does not cancel strong hybridiza-
tion of the surface states with the 8| , 3 / 2Γ ± 〉 band. As a re-
sult, the surface states in strained HgTe films can be resolved
only at large energies, while at the low ones they are indis-
tinguishable from conventional heavy-hole states [13,14].
In thin films of 3D topological insulator the surface states
from the opposite boundaries may be coupled by quantum
tunneling, so that small thickness-dependent gap is opened
up [15–17]. In strained HgTe thin films, the latter arises
deeply inside the heavy-hole band at the energies signifi-
cantly lower than the top of the valence band [4]. In the ul-
trathin limit, the HgTe quantum well transforms into sem-
imetal [2,18] and then to 2D topological insulator [1,19]
with both gapped surface and quantized bulk states.
On the other hand, the electronic states in HgTe QWs are
classified as hole-like nH , electron-like nE or light-hole-like
nLH levels according to the dominant contribution from the
bulk 8| , 3 / 2Γ ± 〉, 6| , 1 / 2Γ ± 〉 or 8| , 1 / 2Γ ± 〉 bands at zero
quasimomentum = 0k [19]. The strong hybridization in
inverted HgTe QWs results in the upper branch of the
gapped surface states being represented by the 1H sub-
band [4]. At large quasimomentum k the wave-functions of
1H subband are localized at the QW interfaces, while at Γ
point of the Brillouin zone they are localized in the QW cen-
ter and are thus indistinguishable from other 2D states.
The gapped surface states in the films of 3D topological
insulators exhibit sizable Rashba-type spin splitting, arising
due to electrical potential difference between the two surfac-
es [20]. Such spin splitting was first observed in QWs of
Bi2Se3 [21], which is a conventional 3D topological insula-
tor with Dirac-like surface states [10–12,21]. The spin split-
© A.A. Dobretsova, Z.D. Kvon, S.S. Krishtopenko, N.N. Mikhailov, and S.A. Dvoretsky, 2019
A.A. Dobretsova, Z.D. Kvon, S.S. Krishtopenko, N.N. Mikhailov, and S.A. Dvoretsky
ting of the gapped surface states also exists in HgTe QWs
and should be naturally connected with the splitting of the
1H subband. Previous experimental studies of 12–21 nm
wide HgTe QWs [22–24] have attributed large spin splitting
of the 1H subband to the Rashba mechanism in 2D systems
[25,26], enhanced by narrow gap, large spin-orbit gap be-
tween the 8| , 1 / 2Γ ± 〉 and 7| , 1 / 2Γ ± 〉 bands, and the heavy-
hole character of the 1H subband. The latter however con-
tradicts the fact that the splitting of other subbands 2H , 3H ,
4H etc. with the heavy-hole character is significantly lower.
In this work, we investigate spin splitting of conduction
band in 18–22 nm HgTe QWs with asymmetrical potential
profile tuned by applied top gate voltage. The beating pat-
tern of Shubnikov–de Haas (ShdH) oscillations, observed in
all the samples at the applied top gate voltage, reveals two
electron concentrations at the Fermi level due to the spin
splitting of the 1H subband. Experimental difference in the
concentrations as a function of the gate voltage is qualita-
tively explained by a proposed toy electrostatic model in-
volving the surface states at the QW interfaces. Self-
consistent Hartree calculations based on eight-band ⋅k p
Hamiltonian [27], being in good quantitative agreement with
the experimental data, clearly show that the large Rashba-
like spin splitting of the 1H subband is caused by the surface
nature of 1H states hybridized with the heavy-hole states.
Experiment
Our experiments were carried out on undoped 22 nm
(#081112) and symmetrically n-doped 18 nm (#130213)
HgTe quantum wells with (013) surface orientation. The
samples were grown by molecular beam epitaxy, the de-
tailed description of their preparation can be found in
[28,29]. The cross section of the structures is shown in
Fig. 1(a). The structures were patterned into Hall bars with
metallic top gate, distances between the contacts 100 and
250 µm and the bar width 50 µm. Electron concentration
of n-doped sample #130213 at zero gate voltage was
11= 7.3 10sN ⋅ cm–2. The experiments were performed at
temperatures from 2 to 0.2 K and magnetic fields up to 8 T.
