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Low-Frequency Admittance of Capacitor with Working Substance "Insulator-Partially Disordered Semiconductor-Insulator"
N.A. Poklonski, I.I. Anikeev, S.A. Vyrko
Belarusian State University, Nezavisimosti Ave., 4, Minsk 220030, Belarus
Received 07.07.2021
Accepted for publication 03.09.2021
Abstract
The study of the electrophysical characteristics of crystalline semiconductors with structural defects is of practical interest in the development of radiation-resistant varactors. The capacitance-voltage characteristics of a disordered semiconductor can be used to determine the concentration of point defects in its crystal matrix. The purpose of this work is to calculate the low-frequency admittance of a capacitor with the working substance "insulator-crystalline semiconductor with point ¿-defects in charge states (-1), (0) and (+1)-insulator".
A layer of a partially disordered semiconductor with a thickness of 150 ^m is separated from the metal plates of the capacitor by insulating layers of polyimide with a thickness of 3 ^m. The partially disordered semiconductor of the working substance of the capacitor can be, for example, a highly defective crystalline silicon containing point t-defects randomly (Poissonian) distributed over the crystal in charge states (-1), (0), and (+1), between which single electrons migrate in a hopping manner. It is assumed that the electron hops occur only from t-defects in the charge state (-1) to t-defects in the charge state (0) and from t-defects in the charge state (0) to t-defects in the charge state (+1).
In this work, for the first time, the averaging of the hopping diffusion coefficients over all probable electron hopping lengths via t-defects in the charge states (-1), (0) and (0), (+1) in the covalent crystal matrix was carried out. For such an element, the low-frequency admittance and phase shift angle between current and voltage as the functions on the voltage applied to the capacitor electrodes were calculated at the t-defect concentration of 3-1019 cm-3 for temperatures of 250, 300, and 350 K and at temperature of 300 K for the t-defect concentrations of 11019, 3-1019, and 1 1020 cm-3.
Keywords: partially disordered semiconductor, low-frequency admittance of capacitor, triple-charged intrinsic point defects.
DOI: 10.21122/2220-9506-2021-12-3-202-210
Адрес для переписки:
Поклонский Н.А.
Белорусский государственный университет, пр-т Независимости, 4, г. Минск 220030, Беларусь e-mail: poklonski@bsu.by; poklonski@tut.by
Address for correspondence:
Poklonski N.A. Belarusian State University, Nezavisimosti Ave., 4, Minsk 220030, Belarus e-mail: poklonski@bsu.by; poklonski@tut.by
Для цитирования:
N.A. Poklonski, I.I. Anikeev, S.A. Vyrko.
Low-Frequency Admittance of Capacitor with Working Substance "Insulator-Partially Disordered Semiconductor-Insulator". Приборы и методы измерений. 2021. - Т. 12, № 3. - С. 202-210. DOI: 10.21122/2220-9506-2021-12-3-202-210
For citation:
N.A. Poklonski, I.I. Anikeev, S.A. Vyrko.
Low-Frequency Admittance of Capacitor with Working Substance "Insulator-Partially Disordered Semiconductor-Insulator". Devices and Methods of Measurements. 2021, vol. 12, no. 3, pp. 202-210. DOI: 10.21122/2220-9506-2021-12-3-202-210
Низкочастотный адмиттанс конденсатора с рабочим веществом «изолятор - частично разупорядоченный полупроводник - изолятор»
Н.А. Поклонский, И.И. Аникеев, С.А. Вырко
Белорусский государственный университет, пр-т Независимости, 4, г. Минск 220030, Беларусь
Поступила 07.07.2021 Принята к печати 03.09.2021
Исследование электрофизических характеристик кристаллических полупроводников с дефектами структуры представляет практический интерес при создании радиационно-стойких варакторов. По вольт-фарадным характеристикам разупорядоченного полупроводника можно определять концентрацию точечных дефектов в его кристаллической матрице. Цель работы - рассчитать низкочастотный адмиттанс конденсатора с рабочим веществом «изолятор - кристаллический полупроводник с точечными ¿-дефектами в зарядовых состояниях (-1), (0) и (+1) - изолятор».
Слой частично разупорядоченного полупроводника толщиной 150 мкм отделен от металлических обкладок конденсатора диэлектрическими прослойками из полиимида толщиной 3 мкм. Частично раз-упорядоченный полупроводник рабочего вещества конденсатора представляет собой, например, сильнодефектный кристаллический кремний, содержащий точечные ¿-дефекты, случайно (пуассоновски) распределенные по кристаллу, в зарядовых состояниях (-1), (0) и (+1) между которыми прыжковым образом мигрируют одиночные электроны. Считается, что прыжки электронов происходят только с ¿-дефектов в зарядовом состоянии (-1) на ¿-дефекты в зарядовом состоянии (0) и с ¿-дефектов в зарядовом состоянии (0) на ¿-дефекты в зарядовом состоянии (+1).
