Piezoelectric effect in p -Si/SiGe/(001)Si modulation doped heterostructures
We present the results of calculations of the piezoelectric effect in a Si/SiGe
 multilayer structure with a narrow quantum well and a wide layer of doped
 Si semiconductor. The proposed theory is a possible explanation of some
 recent experiments on these structures. Предста...
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| Zitieren: | Piezoelectric effect in p -Si/SiGe/(001)Si modulation doped heterostructures / V.K. Dugaev, O.A. Mironov, S.V. Kosyachenko // Condensed Matter Physics. — 2000. — Т. 3, № 4(24). — С. 835-844. — Бібліогр.: 13 назв. — англ. |
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| author | Dugaev, V.K. Mironov, O.A. Kosyachenko, S.V. |
| author_facet | Dugaev, V.K. Mironov, O.A. Kosyachenko, S.V. |
| citation_txt | Piezoelectric effect in p -Si/SiGe/(001)Si modulation doped heterostructures / V.K. Dugaev, O.A. Mironov, S.V. Kosyachenko // Condensed Matter Physics. — 2000. — Т. 3, № 4(24). — С. 835-844. — Бібліогр.: 13 назв. — англ. |
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| container_title | Condensed Matter Physics |
| description | We present the results of calculations of the piezoelectric effect in a Si/SiGe
multilayer structure with a narrow quantum well and a wide layer of doped
Si semiconductor. The proposed theory is a possible explanation of some
recent experiments on these structures.
Представлені результати розрахунків п’єзоелектричного ефекту в
Si/SiGe багатошаровій структурі з вузькою квантовою ямою і товстим
шаром легованого напівпровідного кремнію Si. Запропонована теорія є можливим поясненням деяких недавніх екпериментів на цих
структурах.
|
| first_indexed | 2025-12-07T18:13:51Z |
| format | Article |
| fulltext |
Condensed Matter Physics, 2000, Vol. 3, No. 4(24), pp. 835–844
Piezoelectric effect in p -Si/SiGe/(001)Si
modulation doped heterostructures
V.K.Dugaev 1 , O.A.Mironov 2 ∗, S.V.Kosyachenko 3
1 Chernivtsi Branch of the Institute of Materials Science Problems,
National Academy of Sciences of Ukraine,
5 Vilde Str., 58001 Chernivtsi, Ukraine
2 Department of Physics, University of Warwick, Coventry, CV4 7AL, UK
3 Trade and Economy Institute of Chernivtsi,
1 Central Square, 58000 Chernivtsi, Ukraine
Received April 17, 2000, in final form October 31, 2000
We present the results of calculations of the piezoelectric effect in a Si/SiGe
multilayer structure with a narrow quantum well and a wide layer of doped
Si semiconductor. The proposed theory is a possible explanation of some
recent experiments on these structures.
Key words: SiGe heterostructures, piezoelectricity
PACS: 77.65.Ly, 85.50.+k, 73.61.-r
1. Introduction
The quantum structures with Si/Si1−xGex layers, realized as superlattices, quan-
tum wells (QW) and heterojunctions have been intensively studied recently because
of their extreme technological importance. It is mostly related with huge industrial
productions of Si-based materials together with their numerous applications. The
Si/SiGe-based structures and superlattices can be used as components of faster mi-
crochips, integrated optoelectronic devices, and many others. The combination of
controlled variation of the composition, strain, and thickness of the layers provides
better electronic and optical properties of the structures as compared to any bulk
semiconductor [1,2].
There exists a well-known problem in the theoretical modelling of certain prop-
erties of the semiconducting multilayered structures. The point is that the energy
diagram (spatial variation of the potential) depends on a number of factors such as
real distribution of electrons and holes, and doping profile as well as on the distribu-
∗On leave from Usikov’s Institute for Radiophysics and Electronics, National Academy of Sci-
ences of Ukraine, 310085 Kharkiv, Ukraine
c© V.K.Dugaev, O.A.Mironov, S.V.Kosyachenko 835
V.K.Dugaev, O.A.Mironov, S.V.Kosyachenko
tion of deformations of the crystal lattice. Besides, electron and hole eigenfunctions
and eigenvalues are determined by the electrostatic potential and deformation pro-
files. From the theoretical point of view it leads to a rather complicated problem
of self-consistent solution of Poisson’s equation together with Schroedinger’s and
elasticity theory’s equations.