For magnetotransport measurements the standard lock-in
technique was used with the excitation current 100 nA and
frequencies 6–12 Hz. In this study we were interested in
electron transport when only the first conduction subband is
occupied. Electron concentration was thus in the range
11(1 9) 10− ⋅ cm–2. The electron mobility in this region was
rather high (see Fig. 1(b)) within 10–60 m2/(V·s) for
undoped and 8–20 m2/(V·s) for doped samples.
Let us consider our results obtained for the undoped
structures first. In Fig. 2 longitudinal resistivity xxρ as a
function of magnetic field B is shown for top gate voltages
gV from 0 to 7 V. Due to good sample quality Shubnikov–
de Haas oscillations are already seen at 0.4 T. The key ex-
perimental result is an appearance of oscillation beatings at
gate voltage > 3gV V, whereas at = 0gV V resistivity oscil-
lations are homogeneous. The oscillation beatings give an
evidence of presence of two carrier types in the system with
close concentrations. Fourier analysis of resistivity depend-
ence on inverse magnetic field 1( )xx B−ρ with monotone
background removed indeed shows two nearby peaks (see
Fig. 3(a)). From the Fourier analyzes two electron concen-
trations 1sN and 2sN can be straight calculated by
= /si iN ef h , where we denote by 1f and 2f the lower and
upper frequency positions of the Fourier peaks correspond-
ingly. Note the above expression is written for spin non-
degenerate electrons, this is justified since at considering
gate voltage range only the first conduction subband is
occupied.
Although the Fourier analysis enables finding electron
concentrations reasonably precisely, we found more accu-
rate getting the frequencies from fitting of Shubnikov–de
Haas oscillations by Lifshits–Kosevich formula [30–32]:
0 =1,2
2
= ( )exp cos ,xx i
i i
qii
f
A D X
B B
∆ρ π−π + φ ρ µ
∑ (1)
Fig. 1. (a) The cross section of the structures studied. (b) Transport
mobility dependence on electron concentration for undoped
(#081112) and symmetrically n-doped (#130213) samples.
Fig. 2. (Color online) Longitudinal resistivity xxρ dependences
on magnetic field B at top gate voltages = 0 7gV − V obtained
for undoped 22 nm HgTe quantum well #081112.
186 Low Temperature Physics/Fizika Nizkikh Temperatur, 2019, v. 45, No. 2
Spin splitting of surface states in HgTe quantum wells
where 0ρ is the monotone resistivity part and =xx∆ρ
0( )xx= ρ −ρ is the oscillatory part; ( ) = / sinh( )D X X X is
the thermal damping factor with 2= 2 /B cX k Tπ ω , Bk
being Boltzmann constant and cω being cyclotron frequen-
cy; qiµ are the quantum mobilities; iA and iφ are some con-
stants.
Before fitting the experimental curves we first removed
any residual background, which we extracted from the initial
curves by Fourier filtering. iA , iφ , iµ and if were used as
fitting parameters. We used frequencies achieved from Fou-
rier analysis (see Fig. 3(a)) as starting frequency values. To
increase sensitivity to the low-field data we used the weight
of 10 for data points at magnetic field less than 0.7 T. The
fits were always excellent over the full field range, the ex-
ample of fitting curve for = 7gV V is shown in Fig. 3(c).
Concentrations 1sN and 2sN obtained from the fitting pro-
cess described above as functions of gate voltage are shown
in Fig. 3(b). The sum of two concentrations 1 2s sN N+
matches very well with the total concentration sN obtained
from Hall measurements.
An additional advantage of oscillation fitting is obtain-
ing quantum mobilities qiµ , which are shown in Fig. 3(d)
as functions of the total electron concentration sN , 1qµ and
2qµ are almost the same and do not change in a full con-
centration range from 5·1011 to 9·1011 cm–2, also they are
more than one order smaller than the transport mobility
shown in Fig. 1(b). The difference between transport and
quantum mobilities implies presence of long-range scatter-
ing, which might be electron density inhomogeneities.
