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12
High-Frequency Capacitor with Working Substance "Insulator-Undoped Silicon-Insulator"
N.A. Poklonski, I.I. Anikeev, S.A. Vyrko
Belarusian State University, Nezavisimosti Ave., 4, Minsk 220030, Belarus
Received 10.11.2022
Accepted for publication 14.12.2022
The study of the parameters of capacitors with various working substances is of interest for the design and creation of electronic elements, in particular for the development of high-frequency phase-shifting circuits.
The purpose of the work is to calculate the high-frequency capacitance of a capacitor with the working substance "insulator-undoped silicon-insulator" at different applied to the capacitor direct current (DC) voltages, measuring signal frequencies and temperatures.
A model of such the capacitor is proposed, in which 30 (m thick layer of undoped (intrinsic) crystalline silicon (i-Si) is separated from each of the capacitor electrodes by 1 (im thick insulator layer (silicon dioxide).
The dependences of the capacitor capacitance on the DC electrical voltage U on metal electrodes at zero frequency and at the measuring signal frequency of 1 MHz at absolute temperatures T = 300 and 400 K are calculated. It is shown that the real part of the capacitor capacitance increases monotonically, while the imaginary part is negative and non-monotonically depends on U at the temperature T = 300 K. An increase in the real part of the capacitor capacitance up to the geometric capacitance of oxide layers with increasing temperature is due to a decrease in the electrical resistance of i-Si layer. As a result, with an increase in temperature up to 400 K, the real and imaginary parts of the capacitance take constant values independent of U. The capacitance of i-Si layer with an increase in both temperature T and voltage U is shunted by the electrical conductivity of this layer. The phase shift is determined for a sinusoidal electrical signal with a frequency of 0.3, 1, 10, 30, 100, and 300 MHz applied to the capacitor at temperatures 300 and 400 K.
Keywords: undoped (intrinsic) crystalline silicon, silicon dioxide, capacitance, three-layer flat capacitor. DOI: 10.21122/2220-9506-2022-13-4-247-255
Адрес для переписки: Address for correspondence:
Поклонский Н.А. Poklonski N.A.
Белорусский государственный университет, Belarusian State University,
пр-т Независимости, 4, г. Минск 220030, Беларусь Nezavisimosti Ave., 4, Minsk 220030, Belarus
e-mail: poklonski@bsu. by e-mail: poklonski@bsu. by
Для цитирования: For citation:
N.A. Poklonski, I.I. Anikeev, S.A. Vyrko. N.A. Poklonski, I.I. Anikeev, S.A. Vyrko.
High-Frequency Capacitor with Working Substance High-Frequency Capacitor with Working Substance
"Insulator-Undoped Silicon-Insulator". "Insulator-Undoped Silicon-Insulator".
Приборы и методы измерений. Devices and Methods of Measurements.
2022. - Т. 13, № 4. - С. 247-255. 2022, vol. 13, no. 4, pp. 247-255.
DOI: 10.21122/2220-9506-2022-13-4-247-255 DOI: 10.21122/2220-9506-2022-13-4-247-255
Высокочастотный конденсатор с рабочим веществом «изолятор - нелегированный кремний - изолятор»
Н.А. Поклонский, И.И. Аникеев, С.А. Вырко
Белорусский государственный университет, пр-т Независимости, 4, г. Минск 220030, Беларусь
Поступила 10.11.2022 Принята к печати 14.12.2022
Исследование параметров электрических конденсаторов с различными рабочими веществами представляет интерес для проектирования и создания элементов электроники, в частности для разработки высокочастотных фазосдвигающих цепей.
Цель работы - рассчитать высокочастотную электрическую емкость конденсатора с рабочим веществом «изолятор - нелегированный кремний - изолятор» при различных подаваемых на конденсатор постоянных напряжениях, частотах измерительного сигнала и температурах.
Предложена модель такого конденсатора, в которой слой нелегированного (собственного) кристаллического кремния (/-Si) толщиной 30 мкм отделен от каждого из электродов конденсатора слоем изолятора (диоксида кремния) толщиной 1 мкм.