The main source of deformations in a multilayer structure is non-compatibility
of crystal lattice of constituents. In the case of the non-centro-symmetrical crystals
like, e.g., III-V compounds, there are also deformations produced by the piezoelectric
effect due to internal electric fields accompanying the potential relief. Recent inves-
tigations of piezoelectric coupling in the AlGaN/InGaN heterostructures [4] give
evidence of its strong effect on the properties of the electron gas in these systems.
It is commonly believed that the piezoelectric effect is absent in Si-based struc-
tures since piezoelectric tensor is equal to zero in the corresponding bulk crystals.
On the other hand the crystallic symmetry is changed with the creation of quantum
structures. That change may lead to a nonvanishing nonuniform piezoelectricity.
Recently it was discovered experimentally [5–7] that when a pulsed electric field
is applied to a Si/SiGe structure with an electric-field vector ~E along the growth
direction, the deformation waves appear in such structures. It clearly demonstrates
the existence of the piezoelectricity. Up till now the nature of this effect in the
structures with separate QW is not quite well understood.
The absence of the piezoelectric effect in bulk Si or Si1−xGex solid solutions is
due to their symmetry [8]: the corresponding crystallic class Oh contains the center
of the inversion. It becomes clear from the same considerations that the non-zero
components of the piezoelectric tensor γi,jk can arise if a distinguished direction
arises in the structure.
In the case of a structure with symmetrical Si/SiGe/Si QW (QW for holes being
located on the SiGe solid solution region) there is a distinguished axis perpendicular
to the QW plane, but there is no any definite direction of this axis. Thus, in the
average, piezoelectric tensor has to vanish in this case. The situation changes when
a nonuniform doping is used, as it was in the case of the cited experiments [5–7],
see figure 1. Indeed, if the x axis is chosen along [001] direction, the symmetry class
changes from Oh to C4v, in which case γx,yy = γx,zz, γy,yx = γz,zx, and the γx,xx
components can be nonzero. Obviously, these components should depend on x, so
that the maximum value is reached in a vicinity of the QW vanishing deep in the
bulk of Si and SiGe. The physical nature of the piezoelectric effect becomes even
more clear if we keep in mind a simple model with separated charges where on the
one side there is the QW with a number of carriers (holes, in case of p-doping, as
shown in figure 1), and on the other side there is a layer of uncompensated charged
acceptors. Consequently, when an AC electric field ~E is applied along x, the covers
of this “capacitor” will be moving, thus exciting the acoustic waves.
836
Piezoelectric effect in modulation doped heterostructures
Figure 1. Schematical picture of the Si/SiGe heterostructure.
2. Model and results of calculations
The energy diagram illustrating the position of valence band edge and the filling
with holes is presented schematically in figure 2, for a structure consisting of the
QW and some layer inside Si doped with acceptors. The size-quantized levels are
located within QW near the Si/SiGe interface.
Let us consider a simplified model (figure 3), in which the wide doped region of
width d is located at a distance b (spacer width) from the QW with only one size-
quantization level below the Fermi energy µ by ε0. For simplicity, from now on we
deal with a model with free electrons instead of the holes. Correspondingly, we sup-
pose that the region d is doped with donor impurities. Let the average concentration
of the donors be equal to n0.
The two-dimensional layer of electrons, trapped into QW, can be described by
a distribution of electrons ν δ(x− d− b), where ν is the two-dimensional density of
electrons. The Poisson’s equation for the electrostatic potential ϕ(x) in the region
837
V.K.Dugaev, O.A.Mironov, S.V.Kosyachenko
Figure 2. Energy diagram of the Si/SiGe heterostructure.
0 < x < d is
d2ϕ
dx2
= −
4πe
ε
[n(x)− n0] , (1)
where n(x) is the distribution of free electrons at 0 < x < d and ε is the dielectric
constant.
Within the semiclassical approximation for the electronic gas at 0 < x < d, the
density n(x) and the electrostatic potential ϕ(x) are related by (we put h̄ = 1)
[
3π2n(x)
]2/3
= 2m∗ [µ− eϕ(x)] , (2)
where m∗ is the electron effective mass. The semiclassical approximation is justified
due to the large value of d as compared to the electron wavelength λ. Due to the
electrical neutrality, the total number of the electrons inside the doping region and
the localized electrons trapped into the QW, is constant
∫ d
0
n(x)dx+ ν = n0 d . (3)
We assume that the deviation of concentration from the equilibrium δn(x) =
n(x)− n0 is small, δn(x) ≪ n0. Then the solution of equations (1) and (2) has the
form
ϕ(x) = ϕ1e
−x/Lc + ϕ2e
x/Lc , (4)
838
Piezoelectric effect in modulation doped heterostructures
Figure 3. Simplified model of a structure with narrow quantum well.
