theory and experiment on excitons in silicon diode

8/6/2019 Theory and Experiment on Excitons in Silicon Diode

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Contents

Abstract............1

Acknowledgement..................................2

1. Introduction........................3

2. Significanceof excitons in inorganic semiconductors...................3

3. Three particle theory..............................................6

3.1. Simplified model.........................................................................8

3.2. Simplified theory........................................................................8

3.3. Application of theory to Silicon devices...................................13

3.4.Study of effect of exciton diffusion on device current.14

4. Experiment illustrating the effect of excitons on the photocurrent of

Silicon and GaAs solar cells............................................................20

4.1.Simplified model of the experiment.............................................20

4.2.Experimental Technique..........................................................21

5.Summary...........................................................................................24

6.Appendix............................................................................................25


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Abstract

Kane and Swanson in the year 1992 have done theoretical calculations showing the

existence of a significant number of excitons at certain carrier densities especially at

temperature 77K in Silicon. Later on based on these findings R.Corkish et al proposed the

Three particle theory which includes the contribution of excitons to diffusion current

and its effect on dark saturation and light generated photocurrent in silicon solar cell has

been discussed. Further an experiment performed by I.G.Atabaev et al proving the

contribution of excitons to photocurrent has been discussed. This experiment involves the

study of effect of magnetic field on the photocurrent in Si and GaAs solar cells.


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1.Introduction

An exciton is a nonconducting excited state of a crystal which is neutral and

mobile. There are two types of excitons: Frenkel exciton and Mott Wannier exciton, both

are bound states of an electron and a hole.1

However a Frenkel exciton is the bound state of electron and hole which are

sharply localised that is on a single excited ionic level. And Mott Wannier exciton can be

taught of localised electron and hole levels extended over many lattice constants. Hence

Frenkel exciton and Mott Wannier excitons are opposite extremes of the bound states of

electrons and holes.2

Frenkel excitons are observed in many crystals of aromatic molecules and hence

exciton diffusion is the important transport mechanism in organic solar cells. 1 And Mott

Wannier excitons are found in semiconductor crystals1 however very less attention was

given to the study of exciton transport mechanism in inorganic semiconductors. Later on

theoretical investigation by Kane and Swanson showed that significant numbers of

excitons was observed at low carrier density especially at temperature 77K. Based on these

findings Three particle theory was proposed by R.Corkish et al.


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2. Significance of excitons in inorganic semiconductors:

The exciton density in inorganic semiconductors has been calculated by

considering the reaction between free electrons & holes in equilibrium with excitons:3

n + p exciton

At equilibrium we have,

n p = n* nex (1)

where, nex is the exciton density,

n is the electron density in the conduction band and p is the hole density in the valence

band which are not bound as excitons.

n* is the equilibrium constant.

However in the case of direct band gap semiconductors the exciton lifetime is found to be

short because of radiative electron hole recombination.

From the above equation(1) the importance of excitons can be deduced by calculating theratio (nex/n) which is equivalent to the calculation of (n/ n*).

The value of n* has been calculated by Combescot to be,4

(2)

with n0i=gi mi3/2x 2.4 x 1015 cm-3,

where gi is the degeneracy of the ith particle and mi is the ratio of effective mass to the free

electron mass, m0.

The above equation is written in terms of effective valence density Nv and conduction band

density Nc states as :4

(3)


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where f is given by:

(4)

The order of magnitude of n* for silicon has been calculated. Silicon has six

conduction band minima and both a light and heavy hole valence band , considering the

two spin states the exciton degeneracy has been calculated to be gex=48. And reasonable

assumption for masses has been made to account for f 21 and N cNv 3x 10 17 cm-3.1

Coulomb interaction because of the free carriers leads to screening and hence

influences the binding energy of the excitons. This has been investigated in the case of

ionised donor levels,5

(5)

where Ex is the unscreened exciton energy, rc is the critical radius at which Mott

transition occurs and rs is the Debye screening radius given by:

rs = [(4 e2/ KT)(n+p)] -1/2 (6)

Table 1:

