CHANGE IN THE RESISTANCE OF THE SEMICONDUCTOR
IN THE VARIABLE DEFORMATION FIELD
M. AHMETOGLU (AFRAILOV)1, G. GULYAMOV2, S. H. SHAMIRZAEV2,
A. G. GULYAMOV2, M. G. DADAMIRZAEV2, N. APRAILOV2, F. KOÇAK1
1 Department
of Physics, Uludag University, 16059, Görukle, Bursa, Turkey,
[email protected]
2 Department of Physics, Namangan State Engineering and Technology Institute,
Namangan, Uzbekistan
Received September 12, 2006
In this work the influence of variable deformation on concentration of
nonequilibrium carriers and resistance chances of the semiconductor have been
investigated. The phase shift define by frequency of deformation and life time of
nonequilibrium carriers, have been shown between variable deformation and semiconductor resistance change. It is established that, in a plane resistance-deformation
the phase trajectory forms hysteresis loop. When conductivity varies only due to
electronic processes then the hysteresis loop remains smooth. At the constant
frequency and amplitude, the form of the fluctuation does not change. It is shown that,
the structural changes in the sample leads the hysteresis loop movement in phase space.
Key words: semiconductor, deformation, hysteresis loop.
At influence of deformation changing of the energy band structures leads to
change of concentration of carriers and redistribution of carriers between the
energy valleys [1]. These changes in turn leads to change of resistance of the
sample [2]. If external deformation is variable, change of the resistance of the
sample on time occurs to some backlog from the enclosed mechanical influence.
Let's consider the response of excess concentration of electrons ne on
variable deformation. Deformation can induce generation of the electrons with a
speed ge besides, excess electrons can act from other area of the sample [3].
Concentration can be determined by the equation of a continuity
G
∂ne
n
= ge − e + 1 ∇J n
∂t
τ e
(1)
G
Where J n density of a current of electrons, τ is a lifetime of electrons. For the
homogeneous sample the equation of a continuity is

Paper presented at the 7th International Balkan Workshop on Applied Physics, 5–7 July
2006, Constanþa, Romania.
Rom. Journ. Phys., Vol. 52, Nos. 3– 4 , P. 343–351, Bucharest, 2007
344
M. Ahmetoglu (Afrailov) et al.
2
∂ne
n
= ge − e
∂t
τ
(2)
Thus, the reasons of change of resistance can be the diversified. Generations speed ge, caused by changing of width of a band or heights of a barrier,
rather quickly adapts to external pressure, i.e. practically repeats dependence of
pressure on time. However, concentration of nonequilibrium carriers is defined
not by instant value of the generations speed, and all previous values ge.
If during the time t = T1 speed of generation of excess carriers is zero, this
time the constant deformation ε1 is applied (g(ε1) also will be a constant). The
decision of Eq. (2) for boundary conditions ne = 0 at t = T1 is
( )
T −t ⎤
⎡
ne = ge (ε)τ ⎢1 − exp 1
τ ⎥⎦
⎣
(t > T1 )
(3)
If deformation is stopped at moment T2, that ne is reduce as
( )
( ) ( )
T
T ⎤
⎡
ne = ge (ε)τ ⎢ exp 2 − exp 1 ⎥ exp − t
τ
τ ⎦
τ
⎣
(t > T2 )
(4)
The decision of the Eq. (2) for a case when the deformation is any function of
time as [3]
∫ ge (t0 ) exp ( 0 τ
t
ne (t ) =
t −t
−∞
) dt
0
(5)
Expression (5) can be used to define how the value of nonequilibrium
concentration reacts on varying deformation. We shall assume, that deformation
is varies as
(6)
ε = ε 0 sin(ωt0 )
Then the width of the forbidden zone changes under the same law
ΔEg = Δε
(7)
Where Δε is the deformation potential [2]. Speed of thermal generation has
exponential dependence on width of the forbidden zone and temperature of a
lattice [4].
In this case from the general thermodynamic reasons, generation speed of
nonequilibrium carriers can be presented as
(
ge (t ) = g0 e
Δε( t 0 )
kT
)
−1
Then from expression (5) and (8) we have
(8)
3
Resistance of the semiconductor in variable deformation fiel d
( ) ∫ (e
t
ne (t ) = g0 exp − t
τ
Δε ( t0 )
kT
)
345
t0
− 1 e τ dt0
(9)
−∞
Setting an obvious kind of time dependence of deformation ε(t), it is
possible to receive the time dependence of the concentration ne(t). Then having
excluded time from these equations, it is possible to get a phase trajectory of
process on a plane ne – ε or the equations connecting ne and ε: f ( n, ε ) = 0. The
equation (9) in enough wide area of deformations for function ε(t0) allows to
describe change of concentration of carriers.
