Volodymyr Ya. Degoda1)

Andrii Sofiienko2)

1)Kyiv National Taras Shevchenko University,

Department of Physics

Acad. Glushkova ave., 4, Kyiv 03680,

e-mail: degoda@univ.kiev.ua

2)University of Bergen,

Department of Physics and Technology

Norway, 5020 Bergen, 55 Allegaten str.,

e-mail: asofienko@gmail.com

Kinetics of X-ray conductivity

for an ideal wide-gap semiconductor irradiated

by X-rays

2 2

Why the creation of a general kinetic theory of X-ray conductivity (XRC)

of the semiconductors and dielectrics is necessary?

1. The analysis of all processes of X-ray conductivity will allow to determine the most

relevant aspects for the development of novel detectors of the ionizing

radiation with wide operating temperature range and higher irradiation

stability.

2. To explain the specific experimental results of X-ray conductivity (XRC):

1) An anomalous XRC of ZnSe crystals, when sometimes the decreasing of the

current is observed at higher X-ray excitation;

2) A non-linear current-voltage XRC dependences.

Which characteristics of the semiconductors the kinetic theory of X-

ray conductivity should describe?

1. Spectrometric (amplitude and shape of the current pulse).

2. Integral (voltage-ampere and lux-ampere characteristics at the excitation by X-

rays).

3 3

First, we need to choose a geometric and physical model of the object. The

geometric model can be following as shown below:

In accordance with X-ray absorption law, the spatial distribution of the generated

free electron-hole pairs is following:

0( ) X y

XdF y F e dy 0( , )3

X yXG X

g

hN y t F t e

E

0

0 13

XYXGG

g

hN F e Hd

E

Also we believe that the electrical contacts of the detector are ohmic.

4 4

The material model can be gradually complicated step by step by the incorporation

of the point defects until it becomes adequate to real crystals:

- ideal semiconductor;

- semiconductor with shallow traps;

- semiconductor with shallow and deep traps and recombination centers.

The logic scheme for the development of X-ray conductivity kinetic theory

can have several stages and the following succession can be proposed:

I. Kinetics of X-ray conductivity at the absorption of one photon

1. Free hot carriers are generated after the absorption of X-ray photon;

2. Thermalization of hot carriers during inelastic scattering;

3. Current impulse in an ideal semiconductor;

4. Current impulse in a semiconductor with one type of shallow traps;

5. Current impulse in a semiconductor with different types of shallow traps;

6. Current impulse in a semiconductor with a recombination centers;

7. Current impulse in a semiconductor with shallow and deep traps and

recombination centers.

5 5

II. X-ray conductivity kinetics at the X-ray excitation of the

semiconductors

1. Ideal semiconductor at the low excitation level;

2. Influence of the Coulomb interaction on the concentration of free-carriers;

3. Ideal semiconductor at the high excitation level;

4. Recombination of free electrons and holes;

5. Influence of shallow traps on the kinetics of XRC at the low excitation level;

6. Semiconductor with shallow traps at the high excitation level;

7. Deep traps and recombination centers. The stationary state;

8. The near-contact volume of the charges of XRC;

9. Accumulation of charge carriers on the deep traps;

10. VAC and LAC of the real semiconductor at steady-state excitation;

11. Kinetics of the rise and decay of the current of X-ray conductivity.

6 6

Ideal semiconductor at the low excitation level

A concept of an ideal semiconductor means that such semiconductor doesn't have

any local centers affecting on the charge transport.

For determining the integral characteristics of XRC it is necessary to appoint the

spatial distribution of the generated free electrons and holes and the value of the

additional electric field created by them.

On the first stage we believe that direct recombination of free electrons with free

holes is extremely small (but this assumption can be useful only if there is no

external electric field end carriers drift to the contacts).