Ratio of exciton concentration to electron concentration in silicon at T=300K and T=77K

respectively.6

T=300K T=77K

rs/rc Eex/E n

(1016 cm-3)

n*

(1016 cm-3)

nex /n N

(1016 cm-3)

n*

(1016 cm-3)

nex /n

20.0 0.903 0.13 84.9 0.0015 0.038 2.43 0.015510.0 0.810 0.52 89.6 0.0058 0.151 2.98 0.05034.0 0.563 3.23 103 0.0314 0.941 5.25 0.1792.0 0.250 12.9 124 0.104 3.76 10.6 0.355

1.5 0.111 23.0 134 0.172 6.69 14.5 0.4611.2 0.028 35.9 141 0.255 10.5 17.5 0.600


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1.0 0.0 51.6 143 0.361 15.1 18.6 0.809

From the above table it is clear that the exciton carrier concentration increases and is

significant at concentration just below Mott transition and especially at T=77K.

R.Corkish et al concluded that it is significant to include excitons in the study of transport

mechanism in semiconductors for the following reasons:6

1. The ratio of minority carrier concentration in silicon under low injection at 300K

was plotted as a function of doping density which indicates the occurrence of a

significant number of excitons.

2. The excitons though neutral participate in transport through diffusion and on

reaching the junction dissociates into free charge carriers thereby contribute to

current.

Hence suggesting the need for the inclusion of excitons in solar cell theory R.Corkish,

P.Chan and M.A.Green proposed the three particle theory.


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Fig1: A plot of ratio of exciton concentration v/s minority carrier concentration6

Figure shows that a significant number of excitons exist at T=300K in silicon.6

3.Three particle theory:6

The Three particle theory includes exciton diffusion in transport equations. And

the current continuity equations corresponding to electrons, holes are given by,6

(7)


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(8)

And the excitons being neutral participate in diffusion and the corresponding continuity

equation is given by,6

(9)

where is the flux corresponding to the exciton flow, from the Ficks law of diffusion

excitonic flux is given by:6

= - Dx .nx (10)

where Dx is the diffusion coefficient of excitons, and

nx, n and p represent the excess electron, exciton and hole concentrations respectively.

(11)

Ueh is the net rate of recombination of electrons and holes, Ux is the net rate of

recombination of excitons and are the lifetimes of electrons, holes and

excitons respectively.

The generation rates are given by,

(12)

where Gx0 and Geh0 are the rate of exciton generation and rate of electron generation at the

surface, is the absorption coefficient, z is the distance into the semiconductor.

B is the rate of binding of electrons and holes into excitons and is given by:6

B = b (n p-n*nx) (13)

n* is the equilibrium constant and b is the binding coefficient. The above equationimplicitly implies that electron hole generation recombination rate is slower than carrier

exciton generation rate.


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3.1 Simplified model:6

The theory was developed based on a simplified model of diode and solar cell. The

following assumptions were made:

1. Depletion approximation was considered (that is the total ionised negative charge

per unit area on p-side is equal to the total ionised positive charge per unit area on

n-side).

2. The net current within the depletion region was assumed to be zero (that is the drift

and diffusion currents are approximately equal and opposite of each other).

3. A nondegenerate semiconductor system was considered (that is low level injection

is assumed).

4. Predominant flow mechanism of minority carriers is assumed to be diffusion.

5. Recombination in the depletion region is neglected.

A one sided diode with wide base is considered that is the emitter layer is considered to be

extremely thin and diffusion process only in the base region is studied which is considered

of p-type. However though the predominant flow mechanism of carriers in the bulk region

is by diffusion an electron reaching the depletion region at z=0 will be collected by the

field. Similarly an exciton diffusing to the contact will be dissociated into an electron and

a hole thereby contributing to current. This generalised theory is analysed in one

dimension neglecting all surface effects.

3.2.Simplified theory:6

Diffusion being the predominant transport mechanism for minority carriers and excitons in

bulk the modified continuity equations including electron current density and exciton

current density are given by the coupled pair of differential equations:

(14)

(15)


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where Dx and Dn are the diffusion coefficients of excitons and electrons respectively.