Рассчитаны зависимости емкости конденсатора от постоянного электрического напряжения U на металлических электродах на нулевой частоте и на частоте измерительного сигнала 1 МГц при абсолютных температурах T = 300 и 400 К. Показано, что действительная часть емкости конденсатора монотонно возрастает, а мнимая часть отрицательна и немонотонно зависит от U при температуре T = 300 К. Увеличение действительной части емкости конденсатора до геометрической емкости оксидных слоев при увеличении температуры обусловлено уменьшением электрического сопротивления слоя i-Si. Вследствие этого с увеличением температуры до 400 К действительная и мнимая части емкости принимают постоянные значения, независящие от U. Емкость слоя i-Si при увеличении как температуры T, так и напряжения U шунтируется электрической проводимостью этого слоя. Определен сдвиг фаз для синусоидального электрического сигнала с частотой 0,3; 1; 10; 30; 100 и 300 МГц, подаваемого на конденсатор при температурах 300 и 400 К.
Ключевые слова: нелегированный (собственный) кристаллический кремний, диоксид кремния, электрическая емкость, трехслойный плоский электрический конденсатор.
DOI: 10.21122/2220-9506-2022-13-4-247-255
Адрес для переписки: Address for correspondence:
Поклонский Н.А. Poklonski N.A.
Белорусский государственный университет, Belarusian State University,
пр-т Независимости, 4, г. Минск 220030, Беларусь Nezavisimosti Ave., 4, Minsk 220030, Belarus
e-mail: poklonski@bsu. by e-mail: poklonski@bsu. by
Для цитирования: For citation:
N.A. Poklonski, I.I. Anikeev, S.A. Vyrko. N.A. Poklonski, I.I. Anikeev, S.A. Vyrko.
High-Frequency Capacitor with Working Substance High-Frequency Capacitor with Working Substance
"Insulator-Undoped Silicon-Insulator". "Insulator-Undoped Silicon-Insulator".
Приборы и методы измерений. Devices and Methods of Measurements.
2022. - Т. 13, № 4. - С. 247-255. 2022, vol. 13, no. 4, рр. 247-255.
DOI: 10.21122/2220-9506-2022-13-4-247-255 DOI: 10.21122/2220-9506-2022-13-4-247-255
Introduction
Nonlinear screening of an external stationary electric field in materials occurs when the density of the charge induced by the field in them is not proportional to the total electrostatic potential of the field and charges. An analytical and numerical solution of the problem of nonlinear screening of an impurity ion by a spherically symmetric cloud of mobile charges of opposite sign in covalent crystalline semiconductors is given in [1]. It was also noted there that the concept of nonlinear screening can be justified only when the root-mean-square fluctuations of the potential energy of mobile un-compensated charges are less than the thermal energy for nondegenerate semiconductors and less than the Fermi energy for degenerate semiconductors. With nonlinear screening of an external electrostatic field, alternating layers with nonequilib-rium (due to illumination) electrons and holes can appear in a "semi-infinite" semiconductor [2]. It is also possible that a three-layer structure "negatively charged layer-electrically neutral layer-positively charged layer" appears in a gas plasma located in a strong electric field between metal electrodes (cathode and anode) [3].
In work [4] low-frequency electrical losses in n-type crystalline silicon placed in a capacitor with deep impurity energy levels were studied. The low-frequency admittance and the phase shift angle between the sinusoidal current and voltage in the capacitor with the working substance "insulator-par-tially disordered silicon-insulator" were calculated in [5]. The frequency dependence of the admittance and the nonlinear capacitance on the voltage of the "semiconductor-insulator-metal-insulator-semiconductor" structure, which simulates semiconductor materials with metallic nanosized inclusions, was also theoretically studied [6]. However, the temperature dependences of the admittance and capacitance of such a structure were not calculated. The influence of the process of formation of electrically neutral pairs of defects from electrically charged defects on the chaotic electrostatic potential on the surface of a semiconductor was studied [7], and the conditions for strong localization of a two-dimensional electron gas on the surface in the presence of this potential were determined [8].
In work [9], the differential capacitance of a semiconductor film with an ohmic contact on the back side was calculated. The distribution of the
electrostatic potential induced by an external stationary electric field over the thickness of a semiconductor film deposited on an insulator is considered in [10], and on a metal in [11]. In Refs. [12, 13], the working substance "insulator-intrinsic semiconductor-insulator" of a low-frequency capacitor was considered, however, the calculation of the frequency and temperature dependences of its capacitance was not carried out. Thus, the calculation of the high-frequency electrophysical characteristics of a capacitor with an undoped semiconductor separated by insulating layers from its metal plates is still an urgent task in electronics. This is important for building a theory of phase-shifting electrical circuits with a nonlinear capacitor.