δn(x) = −
em∗kF
π2
(
ϕ1e
−x/Lc + ϕ2e
x/Lc
)
, (5)
where kF is the Fermi momentum, Lc is the screening length in the doped region
Lc = (επ/4kFm
∗e2)1/2 , (6)
and ϕ1, ϕ2 are certain constants. Using equation (3), we can find a relation between
ϕ1 and ϕ2
ϕ2 =
1
exp(d/Lc)− 1
[
π2ν
em∗kFLc
− ϕ1
(
1− exp
(
−
d
Lc
))]
. (7)
Let us find the variation of potential ϕ(x) at x < 0 and x > d (figure 4). The
Poisson’s equation (1) has a nonzero right-hand side for x = d+ b
d2ϕ
dx2
= −
4πe
ε
νδ(x− d− b) (8)
and vanishes for x > d and x < 0 (we neglect a small nonzero carrier concentration
in these regions). Let the external field ~E be applied along the x axis. Then we can
write
ϕ(x) =
−Ex+ A , x < 0
ϕ1 exp(−x/Lc) + ϕ2 exp(x/Lc) , 0 < x < d
Bx+ C , d < x < d+ b
−Ex+D , x > d+ b ,
(9)
839
V.K.Dugaev, O.A.Mironov, S.V.Kosyachenko
Figure 4. Variation of electrostatic potential for the model heterostructure with
external electric field.
where A,B,C,D are constants, which should be determined from the conditions of
matching at x = 0, x = d, and x = d+ b.
The equation determining a charge density in QW for nonzero potential ϕd+b ≡
ϕ(d+ b) is given by
ν =
m∗
π
(µ+ ε0 − e ϕd+b) . (10)
With the aid of equations (4)–(9), and matching ϕ(x) at the interfaces x = 0, x = d,
and x = d+ b, we find
ϕ1 =
1
2 sinh(d/Lc)
[
ELc
(
exp
d
Lc
− 1
)
+
π2ν
emkFLc
]
, (11)
ϕ2 =
1
2 sinh(d/Lc)
[
−ELc
(
1− exp
(
−
d
Lc
))
+
π2ν
emkFLc
]
, (12)
ν = ρ0
[
µ+ ε0 + eE
(
b− Lc
1− cosh(d/Lc)
sinh(d/Lc)
)] [
1 +
4b
aB
+
π
kFLc
coth(d/Lc)
]
−1
, (13)
where ρ0 = m∗/π is the two-dimensional density of states, µ = (3π2n0)
2/3/2m∗, and
840
Piezoelectric effect in modulation doped heterostructures
aB = ε/m∗e2 is the effective Bohr radius.
Using (4), (10)–(12), we can find the distribution of electric field E(x) inside the
region 0 < x < d (in fact, only a part of it depending on the external electric field
will be taken into account here, see below)
E(x) = −
dϕ
dx
= −
E
sinh(d/Lc)
(
sinh
d− x
Lc
+ sinh
x
Lc
− ζ
(b/Lc) sinh(d/Lc)− 1 + cosh(d/Lc)
sinh(d/Lc) [1 + (4b/aB) + ς coth(d/Lc)]
sinh
x
Lc
)
, (14)
where the notation ζ = π/kFLc is used.