From the law of mass action at the equilibrium condition of electron exciton system

np = nxn*

Hence on injection we can replace n by n which corresponds to excess minority

concentration.

Defining z=0 plane at the p-type edge of the depletion region and hence the boundaryconditions are given by:

n(z=0) = npo exp(qVa/KT)

nx(z=0) = nxo

where npo = ni2/NA

ni is the impurity carrier concentration, Va is the applied voltage and nxo is the equilibriumexciton concentration and KT/q = 25.85mV at 300K and at other boundary z= the

electron hole concentrations are said be finite.

expressing equations (14) and (15) in matrix form,

(16)

where,

(17)


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The above equations have been solved using the standard eigenvalue eigenvector

techniques. The eigen values of matrix Mij are calculated considering the characteristic

polynomial corresponding to matrix M as follows:

(18)

where I represents the identity matrix and

The eigenvalues of the matrix are:

(19)

where where

Physical meaning for the above terms in equation (19) can be interpreted by considering

the decoupled case where in b=0, then = M11, M22

where M11 = Ln-2 and M22 = Lx-2 and Ln =(Dnx)1/2 & Lx =(Dxx)1/2 are the diffusion lengths

of the electrons and excitons respectively and x and n represents the lifetime of excitons

and electrons respectively.

In an exciton dominant system where M11>M22 that is 1/Ln2 > 1/Lx2,

M11 must correspond to the larger eigenvalue+.

In an electron dominant system where M11


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1. Homogeneous solution (dark current solution):

Considering the dark current measurements (n=0, x=0) that is the homogenous solution

of the above differential equation:

(20)

(21)

In this one sided model neglecting the majority carrier current the current density is got by

adding the minority carrier current and the also the flow of excitons at z = 0 plane.

(22)

where r = Dx /Dn. Therefore the dark saturation current is

(23)

The above expression is compared to the net current when excitons are neglected, which is

given by

(24)


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2.Inhomogeneus solution:

In order to obtain the light generated photocurrents now consider the condition of

illumination of solar cell in which either or both n & x are finite. That is electron and

optical generation are nonzero. Then the particular solution corresponding to the above

equation is given by

(25)

(26)

where can be calculated from equation(18).

The first term in the above expression is related to the optical generation of electrons and

excitons respectively. The second term speaks about the interaction between the electrons

and excitons. The particular solution that is obtained is added to dark solution and the

boundary conditions for the short circuit are applied to yield:

(27)

(28)


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(29)

on differentiating equations (25) and (26) we get the light generated current density,

(30)

This is then compared with the light generated current density neglecting excitons which is

given by,

(31)

3.3.Application of theory to Silicon devices:6

Applying the theory to silicon devices such as solar cell the following approximations are

made:

1. The total generation rate at the surface G tot= Geh + Gx = 1010cm-3 is assumed for

illuminated case.

2. And also the solar cell is modelled with = 0 thereby simulating constantgeneration rate throughout and this corresponds to the weak absorption of

illuminated light.


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In order to compare the current density due to excitons with the expression which does

not take account of excitons certain approximations are made to the electron exciton

parameters:

1.Electron parameters:

The value of intrinsic carrier concentration is assumed to be n i= 1.00 X 1010cm-3 at

room temperature 300K.7

The lifetime is assumed to vary with doping concentration (Na) according to the

following empirical relationship:8

(32)

Diffusion coefficient (Dn) of an electron has been obtained from an empirical fit to a

model for minority electron mobility as a function of doping density.9

2.Exciton parameters:

Assumptions made about exciton parameters are as follows:

The unscreened exciton binding energy is borrowed from the result obtained from the

wavelength derivative spectral response measurements at 1.8K with a correction to

effective masses at room temperature 300K, yielding an exciton binding energy of around

Ex = 20.6 meV.10

The value of n*(NA) is given by:3

n*(NA) = exp [-Ex (NA)/kT] (NcNv) 1/2/f (33)

where Ex = Ex [1-(NA/nMott) 1/2]2 which is a function of doping density (NA).