Here we note that in [14], the possibility of implementing an electrical analogue of Rayleigh-Be-nard cells in a flat capacitor "insulator-nondegene-rate n-type semiconductor-insulator" by creating a stationary electric potential difference between its metal plates was theoretically studied. These cells represent ring electron currents in the semiconductor cross section, which seems important for the study of synergistic electronic states and processes in device structures.
The purpose of the work is to calculate the high-frequency capacitance of a capacitor with the working substance "insulator-undoped crystalline silicon-insulator" at different applied to the capacitor direct current (DC) voltages, measuring signal frequencies and temperatures. Silicon dioxide is considered as an insulator.
Equivalent circuit of three-layer capacitor
Let a layer of undoped (intrinsic) crystalline semiconductor (i-Si) with thickness Ls and side surface area A be located in the middle between metal plates of a flat electric capacitor and separated from them by insulator layers (SiO2); see Figure 1a. The capacitor is connected to a DC voltage source U. There is no electric charge at the SiO2/i-Si interfaces at U = 0. The x axis of the Cartesian coordinate system is perpendicular to the surface of i-Si layer occupying the space 0 < x < Ls; the y and z axes are parallel to the layer surface. Let us assume that in one part of i-Si layer the potential of the external stationary electric field on the surface is positive 9(x = 0) = +9s / 2, and in the other part it is negative 9(x=Ls) = /2. Then the electric potential difference applied to the semiconductor layer is
Metal
SÍÜ2
Rs
+
yk
i-Si
C
+
ЧН
C
C
г
0
Ls
+
C
Figure 1 - Cross section of a capacitor with a layer of intrinsic crystalline semiconductor (i-Si) of thickness Ls and area A in yz plane; semiconductor is separated from metal plates (electrodes) by insulator layers (SiO2) of thickness of Lox. Across the semiconductor layer (along x axis), an electric potential difference is applied between two electrodes (anode and cathode) parallel to yz plane (a). Equivalent circuit of the device structure (b). Simplified equivalent circuit of the device structure (c)
Us = ^(x = 0) - ^(x=Ls) = 9. The electrodes are parallel to the yz plane, so that the electric potential 9 depends only on x and is independent of y and z. Screening of the external electrostatic field is due to the redistribution of electrons in the c-band and holes in the v-band inside i-Si layer.
The capacitor with the working substance "silicon dioxide-intrinsic silicon-silicon dioxide" contains a parallel RsCsg-circuit of i-Si layer, connected in series with the capacitances of the Cox insulating layers (see Figure 1b). Here, the complex electrical resistance of the semiconductor layer Rs = Rs(Us, ©) = = Rs1(Us, ©) - i Rs2(Us, ©), i = (-1)1/2 is the imaginary unit, Csg = Cs + Cg is the electrical capacitance of the semiconductor, Cox = Asox /Lox and Cg = Ass /Ls are the geometric capacitances of a single SiO2 layer and i-Si layer with static permittivities sox = sroxs0 and ss = srss0; srox and srs are the relative permittivities of SiO2 and i-Si, respectively, s0 = 8.85 pF/m is the electrical constant, Cs = Cs(Us) is the differential capacitance of the semiconductor, Us is the voltage across the semiconductor. In the equivalent circuit (Figure 1c), the capacitance C = C(U, ©) of the capacitor depends on the DC voltage U and the angular frequency © of the measuring signal.
The real C1 and imaginary C2 parts of the complex capacitance C = C(U, ©) = С1(U, ©) + i C2(U, ©) of the equivalent circuit (see Figure 1c) are [15-17]:
Cox
C1 = Re C = -^f [1 + ©Rs2(Cox/2 + 2Csg) +
+ ©2(Rs2 + Rsl)Csg(Cox/2 + Csg)], (1) C I C -©Rs1(Cox /2)2 < 0
C2 = Im C =---< 0,
where
S = [1 + ©Rs2(Cox /2 + Csg)]2 + ©2Rs?(Cox /2 + Csg)2,
U (= Udc) is the stationary electric voltage on metal plates of the capacitor, © is the angular frequency of the alternating current (AC) component of the measuring signal with the amplitude | Uac| << Udc; Rs1 > 0 and Rs2 > 0; see Eq. (13) below.