To show a method of reception ne – ε dependences we shall consider,
Δε << 1 (weak deformation)
kT
Then expression (9) can be presented in the form of
( ) ∫ kTΔε e
ne (t ) = g0 exp − t
τ
t
t0
τ dt
(10)
0
−∞
Let the deformation enclosed to the sample is
ε = ε 0 sin(ωt0 ) (0 < t0 < π/ω)
And other moments of deformation are not present. Then from (10) it is had
( )
t
t
0
ne (t ) = g0 exp − t Δε sin ωt0 e τ dt0 =
τ kT
∫
0
( )
⎡ sin(ωt ) − cos ωt + exp − t ⎤ ,
= g0 Δε
kT 1 + ω2 τ2 ⎢⎣ ωτ
τ ⎥⎦
ωτ2
(0 < t0 < π/ω)
(11)
After the termination of deformation in the Eq. 10 top limit of integral should be
π/ω
2
ne (t ) = g0 Δε ωτ2 2
kT 1 + ω τ
( ) ( ) ( t > ωπ )
⎡1 + exp π exp − t ⎤
⎢⎣
ωτ
τ ⎥⎦
1
(12)
The response of concentration excess electrons on deformation in case ωτ = 1 is
shown in Fig. 1.
It is possible to show that, the density of the nonequilibrium electrons, arising
at sinus wave deformation, contains constant and variable components. Thus, the
variable component of concentration lags behind from the sinus wave deformation on a phase on a θ = arctg(ωτ). Hence, between variable deformation and
change of resistance of the semiconductor there is some always shift of a phase.
346
M. Ahmetoglu (Afrailov) et al.
4
Fig. 1 – a) Dependence of concentration on time (continuous curves),
dependence of deformation on time (dashed lines); b) The phase trajectory of
process of deformation forms the hysteresis loop.
Fig. 2 – a) Dependence of ε on time for rectangular impulses; b) the phase trajectory
of dependence the concentration ne versus deformation ε for a rectangular impulse.
Any periodic deformation can be decomposed in the Fourier series, which
are phase displacement between the components of Fourier series for each
frequency, and the deformation and the concentration.
The form of the dependence of deformation on the time strongly influences
the “phase portrait” of the fluctuating motion of a change in the resistance of
model from the deformation. We analyze the qualitative dependence of the form
of the phase trajectories of the concentration of the excess carriers ne on the
amplitude ε0 and the periods T of the square pulses of deformation.
The square pulse of deformations has an amplitude ε0 and a period T
moreover T = Ti + T0, here Ti is the period of the pulse of deformation and T0 is
the period of the pause between the pulses. If we increase pulse frequency, and
the amplitude to leave of constant the area of loop decreases under the influence
of the square pulse of the deformations.
5
Resistance of the semiconductor in variable deformation fiel d
347
With the amplitude reduction the hysteresis loop decreases as along the
height, thus the width, moreover simultaneously it displaces to the origin of
coordinates (Fig. 3a, b, c). If the amplitude of deformation decreases the
concentrations of minority carriers also decreases. The area of hysteresis loop
increases in the lower frequencies. This increase in the area of loop relates only
with the rectangular voltage pulse. If the shape of pulse is smoother, that with an
increase in the oscillatory period the area of loop can again will decrease.
Fig. 3 – “A phase portrait” of dependence of concentration from duration of
deformation (a) for various rectangular impulses of deformation (b, c).
Thus, the phase trajectories ne – ε strongly depend on the frequency of the
deformation of that applied to the model. Therefore it will conveniently present
phase trajectories for different frequencies in the phase space, moreover as the
third of axis it is convenient to take pulse frequency ν or its period T = 1/ν. As
the example to phase trajectory in phase space ne – ε – ν (ne – concentration, ε0 –
deformation, ν – frequency) let us examine a change in the concentration of
minority carriers under the action of the square pulses of deformation (Fig. 4).
In the three-dimensional space (ne – ε – ν) the phase trajectories for
different frequencies will be determined by the sections of the curved surface of
prism. (Fig. 5).
If deformation not rectangular and smoother with the specific period T, in
this case phase the trajectory of the fluctuating motion of the concentration of
excess carriers will draw the curvilinear conical surface (Fig. 6). With a change
in the parameters of projection system of phase trajectory on the phase plane
n – e forms spiral (Fig. 7).