В работе впервые проведено усреднение коэффициентов прыжковой диффузии по всем вероятным длинам прыжка электрона между ¿-дефектами в зарядовых состояниях (-1), (0) и (0), (+1) в ко-валентной кристаллической матрице. Для такого элемента рассчитаны низкочастотный адмиттанс и угол сдвига фаз между током и напряжением в зависимости от приложенного на электроды конденсатора напряжения при концентрации ¿-дефектов 3-1019 см 3 для температур 250, 300 и 350 К и при температуре 300 К для концентраций ¿-дефектов 1-1019, 3-1019 и 1-1020 см 3.
Ключевые слова: частично разупорядоченный полупроводник, низкочастотный адмиттанс конденсатора, трехзарядные собственные точечные дефекты.
БОТ: 10.21122/2220-9506-2021-12-3-202-210
Адрес для переписки:
Поклонский Н.А.
Белорусский государственный университет, пр-т Независимости, 4, г. Минск 220030, Беларусь e-mail: poklonski@bsu.by; poklonski@tut.by
Address for correspondence: Poklonski N.A. Belarusian State University, Nezavisimosti Ave., 4, Minsk 220030, Belarus e-mail: poklonski@bsu.by; poklonski@tut.by
Для цитирования:
N.A. Poklonski, I.I. Anikeev, S.A. Vyrko.
Low-Frequency Admittance of Capacitor with Working Substance "Insulator-Partially Disordered Semiconductor-Insulator". Приборы и методы измерений. 2021. - Т. 12, № 3. - С. 202-210. DOI: 10.21122/2220-9506-2021-12-3-202-210
For citation:
N.A. Poklonski, I.I. Anikeev, S.A. Vyrko.
Low-Frequency Admittance of Capacitor with Working Substance "Insulator-Partially Disordered Semiconductor-Insulator". Devices and Methods of Measurements. 2021, vol. 12, no. 3, pp. 202-210. DOI: 10.21122/2220-9506-2021-12-3-202-210
Introduction
In the works [1, 2], for the first time, a variant of controlling the hopping electrical conductivity via hydrogen-like donors along a semiconductor film using an external electrostatic field E(x) = -d^/dx perpendicular to the film surface, which does not lead to the appearance of a current and does not violate the electrical neutrality of the film as a whole, was theoretically considered. However, the hopping electrical conductivity longitudinal to the direction of the controlling external electric field was not considered in [1, 2]. The field effect was studied and the quasi-frequency (low-frequency) capacitance and conductivity of silicon crystals with hopping electron migration via point two-level defects with positive and negative correlation energies in three charge states (-1), (0), and (+1) were calculated [3, 4]. However, the electrical capacity and conductivity of the "in-sulator-partially disordered semiconductor-insulator" structure were not investigated in [3, 4]. For the first time, the static capacitance-voltage characteristics of a Z-diode made of crystalline silicon, in which current was carried only by electron hopping via t-defects, were calculated [5]. However, in the diode model constructed in [5], there was no averaging of diffusion coefficients over all probable electron hopping lengths via t-defects in three charge states (-1), (0), and (+1). Taking into account electron hopping via point defects, the temperature and frequency dependences of the dielectric permittivity of silicon irradiated with a large dose of neutrons were studied [6]. The low-frequency electrical capacitance as well as the electric field and potential distribution for the "metal-insulator-intrinsic semiconductor-insulator-metal" structure were calculated [7-9]. However, the capacitance-voltage characteristics for the structure with a disordered semiconductor layer were not calculated in [7-9]. The results of an experiment on measuring the capacitance of a thin-film capacitor (structure Al-Al2O3-Al) were interpreted [10] taking into account quantum effects. A method was described [11, 12] for determining, from the temperature dependences of capacitance and conductivity, the ionization energy and concentration of deep centers in an overcompensated semiconductor placed between insulator plates (40-100 ^m thick polyethylene terephthalate), to which a sinusoidal voltage was applied through copper contacts. However, in [11, 12] the experimental data on the conductivity and capacitance of the studied structure were not compared with theory.
The purpose of this work is to calculate the low-frequency admittance of a capacitor with the working substance "insulator-crystalline semiconductor with point ¿-defects in charge states (-1), (0) and (+1) with hopping migration of electrons between them-insulator".
Model of capacitor with working substance "insulator-partially disordered semiconductor-insulator"
Let a wafer of highly defective crystalline silicon (hd-Si) with a thickness of ds and a surface area A be in the middle between the metal plates of a flat capacitor and separated from them by the layers of insulator (e.g., polyimide) with a thickness of di (Figure 1a). The capacitor is connected to a constant electrical voltage source. The x coordinate axis is perpendicular to the surface of the semiconductor wafer occupying space -ds/2 < x < ds/2, the y and z coordinate axes are parallel to the wafer surface.
