CC BY
7
https://doi.org/10.29013/ESR-22-5.6-37-41
Rasulov Vohob Rustamovich, Associate professor of Fergana State University Rasulov Rustam Yavkachovich, Professor of Fergana State University Mamatova Muhhayyo Adhamovna, Researcher of Fergana State University Karimova Mahliyo Nematjon qizi, Undergradate of Fergana State University Mirzaakbarov Dilshodjon Dovlatboyevich, Researcher of Fergana State University
VOLT-AMPERE CHARACTERISTICS OF A THREE-LAYER SEMICONDUCTOR DIODE OF DOUBLE INJECTION
Abstract. Expressions are obtained for the electron current density distribution over the thickness of the base of a three-layer semiconductor double-injection diode made of semiconductors with deep impurity levels, which is associated with an increase in the bipolar drift mobility of current-carrying carriers. Analytical expressions for the voltage - current characteristics of such diodes are obtained. The minimum voltage at the base of the diode is determined.
Keywords: current density, electrons, three-layer semiconductor diode, double injection, semiconductor, bipolar drift mobility, current-voltage characteristic.
The voltage - current characteristic S - type in diffusion length, the distribution of injected carriers, diodes is observed only when the conductivity of the therefore, the influence ofthe electric field on the dissemiconductor thickness increases with current ac- tribution of nonequilibrium carriers in the diode base cording to a law stronger than linear. This is due to drops sharply with an increase in the injection level, an increase in: a) the rate of thermal generation of although in pure semiconductors the field does not carriers due to the heating of the semiconductor affect the distribution of current carriers. thickness by the flowing current; b) injection coef- For definiteness, a three-layer diode is considered ficient p - n - transition with increasing current; in the work, the base of which is made of silicon with c) carrier mobility in a strong electric field upon scat- an admixture of gold, since the parameters of this tering by ionized impurities; d) lifetime with increas- material are most fully studied experimentally [7]. ing carrier concentration ( t -mechanism) [1-6]. Equation for the distribution of current car-
In this work, the current density distributions over riers in the base of the structure the base thickness and the voltage - current character- Let us consider a one-dimensional three-layer
istic of three-layer diodes, the base of which is made structure, the base ofwhich is made of gold-compen-
of semiconductors with deep impurity levels, is cal- sated silicon, and we assume that the length of the
culated, which is associated with an increase in the base d significantly exceeds the diffusion lengths of
bipolar (group) drift mobility of carriers. In diodes, the minority current carriers in it. Then, in the sta-
the base thickness of which is much greater than the tionary case, the distribution of nonequilibrium
current carriers over the thickness of the base is determined using the following equations
dp Qn
Jp = (p + po)E - eDp^f, Jn = epn (n + n0)E + D—,
s --
, we obtain the following relations useful for
Nf0
further calculations
J = Jn + Jp ,
1 J e dx
dx
P - Po
Ti) (2)
E =
b ■ j
1 + b + 1 - b
where /un, /np and Dn, Dp are the mobility and diffusion coefficients of electrons and holes, respectively, J, J , Jp are the densities of the total, electron and
ba s y +1 ab
(sy +1)2
y + b
(4)
1 + b + -
ab
y ■s +1
■y + b
dy
' n>J p
hole current, E is the electric field strength:
e^p [b(n + no) + (p + po)]
b ^n dp kT dx dx
j • y -
Jp =
y
2 + -
(ey +1)2 (ye +1)
+1
dy
(3)
1 + b + -
ab ye +1
(5)
y + b
e [b(n + no) + (p + &)]' Using the Einstein relation and the notation ten in the following form
b _ , from equations (1, 2) and relation (3) we Mp
express the electric field strength in terms of the concentration of current carriers and, then determining the density of the hole component of the total current, and also substituting the obtained value into equation (3). Having made the following designa-
Thus, the equation for the distribution of minority carriers over the thickness of the base can be writ-
- j
where
y ' + Ç. (y ')2 -
say2 +1
2 1 + b ( 1 + b . ,
y s-+ y I + a | +1
y = ^
(6)
tions j =
J • Lp ,
eDn ■ n0 '
e • LpE r x p
£=-— , £ =—, y = —,
kT L' n
y
1 =
_ n w-
2 +-+-2
V ys +1 (ye +1)2
+1
y
1 + b + -
V
ab ye +1
- b, (7)
+ b
Ç= b-
2(1 + ) ys +1
say
1 + -
ys +1
1 + b +
ab ys +1
+ b (ys +1)2
y
1 + b +
ab
ys +1
+b
-b
y
~ W 142 +-+-2
V ye +1 (ye +1) J
+1
1 + b + -
ab
ye +1
baey
(ye +1)2
and
y
1 + b +
ab
V
+b
b[2 • y (a +1) +1]
1 = r , ,,-r^H" ' C =
ye +1 b [b(1 + b) -1]
(8)
1 =
y[l + b(a +1)] + b {y[1 + b(a +1)] + b}2 b(2 y +1) ^ b(b -1)
y (1 + b) + b'
C =
[y (1 + b) + b ]2
at s y « 1, at y s » 1
(9)
n
o
p
a
a
2
1
2
took into account
bdn dp _ dx dx
b -1 + -
(sy +1)2
dp dx
i P+ N , bp-—+ P + bn0 + p0 =
= n
P + N 2
f
y
1 + b + -
ab ys +1
+ b + p0-
(10)
(11)
Вольтамперная характеристика volt-ampere characteristics
To calculate the voltage - current characteristic, it is necessary to know the electric field strength, which in the case under consideration has the form
Note here that with an increase in the hole con-
E =
b ■ j
1 - b -
a ■b
centration, the quantity n (Z): at y «
Nf0
1 + b +
b ■ a s y +1
y + b
(ys +1)2 ../ (12)
1 + b +
b ■ a
-• y-.