Using equation (5), the distribution of charge density q(x) in 0 < x < d region
(only a part independent on E) can be found
q(x) = −e · δn(x) =
eρ0 cosh(x/Lc)
Lc sinh(d/Lc)
·
µ+ ε0
1 + (4b/aB) + ς coth(d/Lc)
. (15)
The distribution of volume forces acting on the lattice and caused by the external
electric field, within the approximation linear in E, for 0 < x < d (the linear
approximation is the reason why in the equation 14 we can take into account only
the part which is independent on E) is given by
f(x) = q(x)E(x) = −
γeρ0(µ+ ε0)E
2Lc sinh
2(d/Lc)
[
sinh
d
Lc
+ sinh
d− 2x
Lc
+ sinh
2x
Lc
−
γζ
sinh(d/Lc)
(
b
Lc
sinh
d
Lc
− 1 + cosh
d
Lc
)
sinh
2x
Lc
]
, (16)
where
γ =
(
1 +
4b
aB
+ ζ coth
d
Lc
)
−1
. (17)
The stress tensor σxx is related to the distribution of forces f(x) by the equation
[9]
dσxx
dx
+ f(x) = 0 . (18)
All other components of σij are zero. Making use of (15) and (16)–(17), we obtain
σxx(x) = −
∫ x
0
f(x)dx =
γeρ0(µ+ ε0)E
4 sinh2(d/Lc)
[
2x
Lc
sinh
d
Lc
+ cosh
d
Lc
− cosh
2x− d
Lc
+ (1− p)
(
cosh
2x
Lc
− 1
)]
, (19)
where we denote
p =
γζ
sinh(d/Lc)
(
b
Lc
sinh
d
Lc
− 1 + cosh
d
Lc
)
. (20)
841
V.K.Dugaev, O.A.Mironov, S.V.Kosyachenko
The maximum stress arises in the region d < x < d+b, where it does not depend
on x, because the bulk charge is absent inside this region. The corresponding value
is determined by equation (18) with x = d
σ(max)
xx =
γeρ0(µ+ ε0)E
2
[(
d
Lc
+ γζ
)
+ 1−
γςb
Lc
− γζ coth
(
d
Lc
)]
. (21)
From this follows the expression for the deformation tensor (within the isotropic
continuum approximation [9])
uij =
1
Ẽ
[(1 + σ)σij − σσijδij ] , (22)
where Ẽ is the elasticity module, and σ is the Poisson factor. Using (19)-(20), we
find
uxx =
γeρ0(µ+ ε0)E
2Ẽ
[(
d
Lc
+ γζ
)
+ 1−
γζb
Lc
− γζ coth
d
Lc
]
, (23)
uyy = uzz = −
γeρ0(µ+ ε0)σE
2Ẽ
[(
d
Lc
+ γζ
)
+ 1−
γζb
Lc
− γζ coth
d
Lc
]
. (24)
Equations (23) and (24) give us the maximum value of deformations produced by
the external electric field in the spacer region.
3. Discussion
Within the framework of our simplified model we have found the nonzero com-
ponents of the deformation tensor depending on the applied electric field E and
on the geometrical parameters of the structure. It can be a possible explanation
of the piezoelectric effect in nonuniformly doped heterostructures. The obtained
piezomodules correspond to C3v or C4v symmetry classes near the interface. (When
the structure is grown in (111) direction we obtain C3v and when it is grown in the
(001) direction we get C4v).
There is a possibility of other explanations of the non-vanishing piezoelectricity
in the structures under consideration. One of them is related to a reconstruction of
the crystallic lattice in the vicinity of the interface [10,11]. In this case an alloy layer
of lower-symmetry crystallic structure can be formed permitting a bulk piezoeffect.
Our consideration shows that the effect should take place in the structure even
without formating the low-symmetry crystallic structures.
Another model of piezoelectricity in superlattices has been proposed in [12]. It
predicts the generation of high-frequency acoustic waves of the order of hundreds
of GHz. It should be noted that this seems to be a too high value for experiments
under consideration (∼ 225 MHz). Besides, the model [12] deals with a periodical
distribution of charges in the superlattice. Thus, it can not be directly applied to
the heterostructure with a separate QW and a region of doped semiconductor like
in the cited experiments.
842
Piezoelectric effect in modulation doped heterostructures
There is also a model of piezoelectricity in semiconductor superlattices [13] re-
ferring to the charge-density domain motion.
It is hard to compare deformation values for the aforementioned models because
there were no numerical estimations for them. For our model, from the equation (23)
and for the bulk value of SiGe elasticity module, we have obtained that the value of
deformations may be up to 0.1% of the layer thickness.
We believe that a rather simple theoretical explanation of the piezoelectric effect
in non-periodical heterostructures presented here can still be valid even in the case
when some other mechanisms like those cited above are involved.
4. Acknowledgements
This work was partially supported by the Science and Technology Center of
Ukraine under Grant No. 591.
References
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tronics, edited by T. Ando et al., Springer, Berlin, 1998, p. 264–271.
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ture of strained Si/Ge superlattices. // Phys. Rev. B, vol. 47, No. 12, p. 7104–7124.
3. O’Reilly E.P. Valence band engineering in strained-layer structures. // Semicond. Sci.
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Gnezdilov V.P. Observation of piezoelectric-like behavior in coherently strained B-
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6. Khizhny V.I., Mironov O.A., Makarovskii O.A., Braithwaite G., Mattey N.L.,
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(001)SiGe/Si heterostructures. // Acta Phys. Pol., 1995, vol. 88, No. 4, p. 779–782.
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843
V.K.Dugaev, O.A.Mironov, S.V.Kosyachenko
13. Schomburg E., Brandl S., Hofbeck K. et al. Generation of millimeter waves with a
GaAs/AlAs superlattice oscillator. // Appl. Phys. Lett., 1998, vol. 72, No. 12, p. 1498–
1500.