H Metal
0~\
+
hd- Si
Insulator
/
-0
b
RS
Q
S-1
Hb _Q£ ЧЬ* + kjjj -
Q
Q GL
-ds/2 0 ds/2
CL
Figure 1 - Cross-section of capacitor with a wafer of highly defective crystalline silicon (M-Si) of thickness ds separated from the metal capacitor plates by the insulator layers of thickness di. Across the semiconductor wafer an electric potential difference is created by two metal electrodes parallel to the plane yz (a). Equivalent scheme of capacitor with the working substance "insulator-partially disordered semiconductor-insulator" (b). Simplified equivalent scheme of the system (c)
Let us assume that in one part the field potential on the wafer surface is positive 9(x = -ds/2) = , and in the other it is negative 9(x = ds/2) = , then the potential difference applied to the semiconductor is Us = ^(x = -ds/2) - 9(x = ds/2) = 2qs. We will consider electrodes located parallel to the yz plane (so that the field distribution in the wafer along the y and z coordinates will be symmetric). The screening of the external electrostatic field is caused by the redistribution of electrons hopping via defects in the charge states (0, -1, and +1; in units of elementary charge e against the background of a silicon matrix),
с
a
i.e. by the migration of charge states of immobile defects to a distance much greater than the average distance between them.
The capacitor with the working substance "in-sulator-partially disordered semiconductor-insulator" contains series-connected capacitances of insulating layers Ci and a parallel Rs(Cs + Cg)-circuit of the semiconductor wafer (see Figure 1b). Here Ci = = siA/di and Cg = ssA/ds are the geometric capacitances of insulator and semiconductor with static dielectric permittivities s i = sris0 and ss = srss0 (we assume that radiation defects do not contribute to the static dielectric constant of Si crystals), sri = 3.5 and Srs = 11.5 are the relative permittivities of the poly-imide and the silicon crystal lattice, s0 = 8.85 pF/m is the electric constant, Rs = Rs(U) is the semiconductor resistance, Cs = Cs(U) is the differential capacitance of the semiconductor, U is the voltage created by the metal plates of the capacitor.
The real part Ceq = Ceq(U) of the complex electrical capacitance and the active component of the conductivity Geq = Geq(U) of the structure in the equivalent circuit (see Figure 1c) is [11, 13]:
C 1+ Ю2 R2s(Cz + Cs )(Cg + Cs + C /2)
C = 4q 2
Geq =
1 + [vRs (Cg + Cs + С, /2)]2
ю2 Rs (C /2)2
1 + [o,Rs (Cg + Cs + С, /2)]2
(1)
(2)
where U (= Udc) is the constant voltage across the capacitor plates, © is the angular frequency of the variable component of the measuring signal with the amplitude |Uac| < | U |.
From Eqs. (1) and (2) we find the total conduction (admittance) Y = Y(U) and the phase shift 9 = 9(U) between current and voltage of the capacitor with the working substance "insulator-partially disordered semiconductor-insulator":
Y = [G2 + (fflCeq)2]1/2 =
eq/
2
1 + № (Cg + Cs )]2 1 + [oR (Cg + Cs + C/2)]2
\ 1/2
(3)
9 = arctan(-®C/G ) =
= arctan
eq eq Лп2
1 + ш2 RS(Cg + Cs )(Cg + Cs + C /2) vRsC /2
, (4)
where Rs = Rs(U) and Cs = Cs(U). Note that the total resistance (impedance) Z = Z(U) is related to the admittance Y as follows: Z = Y_1.
F
F2
-F
Fi
F(v) = 0
c-band
■Id)
gen rec
|2)-band
e
0 0 +i -i +i о -i 0
_i | l)-band
|a)-
+i
J I ^-i,o At
- IW0,+i
w-band
Figure 2 - Single-electron energy E as function of x coordinate in semiconductor with point two-level (triple-charged) defects of /-type in equilibrium (at U = 0): E^ is the mobility edge of conduction band electrons, eF"- < 0 is the Fermi level in the band gap, counted from the hole mobility edge (Emv) = 0), At = E2 - E1 is the width of energy gap between |1)- and |2)-bands, W0+1 and W-10 are widths of 11)- and |2)-bands. Arrows show hops of single electrons via 11)- and |2)-bands as well as generation [gen: 2(0) ^ ^ (-1) + (+1)] and recombination [rec: (-1) + (+1) ^ 2(0)] electron transitions between them; |d) and |a) are states of shallow hydrogen-like donors and acceptors in the charge states (+1) and (-1), respectively
For Rs > 1/©Cs from Eq. (1), the inverse equivalent capacity of the entire structure is 1 /Ceq = = 2/Сi + 1/^,, + Cg). Since the capacitances of insulator layers Сi and semiconductor Cs + Cg are connected in series, the charge on each of them is equal to Q. Thus, the voltage drops across insulators U = Q/Сi and across semiconductor Us = Q/^, + + Cg) are related to the voltage across the capacitor U = Q/Ceq as follows: U = 2U, + Us. By substituting the charge Q on the capacitor, expressed in terms of Us and Cs + C into U we obtain the voltage across the capacitor U, for which the voltage drop across the semiconductor is equal to Us:
Cs + Cg 2(Cs + Cg) + C U = и,—-g = U,— s g
C
eq
C
(5)
A highly defective silicon crystal (hd-Si) contains point two-level t-type defects in a concentration sufficient to stabilize the Fermi level EF in the energy gap. Defects of /-type in the charge states (+1) and (0) form a |1)-band with the energy levels Ej, and the ones in the charge states (0) and (-1) form a |2)-band in the band gap (energy levels E2), located closer to the c-band than |1)-band (Figure 2). Examples of /-defects are amphoteric impurities (Au, Cu).