The experimental results for symmetrically n-doped
quantum well #130213 are shown in Figs. 4 and 5. Figure 4
shows longitudinal resistivity dependences on magnetic field
( )xx Bρ measured at top gate voltages gV from 0 to –4 V.
Here oscillations are also homogeneous at zero gate voltage
while at < 1gV − V a beating in the oscillations arises
providing two peaks in Fourier transformation of (1/ )xx B∆ρ
(see Fig. 5(a)), xx∆ρ is again the oscillatory part of xxρ .
Since electron mobility in these structures is smaller than in
the undoped ones (see Fig. 1(b)), oscillations arise only at
1B T. Together with the elimination at large B by Zee-
man splitting it enables only one beating being resolved.
Since the beating shifts to larger fields with decreasing gate
voltage at < 3gV − V it disappears due to overlapping with
Zeeman splitting.
We performed the same data processing procedure for
the sample #130213 as we did it for #081112. While fitting
the experimental curves by Eq. (1) we also used the weight
of 10 for data points at magnetic field less than 1.5–2 T to
increase sensitivity to the low-field data. We were succeed
to fit all the curves well over the full field range (see, as
example, Fig. 5(c)), the sum of two concentrations ob-
tained from fitting is in agreement with Hall measurements
(see Fig. 5(b)). Quantum mobilities shown in Fig. 5(d) are
as well as the quantum mobilities in the undoped structure
Fig. 3. (Color online) Results obtained for undoped 22 nm HgTe
quantum well #081112: (a) fast Fourier transformation of
1( )xx B−ρ at gate voltage = 7gV V. (b) Electron concentrations
1sN (red circles) and 2sN (blue triangular) and their sum (green
squares) obtained from Shubnikov–de Haas oscillations and total
electron concentration sN obtained from Hall measurements
(pink line) versus gate voltage. (c) The oscillatory resistivity part
xx∆ρ normalized to the monotone resistivity part 0ρ versus in-
verse magnetic field. Black line shows the result obtained exper-
imentally at = 7gV V while red line is the fitting curve calculated
by Eq. (1). (d) Quantum mobilities 1qµ and 2qµ versus total elec-
tron concentration.
Fig. 4. (Color online) Longitudinal resistivity xxρ dependences
on magnetic field B at top gate voltages gV from 0 to –4 V ob-
tained for symmetrically n-doped 18 nm HgTe quantum well
#130213.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2019, v. 45, No. 2 187
A.A. Dobretsova, Z.D. Kvon, S.S. Krishtopenko, N.N. Mikhailov, and S.A. Dvoretsky
almost the same, do not change in a presented concentration
range and one order smaller than the transport mobility.
Discussion
Beating pattern of Shubnikov–de Haas oscillations at
high gate voltages, while at = 0gV the oscillations are ho-
mogeneous, in both symmetrically doped and undoped
QWs, indicates the origin of the spin splitting being asym-
metry of the QW profile, changing with gV . Let us first
demonstrate that the difference in the electron concentra-
tions extracted from the ShdH oscillations can be qualita-
tively explained by a toy electrostatic model involving the
surface states at QW interfaces. This model was previously
proposed for wide HgTe quantum wells [9], and here we
briefly repeat its derivation.
As for the relative changes in the concentrations, the in-
itial conditions are not important, therefore, for simplicity,
we assume electron concentrations on the top and bottom
surfaces being the same at zero gV . Figure 6 schematically
shows simplified band diagrams and electron distribution
over the surface states for a structure with metallic top gate
at zero and positive gate voltages. In the absence of gate
voltage, the Fermi level remains the same across the struc-
ture. When gate voltage is applied, the Fermi level differs in
the metallic gate and QW layer by geV , where e is the ele-
mentary charge. Since the left surface is closer to the gate,
it partially screens the gate potential from the right surface.
