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Physica B 382 (2006) 65–70

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Phonon mechanism of mobility equilibrium fluctuation and propertiesof 1/f-noise

S.V. Melkonyan�, V.M. Aroutiounian, F.V. Gasparyan, H.V. Asriyan

Department of Physics of Semiconductors & Microelectronics, Yerevan State University, 1 Alex Manoogian St., 375025 Yerevan, Armenia

Received 5 December 2005; received in revised form 17 January 2006; accepted 26 January 2006

Abstract

The main mechanisms of the generation of the equilibrium fluctuations of the electron mobility in homogeneous and non-degenerate

semiconductors are studied. It is proven that the mobility fluctuations are related to energy fluctuations and are conditioned by random

non-elastic scattering and generation–recombination processes. In particular, it is shown that the mobility fluctuations come into

existence as a result of random electron–phonon and phonon–phonon scattering processes. The case of acoustic phonon–phonon

scattering is considered in detail. The spectral density of the electron lattice mobility fluctuations is calculated on the base of a new

phonon mechanism. It is shown that the noise spectrum over a broad frequency range has a 1/f form. The theoretical results for many

samples agree with experimental data.

r 2006 Elsevier B.V. All rights reserved.

PACS: 72.7.+m

Keywords: Electron; Phonon; Scattering; Mobility fluctuation

1. Introduction

The mobility fluctuations of current carriers are fre-quently considered as one of the possible sources of thecurrent 1/f-noise. Despite a mass of publications in the field(see e.g. reviews [1,2]) the origin of the mobility fluctuationsis practically not analyzed in detail. The assumption—firstexpressed by Hooge [3]—that 1/f-noise is generated in thevolume of the semiconductor and is caused by mobilityfluctuation of current carriers, still lacks a theoretical basis.

The aim of this work is to reveal and study the physicalmechanisms of the formation of the electron mobilityfluctuations and compare of the obtained results withexperimental data. Here we consider homogeneous, non-degenerate semiconductors having simple, standard bandstructures with spherical isoenergetic surfaces.

e front matter r 2006 Elsevier B.V. All rights reserved.

ysb.2006.01.521

ng author. Tel./Fax: +37410 555590.

ss: smelkonyan@ysu.am (S.V. Melkonyan).

2. The main sources of mobility equilibrium fluctuations

As is well known (see, e.g. Ref. [4]), in the relaxation timeapproximation the electron mobility is defined as mn ¼

/etk/mnS, where /yS is the averaging symbol over allquantum states of the conduction band, tk is the quasi-momentum relaxation time of the electrons, mn is theeffective mass of electron. Let us present that expression ina more descriptive way for further discussions

mn ¼ mk� �¼Xk

nkmk=Xk

nk; (1)

where nk is the occupancy of a state with wave vector k inthe conduction band. The quantity mk�etk/mn can beinterpreted as the ‘‘local mobility’’ of an electron in state k.The analysis of expression (1) shows that the procedure

of averaging over the quantum states of the conductionband leads to a situation, where, the electron mobility mn

becomes a fluctuating quantity. Indeed, let us assume thatan electron at the instance of time t ¼ 0, has passed fromone quantum state to another in such a way that at least

ARTICLE IN PRESSS.V. Melkonyan et al. / Physica B 382 (2006) 65–7066

one of those transitional states is located in the conductionband. This happened as a result of random generation–re-combination processes (electron transitions of ‘‘band–band’’ or ‘‘conduction band–impurity level’’ types) orintraband scattering. Such random transitions causerandom changes in (equilibrium fluctuations of) occupancy~nk of the quantum levels of the conduction band. Theanalysis of Eq. (1) shows that any change in the electrondistribution over the energy states of the conduction band,inevitably, leads to changes of mn. This is one of the mostobvious causes of the mobility fluctuations ~mn.

The second cause of the electron mobility fluctuations isthe fluctuation of the ‘‘local mobility’’ mk (or relaxationtime tk). The cause of ~mk fluctuations is the stochasticproperties of parameters of the scattering centers, such as,for example, surface roughness, the space distribution ofthe impurity centers and different kind of defects in thevolume of semiconductor, occupancy of phonons states,etc.