The kinetic equations system for an ideal semiconductor simplifies to two

equations for the free electrons and holes and the Poisson equation as follows:

0

G

G

NN D N E N

t

NN D N E N

t

e N NE

7 7

For low excitation level, when the additional electric field of free charge carriers can

be neglected, the following interchange is acceptable: ∂(E·N±)/∂x on E0·∂N±/∂x and

∂E/∂x=0 (the Poisson equation is not used). A solution of the kinetic equations

system for the selected geometry with boundary conditions N±(x=0)=0 and

N±(x=d)=0 for the spatial distribution of the concentration of free electrons and

holes will be following:

The application of the Einstein relation (µ±/D±=e/kT) enables to simplify the

expressions in the exponents and we can see that they are equal for electrons and

holes and even at small values U0 = E0∙d become considerable.

0

0

00

0

0

00

1 exp

,

1 exp

exp 1

,

exp 1

G

G

E x

N DN x E d x

E dE

D

E x

N DN x E x d

E dE

D

8 8

Spatial distributions of electrons (a) and holes (b) in an ideal semiconductor with parameters of

ZnSe when NG = 1010 cm-3∙s-1, d = 1.0 cm and at different external electric field : 0.0 V/cm (1);

0.01 V/cm (2), 0.1 V/cm (3), 1.0 V/cm (4), 10 V/cm (5).

As seen, even if the electric field strength is greater than 1 V/cm then diffusive

motion is weak in comparison with the drift and can be neglected (but the

charge drift velocity is much smaller than the thermal velocity). The unequal

maximum values of the carriers concentrations are determined by the different values

of the diffusion coefficients of free electrons and holes (D-≠D+) and by their effective

masses.

9 9

It is possible to determine a statistically mean lifetime of free electrons and holes as

follows:

The lifetime of free carriers in the sample is inversely related to E0 and is

determined by the drift time of the charge carriers to the electrical contact.

The total current of XRC can be computed as the additive sum of the currents of all

layers of a semiconductor:

1

0 02

26D U

d

0

0 0

0 0 0 0

1 exp2

1 exp

Y Z

РП GG

eU

kTkTi dy dz j j eN

eU eU

kT

Hence, VAC of an ideal semiconductor is conditional by the temperature and total

amount of generated free charge carriers. The obtained dependence has entirely

lost the material parameters and will be the same for all ideal semiconductors.

For the integral characteristics of XRC we have got linear LAC . VAC has a rapid

saturation and when U0 > 10 V it remains practically unchanged at the level of

eNGG.

10 10

The current-voltage characteristics of an ideal semiconductor when

NGG = 1010 s-1 (1,3) and NGG = 108 s-1 (2,4) at 295 К (1,2) and 80 К (3,4).

11

If E0 > 1 V/cm and if the diffusion motion can be neglected in the kinetic

equation, then the equation Eeh(x) is simplified to the following form:

The value of the electric field of free charge carriers can be computed by integrating

of the Poisson equation using the stationary spatial distributions of electrons and

holes:

0 0 0

1

2

x d

ehx

e N N dx e N N dxE x

22

2

0 0 0

11 2 1

2 2

eh GE x eN d x x

E E d d

The maximum value of the electric field Eeh(x) will be at x = d·(µ+/µ++µ-). Next figure

illustrates the relations |Eeh|/E0 at different values of NG. We can define such

concentration of free charge carriers at which their electric field Eeh is equivalent to

the external field E0.

When NG > 1012 cm-3·s-1 then it is necessary to take into account the electrical

field of free electrons and holes.

12 12

The computed ratios Eeh(x)/E0 for the ideal ZnSe (a) in which the electrons

and holes mobilities are different in 25 times and for the semiconductor with

equal mobilities µ- = µ+ = 100 cm2V-1s-1 (b) if NG = 108 cm-3s-1 , d = 1 cm,

U0 = 1 V (1,1’); 3 V (2,2’); 10 V (3,3’); 30 V (4,4’); 100 V (5,5’)

The dependences of max|Eeh|/E0 for ZnSe (full

lines) and for a semiconductor with µ- = µ+ =

100 cm2V-1s-1 (dashed lines) on the applied

voltage to the contacts U0 and NG = 108 cm-3s-1

(1,1’); 1010 сm-3s-1 (2,2’); 1012 сm-3s-1 (3,3’);

1014 сm-3s-1 (4,4’); d=1 cm 13

The value of the free carriers

electric field is directly

proportional to the excitation

intensity. At strong X-ray

irradiation the generated field of

charge carriers is comparable

with the external electric field.