Let us consider silicon under conditions of only hopping electron migration via immobile radiation
e
defects (of t-type) in the charge states (-1) and (0), as well as in the charge states (0) and (+1). The total concentration of defects in the charge states (0), (-1), and (+1) is Nt = N0 + N-1 + N+1.
We assume that |d)- and |a)-centers are completely ionized and their concentrations Nd and Na satisfy the conditions: Nd/Nt < 1 and Na/Nt < 1. Thus, the condition of electrical neutrality of the partially disordered semiconductor has the form:
N+i = N-i,
(6)
where N+1 = Nt+1 and N-1 = Nt-1.
The concentrations of ionized and neutral defects can be written as [14]:
Nz = Ntfz,
(7)
"¡Г" = 1 + P2 eXP
J-1
1 1 1 — = 1 + — exp
fo Pi
= 1 + Pi exp
J+l
r E^ + E2
kT
P2
+—exp
Pi
Ei + E2 + 2EF
kj
E{p + Ei kBT
1
+—exp
P2
-(40)+E2)
kBT
-(4V) + El)
Pi
+ —exp
P2
kBT
'-(Ei + E2 + 2Е<°У
kBT
(8)
where EFv) = Ev - EF is the Fermi level (chemical potential) Ef , counted from the v-band hole mobility edge (E^ = 0) of an undoped crystal [15, 16]; EFv) < 0 for the Fermi level in the band gap; E1 = = E0 - E+1 > 0, E2 = E_j - E0 > 0; kBT is the thermal energy. For dominant radiation defects in silicon (mainly divacancies), following the experimental data from [17-19], we assume: E1 = 225 meV, E2 = 575 meV, i.e. At = E2 - Ex = 350 meV, P1 = Po/P+1 = 1, P2 = = P0/P-i = 1, where PZ is the number of quantum states of the defect in the charge state Z with energy EZ.
With the total concentration of charged radiation defects Nch = N-1 + N+1 with charge ±e randomly (Poissonian) distributed over the crystal, we have equal rms fluctuations W = W-10 = W0+1 of the electrostatic energy, i.e. the widths of |2)- and |1)-bands are [20, 21]:
W-1,0 = W),+i = 1.637
4ns
s V
4n
— ( Nch)eq
1/3
Ф, =0
(9)
where fZ is the probability that the defect is in one of three possible charge states Z = -1, 0, +1.
If we neglect the excited states of radiation defects, then the inverse distribution functions 1/fZ of defects in |1)- and |2)-bands over charge states are [3, 4]:
where the Coulomb interaction of each charged defect only with its nearest charged defect (ion) is taken into account; e is the elementary charge; (Nch)eq = Nt/2 is determined from the condition of maximum effective concentrations N-10 = N0+1 = =N-1N0/Nt=N0N+1/Nt of single electrons hopping via t-defects in the charge states (-1), (0) and in charge states (0), (+1). Then we obtain (N-^ = (N^x =
= Nt/4, (N0)max = Nt/2 and (N-1,0)max = (N0,+1)max =
= Nt/8 [22]. Note that 3 W > At.
For a semiconductor with uniformly distributed point defects of the crystal lattice, the values of the function fZ(§) depend on the coordinate x only through the potential 9(x) and are obtained from fZ
by replacing E<FV) < 0 in Eq. (8) by
Е^(ф(х)) = EFV) - еф(х)
= F(v) -
(10)
that is for ^(x) < 0 the Fermi level EFv)(9) shifts to the top of the v-band and for 9(x) > 0 it shifts to the band gap.
The change in the concentration of charge states Z = -1, 0, +1 of Nz(9) - NZ defects in the electric field with the potential 9(x) is determined by Eq. (7) taking into account Eqs. (10) and (8). In this case, it is assumed that the energy gap At between |1)- and |2)-bands, as well as the width of each band W, do not depend on the potential.