Mott density, n Mott = 1.03 X 1018cm-3 as calculated from the expression derived by Norris

and Bajaj for T


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And the value of f in the expression for n* has been adopted from the calculations made by

Kane and Swanson to be f = 21 and (Nc Nv) 1/2 = 3X 1019.

Lifetime of exciton ( x) is assumed to vary similar to n and and hence calculated fromthe above expression corresponding to x and the exciton lifetime for an intrinsic material

is assumed to be x = 100s, as calculated by the earlier work which can be taken to be the

upper bound.8

Diffusion coefficient of excitons is calculated from an empirical relation obtained by Kane

and Swanson in extrapolating to 300K resulting in D x= 17cm2s-16 and diffusion coefficient

is assumed to be independent of doping density since excitons are neutral particles.

The binding parameter b is assumed to vary with temperature T according to the empirical

relation obtained by Nolle.11

b= 10-3 T-2 + 2.5 X 10-6 T-1/2 + 1.5 X 10-7 (cm3s-1) (34)

b = 3 X 10-7 at 300K.

Here b is considered to be a variable for two reasons as b0 the calculations should yield

a current value which excludes the current contribution due to excitons and also becausethere is no experimental evidence for the value of b.

3.4.Study of effect of exciton diffusion on device current:

1.Study of effect of exciton diffusion on dark saturation current:6

The ratio of dark saturation currents (J0/J0) with excitons included to that excluding the

exciton contribution is calculated by substituting different parameter values calculated

corresponding to the room temperature 300K and the ratios are plotted for different values

of binding coefficient (b) with NA as a parameter.

Fig2: Plot showing the variation of (J0/J0 ) with log b for different values of NA (cm-3):

unfilled squares = 1015 , triangles = 1016 , diamonds = 1017, filled squares = 1018.7


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Graph shows that inclusion of excitons does not affect the saturation current when the

binding coefficient(b) value is very small that is small values of b tend to inhibit the

relaxation of electron hole pairs into excitons and hence as b0 the values relax to those

calculated neglecting excitons.

As the binding coefficient (b) value increases there is a noticeable decrease in the dark

saturation current density at the doping density approximately equal to 1018cm-3 and the

reduction in saturation current is calculated to be upto 30%.

At large value of b and at high doping density (NA) the contribution of excitons is high. In

order to analyse this, a plot of diffusion length v/s doping density (N A) is studied where in

diffusion length depends on the current density. That is from the theory we have

J0/J0 L n/Lx (35)

Fig3: Plot of diffusion lengths v/s doping density for b=10-6.6


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Hence the effect of exciton involvement is noticeable only if exciton diffusion length

differs from that of minority carriers.

2.Effect of excitons on Photocurrent with Gx = 0, Geh 0 :6

Here we consider the formation of excitons only through the relaxation of electron hole

pairs and this corresponds to the region of spectrum where the absorption due by excitonsis negligible compared to the total absorption.

Fig4: Plot of photocurrent density ratio as a function of binding coefficient(b) for different

values of doping density(NA). Unfilled squares=1015, triangles=1016,diamonds=1017,filled

squares=1018.6


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Graph shows the variation of short circuit current ratio (with excitons considered to

excitons neglected) with the binding coefficient (b). It can be seen from the graph that for

larger values of b and for higher doping concentration the short circuit current improves

which has been calculated to be up to 44%.

3.Effect of exciton lifetime on photocurrents with Gx0 :6

Due to the lack of experimental evidence for the assumed value of the lifetime of exciton

an investigation on the sensitivity of the results obtained in this theory to this parameter is

studied.