Then the spatial distributions of

the electrons and holes can be

determined by the full system of

kinetic equations and the

Poisson equation can’t be

neglected.

14 14

The influence of the Coulomb interaction on the

concentration of free charge carriers

The main feature of XRC of the semiconductors at high excitation levels is the

necessity to take into account the Coulomb interaction between free electrons and

holes. Free charge carriers create an additional electric field and move themselves in

the diffusion-drifting way in this self-consistent field.

First we consider the case when there is no external electric field (E0 = 0), and then

we will define the spatial distributions N-(x) and N+(x), we should take into account

the field of free charge carriers Eeh(x) and the Coulomb interaction of free electrons

and holes. It’s obvious that Eeh=0 at any excitation level if µ- = µ+. The electric field of

free harge carriers will be caused only by different concentrations of electrons and

holes.

The Coulomb interaction between electrons and holes will slow down of the drift

motion of the electrons and will speed up the drift of the holes to the contacts. That

is, the internal field Eeh will cause the decrease in the free holes concentration and

will increase the concentration of free electrons and as a result the value of the field

Eeh will slightly decrease until a dynamic balance between the value of the field and

the equilibrium carriers density occurs.

15 15

We can try to find an approximate solutions of the kinetic equations as simple

analytical functions. The changing of the concentration will increase monotonically at

the excitation intensity increment due to the Coulomb interaction and this change will

be given in the form of the addend that has functional dependence as bell-shaped.

Also we use a small difference between following functions:

where 0 < x < d. This allows to use the approximate relation for the Poisson equation:

4 1x x x

Sind d d

2 1 1

8

GN d x

N N SinD D d

And for the value of the electric field

of free charge carriers we have:

3

0

1 1

8

G

eh

eN d xE x Cos

D D d

Spatial distribution of the electric field strength of free charge carriers in

ZnSe at NG = 108 cm-3s-1 and d = 1 cm, U0 = 0 V; the exact calculation

is a dashed line and the approximate calculation is a full line

16 16

For the third addend of the kinetic equation for free carriers we also use the function

the free holes concentration in the form

in the kinetic equation, we get a system with two algebraic equations for defining of

the constant A+ and B+

Hence, such a functional relationship for the spatial distribution of the concentration

of free holes gives the correct tendency for changing their concentration. The same

result can be obtained for the concentration of free electrons, considering that B- = -

B+. Dealing with such functional relationship for the spatial distributions of

concentrations N-(x) and N+(x) we can obtain the values of concentrations that takes

into account the Coulomb interaction of the free charge carriers.

2

.8

GN d x

N x SinD d

2 xN x A d x x B Sin

d

of the carriers concentration in the form Substituting

2 2 4

0

2

1 12

64

G

G

N A D

eN dB

d D D D

2 6

2

0

2

1 1

128

G

G

NA

D

eN dB

D D D

17 17

where the constant b is computed as follows:

2

2

0 0 0

2

2

0 0 0

4 1 1 ,8

4 1 1 ,8

G

G

x x x x N dN x N bSin N b Sin N

d d d d D

x x D x D x N dN x N b Sin N b Sin N

d d D d D d D

2 12 2 2

0 0 0

2 4 2 4 2 4

1 4 1 1 8 1 83 3 2 1 1 1

4 2 2 2G G G

D kT D D kT D D D kT Db

D e d N D e d N D D e d N

The computed relative changing

in the concentrations of free

charge carriers b(NG) by virtue of

their Coulomb interaction as the

function of the intensity of their

generation in ZnSe and Si at

room temperature.