Due to the symmetry of the problem with respect to reflection x ^ -x, we consider only the region -ds/2 < x < 0. The electrostatic potential 9(x) inside the semiconductor at a point with coordinate x satisfies the Poisson equation [23, 24]:
ay
dx2
1_d_ 2 d9
2
dx
р(ф)
(11)
where p(9(x)) = e[N+1(^(x)) - N-1(^(x))] is the volume density of the induced charge; N-1 = N+1 is the electrical neutrality condition of the semiconductor wafer at 9 = 0.
By integrating Eq. (11) over 9, we obtain the electric field strength:
d9 dx
= +
Г р(ф) ^
\1/2
(12)
where for 9 > 0 the "-" sign should be taken, while for 9 < 0 the "+" sign should be taken.
From Eq. (11), taking into account Eq. (12), we
2
e
к
s
s
obtain the charge Qs induced by the external electric field per unit area A of the flat surface of the silicon wafer:
Qp(*)dx,if
A i-dJ2 dx
x=-ds/2
±(-2s, j"^р(ф)d^7 , (13)
C.
A
dQs __
A dy/dx
P(9s )
x_-ds/1
_ eNt [ f+x($s ) - f_Ms )] гф. \1/2
-(2/8s ) J0' р(ф^ф)
(14)
) = — d„
N-i,ofo)M-I + No,+i (ф)Мo dф/dx
dф -
bilities M_10 and M0+1 of electrons hopping via point ¿-defects of the crystal matrix is established by the Nernst-Einstein-Smoluchowski relation (see, e.g., [3, 25]):
D
-1,0
M-i,o
kBT
D,
M,
0,+1 _ e kBT
_ S0,+1 '
(16)
0,+1
where for 9s > 0 the "_" sign should be taken, while for 9s < 0 the "+" sign should be taken.
The differential electrical capacitance per unit silicon surface area A, taking into account Eq. (13), is
The change under the action of the field effect of the hopping electrical conductivity [caused by the migration of single electrons across the wafer thickness via immobile radiation ¿-defects in the charge states (-1) and (0), as well as in the charge states (0) and (+1)] is
where 2_10 > 1, ^0,+1 > 1 are the dimensionless parameters, which are determined by the ratio of the fluctuation spread of ¿-defect levels (with average values of E1 and E2) to the thermal energy kBT; further we assume £,_10 = ^0,+1 = 1.
The diffusion coefficients D_10 and D0,+1 of electrons hopping via ¿-defects in a covalent crystal matrix (see Eq. (16)) can be estimated by averaging over all probable hopping lengths r (cf. [22-27]):
1 2 1 2 D_1,0 = -<r_1,0(r,T)r2>, D0+1 = -<r0,+1(r,T)r2>, (17)
6 6
where r_10(r, T) = vltexp[_(2r/a_1 + W_10/kBT)] and r0,+1(r,T) = Vltexp[_(2r/a0 + W0+JkBT)] are frequencies of electron hopping via ¿-defects in charge states (_1), (0) and (0), (+1) [28]; vlt « 10 THz is the characteristic frequency of crystal matrix phonons; a_1 = ^/(2m0E2)1/2 and a0 = h/(2m0E1)1/2 are the radii of localization of an electron at the ¿-defect in the charge state (_1) and (0), respectively, m0 is the electron mass in vacuum.
From Eq. (17), taking into account the distribution of distances r between ¿-defects [21], we get:
d.
N-^(0) M_1>0 + N0,+1(Q)M0>+1 dty/âx
d9 +
+ ■
ds
N-iMM-1,0 + No,+i (Ф)М0 ,+1 dф/dx
dф -
-ф, 0
ds
N,o(0)M-!,o + No,+1(0) M
0,+l
dq>/ddx
d9, (15)
where N1,0(9) = N^N^/N and N0+1(9) = N)(9)x xN+1(^)/Ni are the effective concentrations of single electrons hopping via ¿-defects in the charge states (_1), (0) and in the charge states (0), (+1); M_10 and M0,+1 are the drift mobilities of electrons hopping via ¿-defects in the charge states (_1), (0) and in the charge states (0), (+1).
The relationship between the hopping diffusion coefficients D_10 and D0 +1 and the drift hopping mo-
2nv lt Neq
D-y> - 3
exp
W
-1,0
V kBT y
4 f 2r
r exp — —
L a—i
-+ N
eq
dr.
Do,+1 -
2nV lt Neq
exp
0,+1
V kBT y
4
r exp
Г2r 4nr3 лг Л
— +-N
v a0 3 у
dr,
(18)
where Neq = (N_1,0)max = (N0,+1)max = N/8.