The change of electron concentration 2sN∆ at the left sur-
face exceeds thus its changing 1sN∆ at the right one. In
their turn, the difference in the concentrations induces an
additional electrical potential growth HgTeeφ between left
and right surfaces, while the Fermi level over the QW layer
remains constant. The difference in the concentrations can
be written as =si Fi iN E D∆ ∆ , where iD ( = 1,2i ) is the den-
sity of states and FiE∆ is the local change of the Fermi en-
ergy for the right (1) and left (2) surface states. 1FE∆ and
2FE∆ are connected thus as 2 1 HgTe=F FE E e∆ ∆ + φ . The
potential difference between the two surface states can be
evaluated from the charge neutrality and the Gauss's law as
HgTe HgTe eff 1 eff HgTe 0= = /sd e N dφ ∆ , where effd is the
effective distance between the opposite surface states and
HgTe is electric field in the well. Here, we neglect a distor-
tion of the QW profile from the linear dependence caused by
distribution of charge carriers in the bulk of QW layer. Fi-
nally, we find
2
2 1 2 1 eff 2 HgTe 0/ = / / .s sN N D D e d D∆ ∆ + (2)
The effective distance between the surface states effd can
differ from the QW width due to localization of the surface
states wave-functions not exactly on the boundaries of HgTe
layer. In addition, the QW width in our samples is compara-
ble with the scale of surface states localization [33] to ex-
clude the interaction between electrons at different bounda-
ries. Parameter effd can be evaluated by fitting experimental
value of 2 1 2 1/ ( / ) / ( / )s s s g s gN N dN dV dN dV∆ ∆ with
Eq. (2). It gives eff = 9d nm for the sample #081112 with
2 1( / ) / ( / ) = 1.43s g s gdN dV dN dV (see Fig. 3), which looks
very reasonable for given QW width.
Fig. 5. (Color online) Results obtained for symmetrically n-doped
18 nm HgTe quantum well #130213: (a) fast Fourier transfor-
mation of 1( )xx B−ρ at gate voltage = 1.5gV − V. (b) Electron
concentrations 1sN (red circles) and 2sN (blue triangular) and
their sum (green squares) obtained from Shubnikov–de Haas
oscillations and total electron concentration sN obtained from
Hall measurements (pink line) versus gate voltage. (c) The oscil-
latory resistivity part xx∆ρ normalized to the monotone resistivity
part 0ρ versus inverse magnetic field. Black line shows the result
obtained experimentally at = 1.5gV − V while red line is the fit-
ting curve calculated by Eq. (1). (d) Quantum mobilities 1qµ and
2qµ versus total electron concentration.
Fig. 6. Simplified band diagram and electron distribution over
surface states for gate voltages = 0gV (a) and > 0gV (b).
188 Low Temperature Physics/Fizika Nizkikh Temperatur, 2019, v. 45, No. 2
Spin splitting of surface states in HgTe quantum wells
Let us obtain the expression for the difference in elec-
tron concentrations at two different surfaces =sN∆
2 1s sN N= − as a function of the total concentration sN .
Now, the initial distribution of electrons over the structure
becomes important. For simplicity, we assume that = 0sN
for symmetric QW profile at = 0gV , and all electrons at
non-zero gV come to the HgTe layer due to the top gate
voltage. Thus, from =si siN N∆ and 1 2=s s sN N N+ , we
get linear dependence of sN∆ on sN :
2 1
2 1
/ 1
= .
/ 1
s s
s s
s s
N N
N N
N N
∆ ∆ −
∆
∆ ∆ +
(3)
Figure 7(a) provides a comparison between experimental
data and estimation within our toy electrostatic model (pre-
sented by green curve) for the undoped sample #081112.
Here, we used = 20 , eff = 9d nm and 2 1= =D D
2* / 2m= π valid for parabolic dispersion of the surface
states. The latter holds since hybridization with heavy
holes modifies the band dispersion of the surface states,
making it close to parabolic. From cyclotron resonance
measurements [34] the effective mass of the surface states
was obtained equal to *
00.026m m≈ , with 0m being free
electron mass.