18 18

Evidently, due to the Coulomb

interaction between free electrons and

holes at NG > 1010 cm-3s-1 their

concentrations become almost equal,

and the value of their electric field

reaches of the maximum value less

than 1 V/cm. It should be noted that

the physical basis of this phenomenon

is the same as the Dember effect. The

Coulomb interaction of the generated

free carriers causes the equalization

of electrons and holes concentrations

in an ideal semiconductor at

increasing of the excitation intensity,

and the diffusion motion of electrons

and holes is determined by the same

diffusion coefficient

D-1 = ½ ·[(D+)-1 + (D-)-1].

It is important for the case of

screening of the external field by free-

carriers.

The maximum value of the electric field

of free carriers as function of the intensity

of their generation in ZnSe and Si at

room temperature

19 19

The ideal semiconductor at the high excitation level

The key feature of XRC of a semiconductor at the high excitation level is the

necessity to take into account the additional electric field created by the free charge

carriers that move in the drift-diffusion way in the self-consistent field. The integrated

XRC is usually experimentally studied at the high values of the external fields

(E0 > 10 V/cm), and that is why for the determining of N-(x), N+(x) and Eeh(x) we can

neglect of the diffusion component in the kinetic equation system

2

2,

G

E x N xN xD N

x x

The kinetic equation system will be simplified:

0

G

G

NN E N

t x

NN E N

t x

e N NE

x

In an ideal semiconductor the

steady state is reached very

quickly, and that is what we are

going to consider.

20 20

After of the integrating with boundary conditions N+(x=0)=N-(x=d)=0 we obtain next

relations:

These relations are confirmed by the change conservation law, by virtue of the fact

that the value of the current density at XRC will be constant for each dy layer along

the direction of the current flux (OX axis):

G

G

G

G

NE N N E x N x x

x

NE N N E x N x d x

x

Gj j j e N x E x e N x E x eN d Const x

From the physical standpoint it is logical that the current density is determined only by

the number of carriers generated in a unit time in the layer, if the field is big enough to

neglect the diffusion losses of the current at the across the contacts and in the case

there are no recombination channels of the electron-hole pairs in the semiconductor

sample. The obtained relation can be inserted into the third equation (Poisson

equation) and for E(x) we get:

0 0

1 1 2 1 12G G

E eN d eN dx EdE x dx

x E x

22

2 2

2 1

0

2 1 1 1 1

2

GeN d x x

E C Cd d

After of the integration:

21 21

The first constant of integration C1 arises from the fact that the average charge of

the free carriers in terms of the sample is non-zero at the different mobilities of the

electrons and holes in semiconductors and dielectrics. And the total field around the

electrical contacts should differ from E0 by the equal value of E(x=0)-E0=E0-E(x=d)

which is determined by the total charge of the free carriers generated in the

semiconductor.

The second constant C2 is determined by two conditions: at the small excitation level

the total electric field tends to the value of the created external field, and the second

condition is caused by the possibility of the significant change of the electric field in the

semiconductor due to the high concentrations of free carriers. The external field is

brought about by the potential difference applied to the electrical contacts,

1

1 1 1

4C

that is a negative quantity when µ- > µ+.

0

0

( )d

E x dx Uthat is why the constant C2 is defined from

Simple approximated relation can be proposed for the spatial distribution of

total electric field E(x) inside a semiconductor as a result of the solving of the

Poisson equation:

24 4

0

2 2

0 0 0 0

11 2 1 2 1

8 2

G GU eN d eN d x xE x

d U U d d

22 22

Normalized spatial dependences of the electric field E(x)d/U0 for an ideal semiconductor with equal (a) µ- =

µ+ = 100 cm2V-1s-1, ε=10 and different (b) µ- = 700 cm2V-1s-1, µ+ = 28 cm2V-1s-1, ε=8.66 (ZnSe) mobilities of

the electrons and holes (d=1 cm, U0=10 V) at different levels of the excitation intensity NG = 1010 cm-3s-1

(1,1’), 1011 cm-3 s-1 (2), 4∙1011 cm-3 s-1 (3), 3∙1010 cm-3 s-1 (2’), 6∙1010 cm-3 s-1 (3’)