П’єзоелектричний ефект в модуляційно легованих
гетероструктурах p -Si/SiGe/(001)Si
В.K.Дугаєв 1 , O.A.Miронов 2 , С.В.Косяченко 3
1 Чернівецький відділ Інституту матеріалознавства НАН України,
58001 Чернівці, вул. І.Вільде, 5
2 Фізичний факультет університету Уорвік,
Ковентрі, CV4 7AL, Великобританія
3 Буковинський фінансово-економічний інститут,
58000 Чернівці, вул. Штерна, 1
Отримано 17 квiтня 2000 р., в остаточному вигляді –
31 жовтня 2000 р.
Представлені результати розрахунків п’єзоелектричного ефекту в
Si/SiGe багатошаровій структурі з вузькою квантовою ямою і товстим
шаром легованого напівпровідного кремнію Si. Запропонована тео-
рія є можливим поясненням деяких недавніх екпериментів на цих
структурах.
Ключові слова: гетероструктури SiGe, п’єзоелектрика
PACS: 77.65.Ly, 85.50.+k, 73.61.-r
844
|
| id | nasplib_isofts_kiev_ua-123456789-120979 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1607-324X |
| language | English |
| last_indexed | 2025-12-07T18:13:51Z |
| publishDate | 2000 |
| publisher | Інститут фізики конденсованих систем НАН України |
| record_format | dspace |
| spelling | Dugaev, V.K. Mironov, O.A. Kosyachenko, S.V. 2017-06-13T12:04:25Z 2017-06-13T12:04:25Z 2000 Piezoelectric effect in p -Si/SiGe/(001)Si modulation doped heterostructures / V.K. Dugaev, O.A. Mironov, S.V. Kosyachenko // Condensed Matter Physics. — 2000. — Т. 3, № 4(24). — С. 835-844. — Бібліогр.: 13 назв. — англ. 1607-324X DOI:10.5488/CMP.3.4.835 PACS: 77.65.Ly, 85.50.+k, 73.61.-r https://nasplib.isofts.kiev.ua/handle/123456789/120979 We present the results of calculations of the piezoelectric effect in a Si/SiGe
 multilayer structure with a narrow quantum well and a wide layer of doped
 Si semiconductor. The proposed theory is a possible explanation of some
 recent experiments on these structures. Представлені результати розрахунків п’єзоелектричного ефекту в
 Si/SiGe багатошаровій структурі з вузькою квантовою ямою і товстим
 шаром легованого напівпровідного кремнію Si. Запропонована теорія є можливим поясненням деяких недавніх екпериментів на цих
 структурах. This work was partially supported by the Science and Technology Center of
 Ukraine under Grant No. 591. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Piezoelectric effect in p -Si/SiGe/(001)Si modulation doped heterostructures П’єзоелектричний ефект в модуляційно легованих гетероструктурах p -Si/SiGe/(001)Si Article published earlier |
| spellingShingle | Piezoelectric effect in p -Si/SiGe/(001)Si modulation doped heterostructures Dugaev, V.K. Mironov, O.A. Kosyachenko, S.V. |
| title | Piezoelectric effect in p -Si/SiGe/(001)Si modulation doped heterostructures |
| title_alt | П’єзоелектричний ефект в модуляційно легованих гетероструктурах p -Si/SiGe/(001)Si |
| title_full | Piezoelectric effect in p -Si/SiGe/(001)Si modulation doped heterostructures |
| title_fullStr | Piezoelectric effect in p -Si/SiGe/(001)Si modulation doped heterostructures |
| title_full_unstemmed | Piezoelectric effect in p -Si/SiGe/(001)Si modulation doped heterostructures |
| title_short | Piezoelectric effect in p -Si/SiGe/(001)Si modulation doped heterostructures |
| title_sort | piezoelectric effect in p -si/sige/(001)si modulation doped heterostructures |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/120979 |
| work_keys_str_mv | AT dugaevvk piezoelectriceffectinpsisige001simodulationdopedheterostructures AT mironovoa piezoelectriceffectinpsisige001simodulationdopedheterostructures AT kosyachenkosv piezoelectriceffectinpsisige001simodulationdopedheterostructures AT dugaevvk pêzoelektričniiefektvmodulâcíinolegovanihgeterostrukturahpsisige001si AT mironovoa pêzoelektričniiefektvmodulâcíinolegovanihgeterostrukturahpsisige001si AT kosyachenkosv pêzoelektričniiefektvmodulâcíinolegovanihgeterostrukturahpsisige001si |