From Eq. (15), taking into account Eqs. (16)-(18), we obtain the resistance of a highly defective crystalline silicon (hd-Si) wafer due to the hopping of single electrons via ¿-defects along its thickness:
Rs - Rs(U(ys)) - A-, Aa
(19)
e
e
o
ф
0
e
X
X
Ф
0
e
0
X
3
e
X
Ф
0
where ds and A are the thickness and the surface area of the hd-Si wafer, o = o(^s) = o(0) + 5o(9s) is the electrical conductivity, and o(0) = e[N-10(0)M-i,0 + + N0,+1(0)M0,+i] is the conductivity at = 0. For the considered low frequencies, the electrical conductivity o is frequency-independent [29, 30].
Note that Eqs. (14) and (19) were obtained under the assumption of quasi-stationary filling of energy levels according to Eq. (8) taking into account Eq. (10), therefore Cs and Rs are the quasi-static (low-frequency) capacitance and resistance of semiconductor. The quasi-stationarity condition is satisfied at <b/2tc < r-10(r, T) and <b/2tc < r0 +1(r, T). In other words, this can expressed by the inequality ©/2n < o/es, where es/o is the Maxwell relaxation time for hopping conduction.
Calculation results and discussion
The calculations were carried out for the following parameter values: semiconductor thickness ds = 150 ^m, insulator thickness di = 3 ^m, relative permittivities of semiconductor (hd-Si) er,s = 11.5 and insulator (polyimide) eri = 3.5, frequency of alternating electric field ©/2n = 1 kHz.
Figure 3a shows the results of calculating the ratio of the low-frequency admittance Y(U) to ©Ci/2 according to Eq. (3) at various values of the voltage U created by metal electrodes on the surface of insulator interlayers, for Nt = 3T019 cm-3 at temperatures T = 250, 300, 350 K. The values of U are related to Us by Eq. (5) and Us = was chosen so that the inequality eUs < At is fulfilled. It is seen that for U = 0 (flat-band mode) the admittance of the capacitor with the working substance "insulator-partially disordered semiconductor-insulator" increases with temperature.
Figure 3b shows the results of calculating the ratio of the low-frequency admittance Y(U) to ©Ci/2 according to Eq. (3) at different values of voltage U created by metal electrodes on the surface of insulator interlayers for temperature T = 300 K at concen-
19
« 1 0.96 0.92
<N
0.
0.84 b 1 0.96
^ 0.92 2
0.8
0.84
T = 300 к
- N\\ 3 //^S^
- \V if
- l\ /
1 i Y i i
-4
-2
0
U, V
Figure 3 - Dependence of admittance 2Y/raQ on electrode voltage U, calculated by Eq. (3): a) for Nt = 3-1019 cm-3 at temperatures T (K): 250 (curve 1), 300 (2), and 350 (3); b) for T = 300 K at /-defect concentrations 11019 (1), 3-1019 (2), and 1 1020 (3)
Figure 4b shows the results of calculating the phase shift angle 9(U) according to Eq. (4) at various values of the voltage U created by metal electrodes for temperature T = 300 K at the concentration of /-defects in disordered silicon Nt = 1T019, 3T019, 1T020 cm-3. It is seen that in the flat-band mode (at U = 0), all other conditions being equal, the phase shift angle modulus is minimum for the concentration of /-defects Nt = 3T019 cm 3 and is maximum for Nt = 1 1020 cm-3.
Note that the value of the Fermi level energy E<Fv) = 400 meV, obtained from the electrical neutrality condition N+1 = N-1, does not depend on the temperature, since EFv) is in the middle between |1)- and |2)-band. This practically coincides with the experi-trations of /-defects in disordered silicon Nt = 11019, mental value of EFv) in silicon [17-19], which con-3T019, 1 1020 cm-3. It is seen that the admittance in- tains a high concentration of radiation defects.
Note that the capacitor with the working substance "insulator-partially disordered semiconductor-insulator" is radiation-resistant, because radiation defects are already present in the semiconductor in large numbers. This suggests that this element is promising for use as a varactor. Also, the dependences of the electrophysical characteristics (Eqs. (1)-(4)) on the potential at the electrodes make it pos-
creases with the concentration of /-defects.
Figure 4a shows the results of calculating the phase shift angle 9(U) between current and voltage according to Eq. (4) at various values of voltage U created by metal electrodes on the surface of insulator interlayers, for Nt = 3T019 cm-3 at temperatures T = 250, 300, 350 K. It is seen that the absolute value of the phase shift angle decreases with temperature.
2
4
a
90
§ 89.5
88.5 88
b
90
СЛ
у
^ 89.5
CD I
89
Figure 4 - Dependence of phase shift angle 0 on electrode voltage U, calculated by Eq. (4): a) for Nt = 3-1019 cm-3 at temperatures T (K): 250 (curve 1), 300 (2), and 350 (3); b) for T = 300 K at t-defect concentrations 11019 (1), 3-1019 (2), and 1 1020 (3)
sible to determine the concentration of t-defects in the disordered semiconductor separated by insulator interlayers from the capacitor plates [11].