If E(x) is known, we can obtain the spatial distributions of free carriers in first

approximation as follows:

24 4

0 2 20 0 0 0

4 4

0 2 20 0 0 0

11 2 1 2 1

2

1 2 1

G

G G

G

G G

N d xN x

eN d eN d x xU

d dU U

N d d xN x

eN d eN d xU

dU U

21

2 12

x

d

23 23

Spatial distributions of the concentrations of

electrons (1,1’) and holes (2,2’) in the ideal ZnSe

semiconductor when NG = 5∙1010 сm-3 s-1 (1,2)

and in the semiconductor with µ- = µ+ = 100

cm2V-1s-1, ε=10 when NG = 4∙1011 cm-3 s-1

(1’,2’); d=1 cm, U0=10 V

The spatial distributions of the

concentrations of the electron and holes

at high excitation level assume a bell-

shaped form and we cannot neglect the

diffusion motion of the carriers anymore,

because the condition

2

2,

G

E x N xN xD N

x x

is not used.

For the high-power

excitations

2

0 0

4

8G

UN

ed

it is necessary to consider the case of

screening of the external electric field by

the free charge carriers. And still should

be remembered that in the ideal

semiconductor at E=0 we can observe

the electrons and holes concentrations

leveling due to their Coulomb

interaction.

24 24

The general kinetic equation system for electrons and holes remains unchanged. The

approximate solution at the high excitation level, when the electric field in the central

part of the sample reaches zero will be following:

where relations for the electrons and holes f1e(x)=f1h(x) will be equal and symmetric at

x=d/2 and also are bell-shaped, and the values of the concentrations will be

determined as the average diffusion coefficient. The functions f2e(x) and f2h(x) describe

the spatial screening distributions of the concentrations of electrons and holes near of

the electrodes. Since f1e(x)=f1h(x) than these spatial distributions of carriers do not

create the electric field and for the Poisson equation can be written the following:

1 2 1 2,

h h e eN x f x f d x N x f x f x

2

0

e

E ef x

x

2

0

h

E ef x

x

0,E 0E

x

Such simplifications allow to write the kinetic equation for the stationary distribution of

the electrons or holes as follows:

- in the central

part of the sample

- Close to the positive electrode,

- Close to the negative electrode

2

1 2

1 220

G

f fD E x f f N

x x

25 25

that can be divided into two parts

and 2

1

20

G

fD N

x

2

2

1 220

fD E f E f

x x

Due to the Coulomb interaction between the free charge carriers of the opposite

signs, in the first equation D+ and D- should be changed on the effective values.

2

11

2

GN d x x

f xd dD

1

0 0 0

2 2 2

1

0 0 0

2 2 2

2exp

4

2exp 1

4

e

h

E kT eU xf

e x e d kT d

E kT eU xf

e x e d kT d

and

The values of the free charge carriers concentrations in the screening layer are

determined only by the values of the applied potential difference and temperature.

And the total electric field in the sample of the ideal crystal at the high excitation level

will be:

2 1 1

e e

kTE x

e d l x l x

where

0

0

exp4

exp 14

e

d eUl d

eU kT

kT

26 26

The value of the electric field, that is close to the contacts, is much larger than the

average value of the electric field U0/d with the screening of the external field by the

free charge carriers at intensive excitation.

Total concentrations of free electrons and holes will be following:

Hence, at the high excitation level, the generated free charge carriers will shield of

the external electric field. As a result, the concentrations of the carriers in the central

part of the sample can be very greater in proportion to the excitation intensity. And the

diffusion kind of motion becomes dominating and also it is the same for electrons and

holes with effective diffusion coefficient D.

The integral current-voltage and lux-ampere characteristics of X-ray conductivity of

the ideal semiconductor will remain linear, because the number of the generated

carriers per unit of time will be equal to the number of carriers that reach the electric

contacts and recombine on them.