The real Y1 and imaginary Y2 parts of the complex admittance Y = Y(U, ©) = Y1 + i Y2 of the capacitor equivalent circuit (Figure 1c) are:
Y1 = Re Y = -©C2, (2)
Y2 = Im Y = ©C1. ()
From Eq. (2) we find the phase shift 9 = 9(U, ©) between the alternating current and the voltage on the capacitor with the working substance "insulator-intrincis silicon-insulator" as
9 = arctan(-Y2/Y1) = arctan(C1/C2). (3)
Formulas (1)-(3) are transformed into formulas from Refs. [5, 18] if R does not depend on the measuring signal frequency ©.
Since the capacitances of the insulator layers (Cox) and i-Si layer (Csg) are connected in series (see Figure 1b), the charge on each of the layers is equal to Q. Then the complex voltage drops on the insulators Uox = Q/ Cox and on the intrinsic silicon Us = = Q/ (Csg + 1 /i©Rs) are related to the complex capacitor voltage U as U = 2 Uox + Us. To find the value of U, we substitute the charge Q on the capacitor, expressed in terms of Us and (Csg + 1 /i©Rs), into Uox and take the modulus of the complex voltage U = |U| = |2Uox + Us|. As a result, we obtain the ratio of the voltage across the entire capacitor U to the voltage across the semiconductor Us = | U^ as
U
Us '
_2_
с
cse+^++(R^2
C R 2
1/2
(4)
where = ©(Rs2 + Rs2).
Let us consider a layer of undoped (intrinsic) crystalline silicon with volume Vs = ALs, which contains equal concentrations of c-band electrons (symbol and index n) and v-band holes (symbol and index p) with bulk concentrations (see, e.g., [19, 20]):
( E(c)
n = y j gn fn dEn = ncFi/2 ^ J = ncFin(4c), 1 С ш
P = V j gp fp dEp = Pv F1
V s J 0
Efv)
квТ
(5)
=Pv Fi/2(nv),
where the energy densities of states of electrons in the c-band and holes in the v-band are
b
c
x
a
gn = Vs(2m„d)3/2£y2/2rc2£3,
gp = Vs(2mpdf2El/2/2n2h3;
mnd, En and mpd, Ep are the effective masses of the density of states and the kinetic energy of c-band electron and v-band hole; h = h/2n is the Planck constant; the Fermi-Dirac distribution functions for electrons and holes are
fn = {1 + exp[(En - EFc))/kBT ]}-1, fp = {1 + exp[(EFv) - Ep)/kBT ]}-1;
EFc) < 0 and EFv) < 0 are the positions of the Fermi level counted from the bottom of the c-band and the
top of the v-band; -(EFc) + E ) = Eg > 0 is the width of the energy gap (/-Si band gap); кв is the Boltzmann constant; T is the absolute temperature;
nc = 2(2nmndkBT)3/2/(2nÄ)3; pv = 2(2ктрйквТ)т/(2кЬ)ъ;
Fermi-Dirac integral (of index 1/2):
F\/2(nc(v)) =
f Vx [1 + exp(x - n^rdx-Л/п J0
Distribution of space charge density p(x) of v-band holes with concentration p(x) and c-band electrons with concentration n(x) along x axis when an external DC voltage U (= Udc) is applied to metal electrodes (plates) of the capacitor taking into account Eq. (5) is given by the expression:
p(x) = e
= e
[p(x) - n(x)] =
pF (ЕЖ
Pv Fl/21 квТ
- ncFi
1/2
EFc)(x)
квТ
(6)
where e is the elementary charge, E^ Xx) = Efv) -
The electrical neutrality condition for /-type silicon in the absence of an external field (U = 0), when EFc) and EFv) do not depend on x coordinate, has the form:
ni = Pi = [ncPv exp(-Eg /квТ )]
1 /2
(7)
The electrostatic potential 9(x) at the point with coordinate x inside /-Si with the volume density of the induced charge p(x) satisfies the Poisson equation [19, 20]:
d29
dx2
p(x) р(ф(х))
£rs£o
(8)
where the charge density p(x) = p(^(x)) is determined by Eq. (6), the value of the potential on the semiconductor surface is determined by the boundary conditions: 9(x = 0) = / 2 and 9(x = Ls) = /2.