In general, nk and mk are fluctuating quantities. Thestatistical relation between fluctuations of ~nk and ~mk isdetermined by the choice of the physical mechanisms oftheir origin. That is why they can be presented as sums offluctuating (marked by the symbol ‘‘�’’) and averagestatistical (marked by the symbol ‘‘-’’) components:

mk ¼ mk þ ~mk; nk ¼ nk þ ~nk.

In case of small fluctuations, when we can limit our selvesto linear approximations, from Eq. (1) we have

~mn ¼Xk

~nkmk þ nk ~mk � ~nkmn

� �,Xk

nk. (2)

Here the following transformation is used:

1Pk

nk

¼1P

k

nk þ ~nkð Þ¼

1Pk

nk

1�

Pk

~nkPk

nk

24

35.

It is important to note that with intraband scattering thequantum states of an electron prior and after the scatteringstay within the same band (conduction band). It is obvious,that with such scattering events the concentration ofelectrons n does not change, i.e. ~n ¼ 0. At the same time,it is obvious that ~nka0.

In contrary to intraband scattering, with generation-recombination processes the transitions of the electronstake place between states of different bands, e.g. conduc-tion and valence bands, or conduction band states and abound state of an impurity level (generation–recombina-tion processes via traps). It is clear, that ~nk fluctuationsoriginate at such random transitions, and as is shown byEq. (2), the mobility fluctuations ~mn originate as aconsequence; note that then ~na0. In general, a fluctuationof the electron concentration is accompanied by a mobilityfluctuation. Thus, the fluctuations of the electron concen-tration and of the mobility are interrelated quantities. Thatis why the investigation of fluctuations in the current or

conductivity is inexpedient. Such investigations are basedon the relation je ¼ snE ¼ enmnE, which does not expressthe main properties and mechanisms of the origin of theconductivity fluctuations. For a more correct study it isnecessary to consider the incorporation of the carrierdistribution function (DF), before the concepts of mobilityand conductivity are introduced.The density of electron current je can be determined via

the electron DF fk in the following way [4,5]:

jeðr; tÞ ¼ �e

4p3

ZBZ

vk f kðr; tÞdk (3)

The integration in Eq. (3) should be carried out in therange of the first Brillouin zone (BZ), r is the radius-vector.The following important circumstance should be men-

tioned separately. As follows from the definition, theelectron velocity vk in a given k state of the conductionband is a constant (not fluctuating) quantity, whereas theoccupancy, i.e. the number of electrons fk dk in that statemay undergo fluctuation. If an electron leaves or passes tosome level of the conduction band as the result of randomintraband or generation–recombination scattering, thensuch a transition will show up in a form of a DF fluctuation~f k in Eq. (3). That is why from Eq. (3) for the fluctuation ofthe electron current density we have

~jeðtÞ ¼ �e

4p3

ZBZ

vk ~f kðtÞ dk. (4)

In this simplified analysis, only the temporal fluctuationsare considered: it is assumed that in Eqs. (3) and (4) thequantity fk is coordinate-independent due to averagingover the sample volume [6].As is known [4], at weak electrical fields the DF of the

electrons can be presented in the form of a sum ofequilibrium and non-equilibrium components:

f k ¼ f 0k þ

e

mn

Etkdf 0

k

dvk. (5)

Where f 0k is the equilibrium Fermi-Dirac DF. Splitting tk,

f 0k and E in average and fluctuating components, we obtain

from Eqs. (5) and (4), for the ac fluctuation of currentdensity ( ~E ¼ 0) in the linear approximation:

~jeðtÞ ¼ �e

4p3

ZBZ

dkvk ~f0

k þe

mn

E tkd ~f

0

k

dvkþ ~tk

df0

k

dvk

" #( ): (6)

Note that fluctuation ~f0

k follows from ~nk fluctuation, as it isdescribed above. Representing the functions ~f

0

k, ~tk as thesum of symmetric ( ~f

0;s

k ðtÞ, ~tskðtÞ) and asymmetric ( ~f

0;a

k ðtÞ,~takðtÞ) components relatively to k (e.g., ~tsk ¼ ½~tk þ ~t�k�=2,~tak ¼ ½~tk � ~t�k�=2), we obtain from Eq. (6)

~jeðtÞ ¼ ~j0ðtÞ þ ~jEðtÞ,

ARTICLE IN PRESSS.V. Melkonyan et al. / Physica B 382 (2006) 65–70 67

where

~j0ðtÞ ¼ �e

4p3

ZBZ

dkvk ~f0;a

k ,

~jEðtÞ ¼ �e2

4p3mn

ZBZ

dkvkE tkd ~f

0;s

k

dvkþ ~tsk

df0

k

dvk

" #. ð7Þ

As is clearly seen, the symmetric components of the

equilibrium fluctuations ~f0;s

k , ~tsk appear in the current

fluctuations (item ~jE in Eq. (7)) only in the presence of theexternal electric field Ea0; their input into the currentfluctuations disappears at E ¼ 0.