12

0 0

2 2

12

0 0

2 2

21 exp 1

42

21 exp

42

G

G

N d kT eUx x xN x

d d e d kT dD

N d kT eUx x xN x

d d e d kT dD

27 27

Recombination of free electrons and holes

In an ideal semiconductor model, there is only one fundamental possibility for the

disappearance of the generated free charge carriers in the sample. It is the

recombination of free electrons and holes. The probability of this process (r0) is very

small and in the real semiconductors with very low concentration of recombination

centers (~1012 cm-3) such process can be neglected in contrast to the probability of

recombination on the recombination centers. Also, we consider the potential effect of

the direct recombination of electrons and holes in the spatial distribution of the free

carriers equilibrium concentrations. This process leads to the occurrence of the

summand r0N-N+, that basically changes kinetic equations for free carriers:

2

02

2

02

0

G

G

N NN D E N r N N

t x x

N NN D E N r N N

t x x

e N NE

x

Parameter r0 is the product of the recombination cross-section of free electrons on

their thermal velocity: r0 = σ0∙vT ≈ 3·10-14 cm3 s-1.

28 28

Hence, the direct recombination of free electrons and holes has influence upon the

spatial distributions of carriers only at high excitation levels (NG > 1014 см-3 с-1), that

causes the reduction in current of XRC. The intensity of current is directly proportional

to N±max and the relation of LAC for current XRC will follow the relation N±

max(NG).

This equation can not be solved by using of simple analytical functions, but we can

try to find the approximate solution in the following form:

Obviously, this process may be observed only at the high excitation levels, when the

concentrations of free carriers in the ideal semiconductor increase up to ~1014 сm-3.

At such concentrations of N+(x) and N-(x) the screening of the external electric field is

observed, so we should consider the kinetic equation:

2

21

0 120

G

fD r f N

x

2

1

xf x A d x x C Sin

d

and

2

~4

AdC

As a result here is an approximate solution for the spatial distributions of the

concentration of free charge carriers:

2 2 6

2

302

12 256

G GN d x x N d x

N x r Sind d dD D

29 29

The calculated relations N±max(NG) (a) and the spatial distributions of the

concentrations N±(x) (b) when NG > 1016 cm-3 s-1 without (1) and with (2) the direct

recombination of free electrons and holes (r0 = 3·10-14 cm3s-1) when d = 1 cm, D =

1.35 cm2s-1.

30 30

Conclusions

Analysis of the X-ray conductivity for an ideal semiconductor shows the necessity

to consider thef X-ray excitation according to its intensity: low, medium and high

level.

At the low excitation level, when the time interval between the two successive

acts of absorption of X-ray quanta is greater in many times than the drift time of

the charge carriers to electric contacts of the sample, we can consider the

kinetics of X-ray conductivity as the additive sum of current pulses when one X-

ray photon is absorbed. In this case, the integral characteristics are determined

by the additive sum.

At the medium excitation level of an ideal semiconductor when the macroscopic

interaction between free charge carriers generated by the absorption of different

X-ray photons occurs, and this interaction is caused by change of the electric

field E(x) ≠ Const, the consideration of spectrometric characteristics makes no

sense. And the integral characteristics are determined by the spatial distributions

N+(x), N-(x) and E(x) provide for LAC the linear dependence on the excitation

intensity, and for VAC it’s the characteristic dependence on the current saturation

of the XRC.

31 31

At the high excitation level there is a screening of external electric field by the

charge carriers in the middle of the sample and the two barriers around

electrical contacts occur. The barriers are transparent for the carriers of one

sign and nontransparent for the carriers of the opposite sign. At that the integral

characteristics are also determined by the spatial distributions N+(x), N-(x) and

E(x) and provide the linear dependence for LAC and the indicative VAC for the

ideal semiconductor.

At extremely high excitation levels (NG > 1015 см-3 с-1) the process of the direct

recombination of free electrons and holes can be observed that changes the

linear dependence of LAC on the excitation intensity into sublinear. At that, the

character of VAC remains unchanged but the level of current saturation of the

XRC decreases.

32 32