Figure shows the plot of photocurrent ratios as functions of doping density(N A) with

lifetime of excitons as parameter for a fixed value of b = 10-7 cm-3 s-1. The different values

of exciton lifetimes chosen are as follows

= (x(NA ,(n(NA x(NA) =10 ,(n(NA x(NA) = 0.1 (n(NA

Fig 5: Plot of dark saturation current density ratio as a function of doping density;

Ratio of exciton to carrier lifetime is the parameter varied with the values: 0.1-

triangles, 1- squares, 10- diamonds.6


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From the graph it can be observed that there is a large divergence of the ratio

from unity when lifetimes are equal at greater values of binding coefficient and at high

doping levels. This is because of our assumption that Dx Dn, else if Dx = Dn this was

assumed then our results would have been Jo/Jo = 1 and JL/JL = 1. This implies that the

fundamental reason for the changes due to inclusion of excitons is the difference in the

lifetimes of excitons and electrons.

From the graph it is evident that when x > n, the current ratios improves and

for

>(x(NA n(NA), there is a decrement in current ratios. From this it can be concluded that

the beneficial effects of inclusion of excitons in the theory is due to the dependence ofcurrent ratios on exciton diffusion lengths.

Hence there is a need for experimental determination of exciton lifetimes.

4.Study of effect of inclusion of exciton diffusion on light generated photocurrent with Gx

0 and Geh 0:6

This corresponds to the region of absorption where in absorption by exciton states

correspond to the significant fraction of total absorption.

The short circuit current ratio JL/JL was plotted as a function of the binding coefficient b

with Gx0/Gtotal0 as a parameter varying between zero and unity.

Fig 6: Plot of photocurrent ratio JL/JL as a function of binding parameter(b) for a fixed

doping density 1018cm-3. Fraction of optical generation of excitons is the parameter

considered: unfilled squares, 0% ; triangles, 10%; diamonds, 50%; filled squares, 100%.6


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From the graph it can be observed that for small value of binding coefficient there

is a large increase in short circuit current and a large fraction of total generationcorresponds to excitons rather than carrier pairs as smaller value of b prevents the

dissociation of excitons into free carriers. And for larger values of b there is a decrease in

the current density ratio indicating that excitons dissociate into free carriers.

4. Experiment illustrating the effect of excitons on the photocurrent of Silicon

and GaAs solar cells.12

Above theory thus concludes that excitons contribute significantly to the light generated

photocurrent and to the reduction of dark saturation current. The inclusion of this theory in

solar cells by studying the effect of magnetic field on the photocurrent has been performed

by I.G.Atabaev et al. In the experiment a magnetic field of particular strength has been

applied to solar cell and the change in the photocurrent is observed. And the results are

interpreted considering the simplified model of the experiment.

4.1.Simplified model of the experiment:12

The following assumptions were made:

1. Excitons are assumed to be generated only near the surface of the semiconductor

by choosing the appropriate wavelength for illuminating.

2. In presence of magnetic field all the excitons are assumed to decay into free charge

carriers and hence the corresponding current measured in the presence of the

magnetic field is attributed completely due to free charge carriers.

However this is an ideal case and in reality the excitons have finite diffusion length and

they gradually decay into free charge carriers but they tend to have a small diffusion length

in the presence of the magnetic field. In the absence of the magnetic field the excitons are

considered to contribute significantly to current and hence the diffusion length of excitons

and that of electrons and holes can be considered to be proportional to the photocurrents


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Fig 7b:Plot of I v/s intensity for GaAs (sample2)12

The figure shows the photocurrent variation with intensity for two different samples of

GaAs sample 1 having efficiency 23% and sample 2 having efficiency 18% .

From these measurements the following observations were made:

I = [ Iph(H) Iph] Leh is valid

in the surface region of the solar cell. And hence the contribution due to exciton is 6-8%.

At liquid nitrogen temperature the contribution of excitons is found to increase by 10-15%.

Therefore the condition Lex> Leh is satisfied in the case of GaAs solar cells.

Similarly the measurements were repeated with two different samples of Si: sample1 with

efficiency 15% and sample2 with efficiency 11% in the presence of magnetic field. And

corresponding graphs were plotted.