From the solution of the Poisson equation (8), we obtain the charge Qs induced by the external electric field per unit area A of the flat surface of /-Si layer:
Ôs С Ls A = I Р(ф(х)) dx.
A Ls/2
(9)
The differential capacitance per unit area A of /-Si surface, taking into account Eqs. (6)-(9), is
Cs
A
_dô
A dфs ^ Ôs(^s /2)
s = a 8s P(-Фs /2)
(10)
- e<p(x) and EF (x) = -[Eg+EF )(x)] are the positions of the Fermi level relative to the top of the v-band and the bottom of the c-band in an electric field with potential 9(x).
Note that for a nondegenerate gas of c-band electrons and v-band holes the following inequalities hold: EFc)(x) < 0, | EFc)(x)| > 3kBT and EFv)(x) < 0, | EFv)(x)| > 3kBT. Under these conditions, the Fermi-Dirac energy distribution functions of electrons and holes transform into the Maxwell-Boltzmann distribution functions [20, 21]. In this case, the space charge density distribution (6) over the thickness of /-Si layer takes the form (see, e.g., [22, 23]):
p(x) = - 2enisinh(e^(x)/kBT),
where ni = pi is the concentration of c-band electrons (equal to the concentration of v-band holes) in /-Si layer at U = 0.
where the volume charge density p(-^s /2) is determined by Eq. (6), and the charge 0s(-^s /2) is determined by Eq. (9).
Stationary (DC) electrical conductivity of /-Si layer, due to the migration of electrons and holes across the layer thickness, is
Odc = Odcn + Odcp,
eM« С Ls ,4, eMp С Ls /
= L I n(x)dx, Gdcp = L I P(x)dx,
Ls «Z 0 Ls «Z 0
^dcn
(11)
where odcn and odcp are the electron and hole DC electrical conductivities, n(x) and p(x) are the concentrations of c-band electrons and v-band holes given by Eqs. (5), and are the drift mobilities of electrons and holes.
Here we note that the drift mobilities of electrons and holes in undoped silicon are limited by phonon scattering and, under isothermal conditions, do not depend on the coordinate.
AC electrical conductivity of /-type crystalline silicon according to the Drude-Lorentz model, taking into account Eq. (11), is (see, e.g., [19, 20]):
8
s
Table
Parameters of intrinsic (undoped) crystalline silicon
T, K Eg, eV mnd /m0 mpd /m0 m„G/m0 mpG/m0 cm2-V-l-s-i Цp, cm2-V-l-s-i -3 n¡, cm 8rs Reference
300 l.l2 í.062 - 0.259 - i750 450 - ii.6 [24]
l.l242 í.062 i. 16í 0.259 0.708 i450 505 i.O2-iOi0 ii.9 [25]
l.l2 l.O8 0.8l 0.2б - i400 450 ii0i0 ii.7 [2б]
l.l2 i.09 i.OOi - - - - 9.65-í09 - [27]
l.l2 i.09 - - - l430 480 i.O7-iOi0 - [28]
400 l.O95 - - - - - 25 i - - [24]
i.098 - i.239 - 0.74б 722 25 i - - [25]
i.097 - - - - 759 2бЗ 1.65-1012 - [26]
- i.i09 í.068 - - - - i.87-iOi2 - [27]
í.0968 i.ii - - - б97 2З4 6.00-i0i2 - [28]
Note: Eg is the band gap, mnd and mpd are the effective masses of the density of states of c-band electrons and o-band holes, mno and mpo are the effective masses of the electrical conductivity of electrons and holes, and are the drift mobilities of electrons and holes, n is the intrinsic electron concentration equal to the intrinsic hole concentration p1, ers is the relative permittivity of intrinsic silicon at absolute temperatures T = 300 and 400 K; here m0 is the electron mass in vacuum.