Our toy electrostatic model is seen perfectly reproducing
the slope of the experimental behavior of ( )s sN N∆ . More-
over, it can fit experimental data if one assumes the residual
concentration of 114 10⋅ cm–2 in the absence of gate voltage.
Note that this value is twice higher than it was measured for
the sample #081112 at = 0gV (see Fig. 3). The difference
between theoretical estimation and experimental values
gives the evidence of the importance of microscopic details
of the surface states, which were completely ignored with-
in our toy model.
Therefore, we also perform self-consistent calculations of
Poisson and Schrödinger equations with 8-band ⋅k p Hamil-
tonian [27]. These calculations take into account all micro-
scopic details of the surface states and thus allow obtaining a
realistic QW profile. As it is done for a toy electrostatic
model, here we also assume that all electrons at non-zero
gV come to the HgTe layer due to the top gate. At the final
iteration of solving self-consistently Poisson and Schrö-
dinger equations, we obtain energy dispersions ( )E k (k is
a quasimomentum in the QW plane). Then, for a given
value of sN , we find the position of Fermi level and obtain
the values of Fermi wave-vectors 1k and 2k . Finally, we
find electron concentrations by 2= / 4si iN k π. Theoretical
values of ( )s sN N∆ found from self-consistent calculations
are shown in Fig. 7(a) by blue curve and are in a good
agreement with the experimental data.
Figure 7(c) provides an energy dispersion of the surface
states at 11= 9 10sN ⋅ cm–2, where they are represented by
1H subband due to hybridization with the states of heavy-
hole band. Surface state connection with the 1H subband is
also supported by Fig. 7(d). The figure shows theoretical
QW profile and wave-functions of the states at the Fermi
level (see green curves). Spin-split states corresponding to
1k and 2k wave-vectors are clearly seen to localize at the
opposite boundaries of HgTe QW. Large overlapping be-
tween the surface states in our samples also explains only
qualitative agreement of the experimental data with our toy
electrostatic model. We note that hybridization of the sur-
face states with the heavy-hole band is partially included in
the toy model by using expression for the density of states
2*= / 2D m π , which is inherent for parabolic spectrum.
The dashed black curves show dispersion of the surface
states neglecting hybridization with the heavy holes. The
surface states mixing with the 8| , 3 / 2Γ ± 〉 band is indeed
seen transforming the linear dispersion of surface states
into parabolic. Interestingly, the spin splitting of the sur-
face states is significantly suppressed if the hybridization is
included.
( )s sN N∆ obtained experimentally for the n-doped struc-
ture #130213 is shown in Fig. 7(b). This pattern contradicts
Fig. 7. (Color online) (a) and (b) show the difference between elec-
tron concentrations 2 1=s s sN N N∆ − as a function of total concen-
tration sN obtained experimentally (red circles) for samples
#081112 (a) and #130213 (b). In (a) green line corresponds to cal-
culations within toy electrostatic model, while blue line shows self-
consistent calculations of Poisson and Schrödinger equations with
eight-band Kane model Hamiltonian. (c) and (d) show results of the
self-consistent calculations of Poisson and Schrödinger equations
for electron concentration 11 2= 9 10 cmsN −⋅ . All electrons are
assumed coming to the well due to top gate voltage. (c) shows the
energy spectrum, where black dashed lines correspond to surface
states without hybridization with heavy holes. (d) shows HgTe
quantum well potential profile (blue and red lines are 8Γ and 6Γ
bands correspondingly) and squared absolute values of wave
functions of electron states at the Fermi level (green lines).
Low Temperature Physics/Fizika Nizkikh Temperatur, 2019, v. 45, No. 2 189
A.A. Dobretsova, Z.D. Kvon, S.S. Krishtopenko, N.N. Mikhailov, and S.A. Dvoretsky
our expectations of the spin splitting increasing with the
absolute gate voltage value and decreasing thus electron
concentration. The reason is likely the presence of only one
beating in the ShdH oscillations and thus less precise elec-
tron concentration determination. As seen in Fig. 5(b) it is
not crucial for determination of the total electron concen-
tration however seems significant for that of electron con-
centration difference.