y + b
takes the values
b (205y +1)
i1+\e)>
b (esb -1)
and at y »
Nfo°
b (b -1)
i1 + d)t
(1 + 05b)y + b [[(1 + dSb)y + b]2
b(2y+1)
where E = Lp e По kT
ys +1
E. Then the voltage drop in the
structure is determined by the relation
U = J Edx = |
P-j
takes the values
(1 + b )y + b
1 + B +
ab
ba ys +1
-dx +
y + B
2 . From wich we get that 1 < ^ < 2,
[(1 + b )y + b ]
and Z ^ 1 at y » 1 (see, for example, [8]).
Above and below x is given in units of the effec-
g PT
tive diffusion length of holes L - J — , and the
kT
electric field strength in units of--
1 - B— 2
г (y£ +1)2 , ,
+J=-Yu i -y - dx = U 1 +U 2
(13)
Here
, n ab
1 + B + - y + B
ys +1
Уг +-
b
U 2 =i
P
2 j ■ b ■
U1 = 1 -1+., (14)
1 1 + b ( +1) y +_b_
Уо 1 + b(a +1)
1 + b (a +1)
-dx = -
2 ■ j ■b
I
dx
0 ^ + Be-ix +_b_ 1 + b(a +1) 0 Aekx + Be^
1 + b (a +1)
2 • j-b
b[ ] U + k '
Jl
A 1 «1 B
• cosec-
л • k2 k^ ^ к
-К-1)"
•2 n =0
BJ (k1 + k2)n + k1
-K-1)+
f A 1-"-1 e(k1 + k2 )+k2 ]d
B J n (k1 + k2 ) + k2
or
U 2 =
л-b
VT+î - j
■e
d_
2^
f1 + b{a +bl Jf
+1
j+1 + j j+1 - j
nk,
■cosec
1 -a(a +1) +
yd
1 + a + -
yd
y[1 + b (a +1)]
1 + a + -
y0
j+ï - j
(15)
where
A -
B =
k1 (e-k2da- b ) k2 (da- b) '
2B(eM - e)k2 [1 + B(a + 1)]( ek2d - b) '
2 '
Thus, for k1d and k2d << 1 it is easy to obtain the following expressions
a
0
1
1
n=0
U2 =
2-b-n
[1 + b(a +1)] k1 + k2
(k1 + k2 )d
b1 (1 + k2d )- c Or
b1 (1 -k1d)-c ^ b1 (1 + k2d )- c k2
k1+k2
-cosec-
U2 =
2b-n ■d [1 + b( a + 1)]b1
-Jj2 +1 - J
1 + ( J + J J J +1) d-
1 + [j+î + j] d-
4.
j 2+1+J
1 + [ j-4f+1 + j ] d-
j+1 - j
Vj 2 +1+J +1
■ cos e-
n(yj j2 +1 + j )
2jf
+1
At dj << 1
/
n-b
U2 =
1 + a + —
V yo
[1 + b (a +1)]2 y
y[1 + b (a +1)] " 1 " 1 + a + — yd
•J1 -a 1 + b (a +1) + £ - yd-_ 1 + a + — [ yo -
where yL = y(x = L) . Note that for e M « 0
b1ekld >> c and eM >> efrom the last relations can be obtained the results of [8]. But, unfortunately, the last approximation is relatively crude: in particular, for a type structure p+ - n - p+, it gives the
nb ■ d
and it
following result, U - - - ,_
g , ^2 [1 + b(a +1)] V^bT- c is imaginary. So, when calculating symmetrical struck + k tures, for example, p+ - n - p+, p+ - p - p+ , n+ - n - n+, n+ - p - n+ it is not necessary to use the last approximation.
==' Conclusion
c
'b In conclusion, we note that the rate of hole re-
combination, i.e. the total number of holes captured per unit of volume per unit of time is described by
(16) the following expression: Rp =(\V_0 f_ + W o f^p . We also note the following. At low hole concentrations, the quantities f_ and f0 in Eqs. (2) and (9) can be replaced by their equilibrium values. In particular, at n -65p (9) takes the form
R = (f0W_Q + f00W0 A p .Athole concentrations sat-
p N isfying condition p >-, the impurity states of
gold are recharged, and "now there should be more electrons than holes, just by the amount of the formed uncompensated charge, and the recombina-
(17) tion rate will be determined by the corresponding stationary values of the fractions of minus and zero-charged atoms of deep impurities in the equilibrium state. Uus, the difference between the electron and hole concentrations increases from the equilibrium electron concentration to N (Of0 - fQQ ) .^is means that in this case the nonequilibrium concentration of nonequilibrium electrons can exceed the concentration of equilibrium ones by several orders of magnitude. In this case, the lifetime of holes will change
1 6 +1 from t1 - -to Tt = -
f°W_o + fo°Wc
0+
W_o+6Wo+
References:
i.
2.
3.
Gurin N. T., Korneev I. V., Maksin A. N., Novikov S. G. Position-sensitive combined device with N-shaped voltage - current characteristic // Proceedings of the IV Russian seminar on fiber lasers.- Ulyanovsk: UlGU. 2010.- P. 125. (in Russian).
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k
k
d
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