Oac = Ol + i O2 = Ol„ + Olp + í(02n + Op),
Oln = Odcn/[1 + (rax«)2], O2n = Odc„©X„/[1 + (®T„)2], (12)
Olp = Odcp/[1 + (Wtp)2], O2p = Odcp©Xp/[1 + (raTp)2],
where Tn = ^nmno/e and tp = ppmpo/e are the quasi-momentum relaxation times of average c-band electron and average v-band hole when they scatter on phonons of /-type silicon crystal lattice, mno and mpo are the effective masses of the electrical conductivity of c-band electrons and v-band holes, ra is the angular frequency of the measuring signal.
From Eq. (12) we obtain the complex electrical resistance of undoped silicon in the form:
Es = L = Rsi - i Rs2, A Gac
Esi =
Ls
Gi
Es2 = "
G2
(i3)
Ubd is the experimentally measured threshold breakdown voltage of SiO2 of thickness Lox. Thus, the voltage U must satisfy the condition U < Ubd, where Ubd ~ 0.4-1 kV for silicon dioxide of thickness Lox = = 1 pm at a stationary external breakdown electric field strength Ebd = 4-10 MV/cm [29-31].
Figure 2 shows the dependence of the voltage Us across /-Si layer of thickness Ls = 30 pm according to Eq. (4) on the voltage U (created by metal electrodes of the capacitor on the surface of each of two SiO2 interlayers of thickness Lox = 1 pm) for measuring
0.б
2 2 s2 2 2
A Oi + o2 A Oi + o2
where o1 = o1n + o1p and o2 = o2n + o2p.
Calculation results and discussion
The calculations of capacitances C1 and C2 were performed for the parameters [24-28] of undoped silicon (/-Si) indicated in Table for its thickness Ls = 30 pm and measuring signal frequencies ra/2n = = 0.3, 1, 30, 100, and 300 MHz. Relative permittivity of silicon dioxide (SiO2) Srox = 3.9 and its thickness Lox = 1 pm. In the calculations it was assumed that the capacitor is under isothermal conditions.
The electrical voltage of the external field on the whole structure should not exceed breakdown field voltage of silicon dioxide (SiO2) Ubd, where
V,
0.4 -
0.2 -
0
0.5
i
i.5
U, V
Figure 2 - Dependence of electrical voltage Us across /'-Si of thickness Ls = 30 ^m on stationary voltage U on the capacitor electrodes for Lox = 1 ^m according Eq. (4), at frequency œ/2rc, MHz: 0 (curves 1, 1'), 0.3 (2, 2') and 1 (3, 3') at T, K: 300 (blue lines 1-3) and 400 (red lines 1'-3' )
3 -
2 -
0 -
-1
0
-3
-4
-5 CD
<N
-6 ^
-7
0.2
0.4
Us, V
0.6
1 1', 3
' 2' __
0.8 L— -
К ^0.6 о 2 _ 1 -
0.4 2_^ -
0.2 i i i
0
0.5
1
U, V
Figure 3 - Dependence of decimal logarithm of real lg(Rs1A, Ohm-cm2) (curves 1,1') and imaginary lg(Rs2A, Ohm-cm2) (curves 2, 2') parts of complex resistance of /-Si layer at frequency ra/2n = 1 MHz on electrical voltage Us across the layer; calculation by Eqs. (13) for Ls = 30 pm at T, K: 300 (blue lines 1, 2) and 400 (red lines 1', 2')
-0.1
-0.2
-0.3
-0.4
1.5
Figure 4 - Dependence of real 2C1/Cox (curves 1, 1', 3) and imaginary 2C2/Cox (2, 2') parts of complex capacitance on voltage U at the capacitor electrodes, calculated by Eqs. (1) for Ls = 30 pm and Lox = 1 pm at T, K: 300 (blue lines 1, 2) and 400 (red lines 1', 2') and ra/2n, MHz: 0 (curve 3) and 1 (curves 1, 1', 2, 2')
0
1
signal frequencies ®/2n = 0, 0.3, and 1 MHz at temperatures T = 300 and 400 K. It is seen that at constant voltage across the capacitor contacts (U = const), the voltage drop Us across /-Si layer decreases with temperature.