Fig 8a: Plot of photocurrent v/s intensity corresponding to Si sample1 (shallow p-n

junction)12

Fig 8b: Plot of photocurrent v/s intensity corresponding to Si sample 2(deep p-n junction)12


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Figure8a shows that the Si solar cells exhibit behaviour different from that of

the GaAs solar cells. where I amounts to 3% but has opposite sign. It has been reasoned

out to be because of the difference in the mechanism of generation and accumulation of

photo induced charge carriers in them. In the case of GaAs solar cell the photo carriers are

generated near the surface of the solar cell. But in the case of silicon this assumption is not

valid because of its greater absorption coefficient and hence the generation of charge

carriers occurs in the bulk. However in the case of Si sample 2 with deep pn junction

generation is predominant in diffusion region and the diffusion length of excitons in this

region is comparable to that of the excitons. Whereas in the case of shallow p-n junctions,

the generation and recombination is predominant in the base region of the solar cell and

hence the diffusion length of excitons is lesser than that of free carrier diffusion length.

And hence the observed change the photocurrent in the the presence of the

magnetic field is evidently because of the decrease in the exciton diffusion length as the

Lorentz force acting on the excitons tend to decouple the bound system into free electron

hole pairs.13

Therefore the results obtained from this experiment support the three particle theory.


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5.Summary:

Kane and Swanson in the year 1992 made a theoretical investigation on the

exsistence of excitons at different temperatures. They arrived at the conclusion that

significant carrier density of excitons exists especially at temperature below 77K .

R.Corkish et alproposed the Three particle theory based on the conclusion

drawn from the work of Kane and Swanson. They included the contribution of excitons to

diffusion current due to excitons in transport mechanism . By considering the simplified

model of silicon diodes and solar cells the contribution of excitons to reduction in dark

saturation and increase in light generated photocurrent has been demonstrated.

An experiment conducted by I.G.Atabaev et al demonstrating the contribution

of excitons to photocurrent in Si and GaAs solar cells has been discussed which involves

the study of effect of magnetic field on the photocurrent.

Calculation of the parameters such as lifetime of excitons, diffusion length and

diffusion coefficient enables the optimisation of these parameters to effectively utilise theexcitonic photocurrent contribution and hence improve the efficiency of the solar cell.

6.References:

1)N.W.Ashcroft and N.D.Mermin, Solid state physics (Brooks/cole,1976)

2) C.Kittel,Introduction to solid state physics(John Wiley & Sons,1960)

3) D.E.Kane, R.M.Swanson,The effect of excitons on apparent band gap narrowing and transportin semiconductors, Journal of applied physics,73(7),(1993) 1193-1197.


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4) M.Combescot,Thermodynamics of an electron-hole system in semiconductors,

Physica Status Solidi B, 86(1), (1978) 349-358 .

5) J. R. Lowney, A. H. Kahn, J. L. Blue, and C. L. Wilson ,Disappearance of impuritylevels in silicon and germanium due to screening , Journal of applied physics,52, (1981)

4075.

6) Richard Corkish, Daniel S.-P. Chan, and Martin A. Green,Excitons in silicon diodes and

solar cells: A three-particle theory,Journal of applied physics,79 (1),(1996)195-203.

7) B. Sproul and M. A. Green ,Improved value for the silicon intrinsic carrier

concentration from 275 to 375 K , Journal of applied physics,70, (1991) 846.

8) J. G. Fossum, Computer-aided numerical analysis of silicon solar cells,Solid stateelectronics,19(4), (1976)269-277.

9) W. Stephens and M. A. Green , Minority carrier mobility of Czochralskigrown silicon

by microwavedetected photoconductance decay , Journal of applied physics,. 74, (1993)

6212.

10) A. Hangleiter andR. Hcker ,Enhancement of band-to-band Auger recombinationby electron-hole correlations,Phys. Rev. Lett. 65, (1990) 215218.

11) E. L. Nolle, Sov. Phys. Solid State9, (1967)90

12) G. Atabaev, N. A. Matchanov, M. U. Khazhiev, and D. Saidov, On the possibility of

determining the diffusion length of excitons in semiconductors from photomagnetic

measurement data ,Physics of the Solid State, 47(11), (2005) 20252029.

13) K. Zeeger, Semiconductor Physics (Springer, Wien,1973).
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