Conclusion
To sum up we have investigated Rashba-like spin split-
ting of the conduction 1H band in 18–22 nm HgTe quantum
wells. Beating pattern of Shubnikov–de Haas oscillations,
arising with applying top gate voltage in both undoped and
symmetrically n-doped structures, provides two close elec-
tron concentrations. We have qualitatively described the
evolution of the difference between these concentrations
with gate voltage by a toy electrostatic model involving
electron states localization at the well interfaces. The quan-
titative agreement between the experimental data and theo-
retical calculations was achieved by self-consistent solving
Poisson and Schrödinger equations with eight-band ⋅k p
Hamiltonian, which takes into account microscopic details
of the surface states omitted in our toy model. Comparison
of the toy electrostatic model with the rigorous self-
consistent calculations clearly shows large spin-splitting of
1H subband in the HgTe quantum wells being due to the
surface nature of its states.
Acknowledgments
This work was supported by the Russian Foundation for
Basic Research (projects no. 17-52-14007) and Govern-
ment of Novosibirsk region (projects no. 17-42-543336).
_______
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___________________________
Спінове розщеплення поверхневих станів
в квантових ямах HgTe
А.А. Добрецова, З.Д. Квон, С.С. Кріштопенко,
М.М. Михайлов, С.А. Дворецький
Виявлено появу биття в осциляціях Шубнікова–де Гааза в
зоні провідності HgTe квантової ями завтовшки 18–22 нм
при прикладенні верхньої затворної напруги. Аналіз биття
вказує на два типи електронів з різними концентраціями на
рівні Фермі, що виникають внаслідок рашба-подібного спі-
нового розщеплення першої підзони провідності H1. Різниця
двох концентрацій ΔNs як функція затворної напруги якісно
пояснюється запропонованою спрощеною електростатичною
моделлю поверхневих станів, локалізованих на гетерограниці
квантових ям. Експериментальні значення ΔNs також знахо-
дяться в хорошій кількісній згоді з самоузгодженими розра-
хунками рівнянь Пуассона та Шредінгера для восьмизонного
k · p гамільтоніана. Отримані результати наочно демонстру-
ють, що велике спінове розщеплення першої підзони провід-
ності обумовлено поверхневою природою станів H1, гібриди-
зованих із зоною важких дірок.
Ключові слова: спінове розщеплення, ефект Рашби, поверх-
неві стани, осциляції Шубнікова–де Гааза, квантові ями.
Спиновое расщепление поверхностных состояний
в квантовых ямах HgTe
А.А. Добрецова, З.Д. Квон, С.С. Криштопенко,
Н.Н. Михайлов, С.А. Дворецкий
Обнаружено появление биений в осцилляциях Шубнико-
ва–де Гааза в зоне проводимости HgTe квантовой ямы тол-
щиной 18–22 нм при приложении верхнего затворного на-
пряжения. Анализ биений указывает на два типа электронов
с различными концентрациями на уровне Ферми, возникаю-
щих вследствие рашба-подобного спинового расщепления
первой подзоны проводимости H1. Разность двух концентра-
ций ΔNs как функция затворного напряжения качественно
объясняется предложенной упрощенной электростатической
моделью поверхностных состояний, локализованных на ге-
терограницах квантовых ям. Экспериментальные значения
ΔNs также находятся в хорошем количественном согласии с
самосогласованными расчетами уравнений Пуассона и Шре-
дингера для восьмизонного k · p гамильтониана. Полученные
результаты наглядно демонстрируют, что большое спиновое
расщепление первой подзоны проводимости обусловлено
поверхностной природой состояний H1, гибридизованных с
зоной тяжелых дырок.
Ключевые слова: спиновое расщепление, эффект Рашбы,
поверхностные состояния, осцилляции Шубникова–де Гааза,
квантовые ямы.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2019, v. 45, No. 2 191
https://doi.org/10.1103/PhysRevB.39.1120
https://doi.org/10.1103/PhysRevB.71.155310
https://doi.org/10.1103/PhysRevB.92.165314
Introduction
Experiment
Discussion
Conclusion
Acknowledgments
|