With an increase in the time the concentration of equilibrium carriers, it
also slowly changes. This change in carrier concentration affects the resistance
of film and the period of the life of nonequilibrium electrons. This indicates, that
348
M. Ahmetoglu (Afrailov) et al.
6
Fig. 4 – “A phase portrait” changes of concentration n at constant amplitude ε0 (a)
and for three various on duration of rectangular impulses T1 > T2 > T3 (b, c, d).
Fig. 5 – The phase trajectory
(ne – ε – ν).
Fig. 6 – The phase trajectory of excess carriers in (ε – n – ν) space,
lies on the conical surface.
7
Resistance of the semiconductor in variable deformation fiel d
349
Fig. 7 – With an increase in
the oscillatory period the
helix increases its amplitude.
the parameters of the system slowly changes and therefore hysteresis loop “sails”
in phase the space of ΔR – ε and trajectory in the experiments is not locked by
phase. Evidently, floating hysteresis loop into ΔR – ε to plane it is possible to
explain with a change in the parameters of film with repeated the application of
cyclic deformation. From other side there is a phase shift between the stress and
the strain which it is determined by Debye's losses. In this case, it is necessary to
use Debye's formulas for the pliability, which must be connected with the
processes of relaxation [5]. Evidently, the thermodynamic theory of ordering [5]
can explain the shifts of hysteresis loop in the phase plane of ΔR – ε resistancedeformation. Possibly, change and shift of hysteresis loop with an increase in the
number of cyclic deformations it is connected with the structural changes in the
film may be connected with the fatigue of material. However, this is not still
accurately established. One should separately emphasize that, the form of hysteresis loop strongly depends on the temporary dependence of the deformation of
ε(t). For example, with a very slow change in the pressure the resistance of
model completely manages after changes in the pressure and phase displacement
between the deformation and the resistance is absent. In this case, the area of
hysteresis loop is turned into nul. If there are no changes in the lattice a slow
change in the deformation hysteresis is not observed. With an increase in the rate
of change in the deformation line of the dependence of resistance on the pressure
phase trajectory is converted into the hysteresis loop or the helix (Fig. 8).
With the high frequencies of a change in the pressure again the area of loop
approaches zero. Evidently, when the time of a change in the deformation of the
order of the characteristic of the relaxation time of resistance τ, that the area of
hysteresis loop is maximum.
A change in the hysteresis loop with the frequency it must contain important information about the relaxation processes in the films. If hysteresis loop
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M. Ahmetoglu (Afrailov) et al.
8
Fig. 8 – With an increase in the frequency first the area of hysteresis loop
grows with the high frequencies the hysteresis loop it will become the
horizontal section (ν1 ≈ 0 < ν2 < ν3 < ν4).
was formed only due to the electronic processes and the structural changes in the
film, this does not occur, then the hysteresis loop should be closed and it in due
course in ΔR – ε plane should not moves.
Really due to mechanical pressure there are irreversible changes in a crystal
lattice subsequently it the characteristics of a sample with time are slowly vary.
As a rule electronic processes in the model, it flows considerably more
rapid, than change in the crystal structure. For the analysis of experimental
results and theoretical studies conveniently introduces two characteristic time τe –
the time of electronic processes and τi – time caused with the change in the
lattice. Moreover τe << τi.
Thus, rapid processes are caused electronic by processes, and slow with the
changes in crystal lattice. Changes in crystal lattice must give disruption in the
hysteresis line and with the periodic repetition hysteresis loop it must it will
become the helix (Fig. 9).
9
Resistance of the semiconductor in variable deformation fiel d
351
Fig. 9 – a) for the locked trajectories relaxation is only electronic; b) the disruption
of loop is caused by structural changes in the model.
On the basis of the lead (carried out) researches, it is possible to draw the
following conclusions:
– Existence hysteresis loops of a phase trajectory in a plane resistance–
deformation (R – ε) it is caused relaxation by processes in a sample;
– Closed relaxation the loop is caused only by electronic processes and
change in a crystal lattice do not occur.
– Open screw trajectory on (R – ε) planes it is caused by irreversible structural changes in a crystal lattice.
REFERENCES
1. G. L. Bar, G. E. Pikus, Symmetry and deformation effects in semiconductors, Nauka, 1972, 584.
2. P. S. Kireev, Physics of semiconductors, VSH, 1975, 584.
3. J. Blekmor, The theory of the recombination speed in semiconductors.
4. Problems in thermodynamics and the static physics, Edited by Lansberg, MIR, Moscow,
1974, 399.
5. B. Ridli, Quantum processes in semiconductors, MIR, Moscow, 1986, 304.
6. A. Nowick, W. Heller, Adv. Phys. 12, 251 (1963).