Figure 3 shows the dependences of the real Rs1 and imaginary Rs2 parts of the electrical resistance of /-Si layer according to Eqs. (13) on the DC voltage Us across this layer for measuring signal frequency ®/2n = 1 MHz at temperatures T = 300 and 400 K. It is seen that the electrical resistance of /-Si layer decreases with temperature and voltage.
Figure 4 shows the results of calculating the ratio of the real Ci(U, ©) and imaginary C2(U, o>) parts of the complex capacitance to the capacitance of the insulating layers Cox/2 according to Eq. (1). The calculation was performed for different values of voltage U (created by metal electrodes of the capacitor on the surface of SiO2 insulating interlayers, each of thickness Lox = 1 pm) for semiconductor of thickness Ls = 30 pm at absolute temperatures T = 300 and 400 K and measuring signal frequencies ®/2n = = 0 and 1 MHz. As the temperature T increases, the concentrations of electrons and holes increase (see Table), which leads to a decrease in the electrical resistance of /-Si layer and, as a result, to an increase in capacitances C and C2. As the voltage U increases, the capacitance C1 increases monotonically, while the capacitance C2 changes non-monotonically. This is due to the displacement of the majority of
90
75
сл
0
sa
ад
<D ТЗ
CD
1 60
45
0 0.5 1 1.5
U, V
Figure 5 - Dependence of measuring signal phase shift angle 0 at frequency œ/2rc on stationary voltage U at the capacitor electrodes, calculated by Eq. (3) for Ls = 30 pm and Lox = 1 pm at T = 300 K (blue lines); œ/2rc (MHz) = = 0.3 (curve 1), 1 (2), 10 (3), 30 (4), 100 (5), 300 (6) and at temperature T = 400 K (red line 2' for œ/2rc = 1 MHz)
electrons to SiO2 insulating layer adjacent to the anode, and the majority of holes to SiO2 insulating layer adjacent to the cathode.
Figure 5 shows the results of calculating the dependences of the phase shift angle 0 between the alternating current and measuring signal voltage according to Eq. (3) on the constant voltage U (created by metal electrodes of the capacitor on the surface of SiO2 interlayers) at the absolute temperature
T = 300 K for frequencies ®/2n = 0.3, 1, 10, 30, 100, and 300 MHz and at T = 400 K for ®/2n = = 1 MHz. It is seen that with an increase in temperature T, other things being equal, the phase shift angle modulus 191 increases. With an increase in the voltage U on metal electrodes, the calculation gives a non-monotonic dependence of the phase shift angle 9 on U.
Note that, according to Eq. (3), the maximum value of -n/4 (= -45°) is reached by the quantity -n/2 < 9 < 0 at such an angular frequency © when the equality Y2 = Y (or Q = -C2) is satisfied.
Conclusion
A theoretical model of high-frequency capacitor with the working substance "silicon dioxide-intrinsic silicon-silicon dioxide" is developed. A 30 pm thick semiconductor layer of /-type crystalline silicon is separated on both sides from metal plates of the capacitor by 1 pm thick insulating layers of silicon dioxide.
The numerical calculation shows that the real part of the capacitance increases monotonically with voltage at the capacitor electrodes while the imaginary part is negative and non-monotonically depends on the voltage at measuring signal frequency of ©/2n = 1 MHz at the temperature T = 300 K. It is shown that at the temperature T = 400 K the real and imaginary parts of the capacitance of the capacitor take constant values, independent of voltage, which is due to a decrease in the electrical resistance of /-Si layer. The capacitance of /-Si layer with an increase in both the temperature T and the DC voltage U for frequencies ©/2n < 1 MHz is shunted by the electrical conductivity of this layer.
The dependence of the phase shift angle on the voltage of a sinusoidal electrical signal applied to the capacitor for frequencies of 0.3, 1, 10, 30, 100, and 300 MHz at temperatures 300 and 400 K is calculated. It is shown that as the signal frequency increases, the minimum of the phase shift angle modulus shift towards higher voltages, and at a frequency of 1 MHz, with an increase in temperature to 400 K, it reaches a constant value. Note that if a narrow-gap /-InSb is used as an undoped (intrinsic) semiconductor, then such a capacitor will operate at cryogenic temperatures.
Acknowledgments
This work was supported by the Belarusian National Research Program "Materials Science,
New Materials and Technologies" and Grant for Young Researchers by the Ministry of Education of the Republic of Belarus.
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