Conclusion
The structure "insulator-partially disordered semiconductor-insulator" is proposed as a working substance of a capacitor. The semiconductor layer with a thickness of 150 pm is separated from the metal electrodes of the capacitor by insulating layers of polyimide with a thickness of 3 pm. The semiconductor layer is a highly defective silicon crystal containing radiation point two-level t-defects in three charge states (-1), (0), and (+1) with hopping migration of single electrons via them, i.e. defects form |1)- and |2)-bands in the band gap.
The calculation gives a nonmonotonic dependence of the low-frequency admittance and the phase angle between current and voltage on the electric potential at the metal plates. At the concentration of t-type radiation defects equal to 3-1019 cm-3, with an increase in temperature from 250 to 350 K, the admittance increases by about 12%. With an increase in the concentration of t-defects from 1-1019 to 1-1020 cm-3 at temperature of 300 K, the admittance
of the capacitor increases by about 13%. In the calculations, for the first time, the diffusion coefficients were averaged over all probable electron hopping lengths via t-defects in the charge states (-1), (0) and (0), (+1) in the covalent crystal matrix. Note that the considered element is radiation-resistant, since the semiconductor layer already contains radiation point defects in a high concentration.
Acknowledgments
The work was supported by the Belarusian National Research Program "Materials Science, New Materials and Technologies" and by the European Union Framework Programme for Research and Innovation Horizon 2020 (grant No. H2020-MSCA-RISE-2019-871284 SSHARE).
References
1. Zvyagin I.P. Controlling the jump-over conductivity of compensated semiconductors with the aid of an electrical field. Sov. Phys. Dokl., 1977, vol. 22, no. 11, pp. 647-648.
2. Poklonskii N.A., Vyrko S.A. Screening of an electric field and the quasi-static capacitance of an induced charge in semiconductors with hopping conductivity. Russ. Phys. J., 2002, vol. 45, no. 10, pp. 1001-1007. DOI: 10.1023/A:1022867001332
3. Poklonski N.A., Vyrko S.A., Zabrodskii A.G. Calculation of capacitance of self-compensated semiconductors with intercenter hops of one and two electrons (by the example of silicon with radiation defects). Semiconductors,, 2008, vol. 42, no. 12, pp. 1388-1394.
DOI: 10.1134/S1063782608120038
4. Poklonski N.A., Vyrko S.A., Zabrodskii A.G. Field effect and capacitance of silicon crystals with hopping conductivity over point radiation defects pinning the Fermi level. Semiconductors, 2007, vol. 41, no. 11, pp. 1300-1306. DOI: 10.1134/S1063782607110048
5. Poklonski N.A., Kovalev A.I., Vyrko S.A. [Low-frequency electrical capacitance of semiconductor diode with hopping conductivity via triple-charged defects]. Doklady Natsional'noi akademii nauk Belarusi [Doklady of the National Academy of Sciences of Belarus], 2017, vol. 61, no. 4, pp. 52-59 (in Russian).
6. Zukowski P.W., Kantorow S.B., Kiszczak K., M^czka D., Rodzik A., Stelmakh V.F., Czarnecka-Such E. Study of the dielectric function of silicon irradiated with a large dose of neutrons. Phys. Status Solidi A, 1991, vol. 128, № 2, pp. K117-K121.
DOI: 10.1002/pssa.2211280243
7. Djuric Z., Smiljanic M. Static characteristics of metal-insulator-semiconductor-insulator-metal (MISIM)
U, V
structures—I. Electric field and potential distributions. Solid-State Electron., 1975, vol. 18, no. 10, pp. 817-825. DOI: 10.1016/0038-1101(75)90001-5
8. Djuric Z., Smiljanic M., Tjapkin D. Static characteristics of the metal-insulator-semiconductor-insulator-metal (MISIM) structure—II. Low frequency capacitance. Solid-State Electron., 1975, vol. 18, no. 10, pp. 827-831. DOI: 10.1016/0038-1101(75)90002-7
9. Ng K.K. Complete Guide to Semiconductor Devices. New-York, Wiley-IEEE Press, 2002, xxiv+740 p. DOI: 10.1002/9781118014769
10. Krupski J. Interfacial capacitance. Phys. Status Solidi B, 1990, vol. 157, no. 1, pp. 199-207.
DOI: 10.1002/pssb.2221570119
11. Berman L.S., Klinger P.M., Fistul' V.I. Determination of the parameters of deep centers in an overcom-pensated semiconductor from the temperature dependence of the capacitance and active conductance. Sov. Phys. Semicond., 1989, vol. 23, no. 11, p. 1206-1208.
12. Elfimov L.B., Ivanov P.A. Surface capacitance of a semiconductor with a deep dopant (in the example of p-6H-SiC<B>). Semiconductors, 1994, vol. 28, no. 1, pp. 97-100.
13. Poklonski N.A., Gorbachuk N.I. [Fundamentals of impedance spectroscopy of composites]. Minsk, BSU, 2005, 130 p. (in Russian).
14. Poklonskii N.A., Stelmakh V.F., Tkachev V.D., Voitikov S.V. Screening of electrostatic fields in semiconductors by multicharged defects. Phys. Status Solidi B, 1978, vol. 88, no. 2, pp. K165-K168.
DOI: 10.1002/pssb.2220880266
15. Mott N. Impurity bands in silicon and germanium. Disordered Semiconductors, ed. by M.A. Kastner, G.A. Thomas, S.R. Ovshinsky. New York, Plenum Press, 1987, pp. 3-10.
DOI: 10.1007/978-1-4613-1841-5_2
16. Mott N. The mobility edge since 1967. J. Phys. C: Solid St. Phys, 1987, vol. 20, no. 21, pp. 3075-3102. DOI: 10.1088/0022-3719/20/21/008
17. Lugakov P.F., Lukashevich T.A., Shusha V.V. Nature of the defect determining the Fermi level stabilization in irradiated silicon. Phys. Status Solidi A, 1982, vol. 74, no. 2, pp. 445-452.
DOI: 10.1002/pssa.2210740209
18. Kuznetsov N.V., Soloviev G.G. [Radiation resistance of silicon]. Moscow, Energoatomizdat Publ., 1989, 96 p. (in Russian).
19. Brudnyi V.N. Charge neutrality in semiconductors: defects, interfaces, surface. Russ. Phys. J., 2013,
vol. 56, no. 7, pp. 754-756. DOI: 10.1007/s11182-013-0095-4.
20. Poklonski N.A., Vyrko S.A., Poklonskaya O.N., Zabrodskii A.G. Transition temperature from band to hopping direct current conduction in crystalline semiconductors with hydrogen-like impurities: Heat versus Coulomb attraction. J. Appl. Phys., 2011, vol. 110, no. 12, pp. 123702 (1-7). DOI: 10.1063/1.3667287
21. Poklonski N.A., Vyrko S.A., Poklonskaya O.N., Zabrodskii A.G. Role of electrostatic fluctuations in doped semiconductors upon the transition from band to hopping conduction (by the example of p-Ge:Ga). Semiconductors, 2016, vol. 50, no. 6, pp. 722-734.
DOI: 10.1134/S1063782616060191
22. Poklonskii N.A., Lopatin S.Yu. Stationary hopping photoconduction among multiply charged impurity atoms in crystals. Phys. Solid State, 1998, vol. 40, no. 10, pp. 1636-1640. DOI: 10.1134/1.1130623
23. Grundmann M. The Physics of Semiconductors. An Introduction Including Nanophysics and Applications. Berlin, Springer, 2021, xxxvii+889 p.
DOI: 10.1007/978-3-030-51569-0
24. Poklonski N.A., Vyrko S.A., Podenok S.L. [Statistical physics of semiconductors]. Moscow, KomKniga Publ., 2005, 264 p. (in Russian).
25. Poklonski N.A., Vyrko S.A., Kovalev A.I., Dz-eraviaha A.N. Drift-diffusion model of hole migration in diamond crystals via states of valence and acceptor bands. J. Phys. Commun., 2018, vol. 2, no. 1, pp. 015013 (1-14). DOI: 10.1088/2399-6528/aa8e26
26. Baranovskii S., Rubel O. Chapter 9. Charge transport in disordered materials. Springer Handbook of Electronic and Photonic Materials, eds. S. Kasap, P. Capper. Berlin, Springer, 2017, pp. 193-218.
DOI: 10.1007/978-3-319-48933-9_9
27. Shklovskii B.I., Efros A.L. Electronic Properties of Doped Semiconductors. Berlin, Springer, 1984, xii+388 p.
28. Klimkovich B.V., Poklonskii N.A., Stelmakh V.F. Alternating-current hopping electrical-conductivity of covalent semiconductors with deep-level defects. Sov. Phys. Semicond, 1985, vol. 19, no. 5, pp. 522-524
29. Dyre J.C., Schrader T.B. Universality of ac conduction in disordered solids. Rev. Mod. Phys, 2000, vol. 72, no. 3, pp. 873-892.
DOI: 10.1103/RevModPhys.72.873
30. Long A.R. Frequency-dependent loss in amorphous semiconductors. Adv. Phys., 1982, vol. 31, no. 5, pp. 553-637. DOI: 10.1080/00018738200101418