A fluctuating current (item ~j0 in Eq. (7)) is determined by

the asymmetric component ~f0;a

k , the input of that component

does not depend on E. That is why it can be named a ‘‘pureequilibrium’’ component of the current fluctuation, which isdetermined by electrons quasi-momentum fluctuations.

The conductivity is defined as a proportionality coefficientin the j�E relation. Therefore, it is necessary to consider thesecond component ~jE in Eq. (7) for the determination of theconductivity fluctuation it is only that component that can beinterpreted as conductivity (or mobility) fluctuation.

3. Intraband fluctuations of the electron lattice mobility

Usually, in semiconductors several scattering mechan-isms are functioning at the same time. They are character-ized by relaxation times, which depend on concentrationsof ‘‘dynamic’’ (phonons) and ‘‘static’’ (ionized and neutralimpurities, vacancies, dislocations and, etc.) defects. Con-centrations of static defects and their relaxation times, ingeneral, cannot fluctuate randomly. Contrary to this,concentrations of dynamic defects, i.e. of phonons (oroccupancies of phonon modes) and therefore electron–pho-non relaxation times sk,e–ph are fluctuating quantities. Thesources of symmetric fluctuations (energy fluctuations) ofphonon occupancy are principally non-elastic electron–pho-non, phonon–phonon scattering and electrons g–r transitions(phonon–driven recombination via impurity centers). Ran-dom perturbations of phonon energy can be observed in theelectron system by observing electron–phonon interactions assymmetric fluctuations ~ns

k and ~msk.In particular, here we consider the phonon–phonon

fluctuations of the electron lattice mobility m1. Therefore inEq. (7) the quantity tk is the relaxation time of the electron-phonon scattering sk,e–ph. As at the intraband scatterings~n ¼ 0, then the second item in Eq. (7) can be represented inthe following phenomenological way

~jE ¼ ~slE ¼ en ~mlE. (8)

Then from Eqs. (7) and (8) we obtain

~mlðtÞ ¼ ml

RBZ dkðEkÞ

2 tk;e�phd ~f0;s

k

.dek þ ~tsk;e�phdf

0

k

.dek

h iRBZ dkðEkÞ

2tk;e�ph df0

k

.dek

h i .

(9)

Here ek is the electron energy, E ¼ E=E is the unit vector.

From Eq. (9) it is clearly seen that the electron mobilityfluctuations are exclusively conditioned by the symmetricfluctuations ~f

0;s

k and ~tsk;e�ph, i.e. by the energy fluctuationsof electrons and phonons, respectively.As the analysis shows, dropping the functions ~f

0;s

k ðtÞ and~tsk;e�phðtÞ in Eq. (9) depend on the mechanisms of theirorigin. In particular, here we consider equilibrium fluctua-tions formed as a result of random acoustic phonon–phonon scattering. Such fluctuations are well studied inRef. [6]. In our introduction of the new phonon mechanismof the origin of the electron and phonon energy fluctua-tions, the expression for the Fourier-component of theelectron DFs fluctuation ~f

0;s

k ðoÞ was obtained:

~f0;s

kj j�k1ðoÞ ¼ �ðek � ebÞ

@f0

k

@ek

l0q0

3ffiffiffiffiffiffiffiffiffiffiffiffiffiffii2poBp tanh qm

ffiffiffiffiffiioB

r !, (10)

where

l0 ¼ 3�4pkBTq3

0ð1þ xÞ� � Z

qj j�q0

dq_oqGsq,

x ¼ 6 kT=q0

� �3eF=kBT

Z k1=kT

0

dxx4ðx2 � eb=kBTÞe�x2

,

eb ¼ _2�2mn

� � Z k1

0

dkk4e�ðk=kTÞ2

Z k1

0

dkk2e�ðk=kTÞ2

,

kT is the wave vector of the thermal electrons, F is theFermi level, oq ¼ v0q, q and v0 are the wave vector andvelocity of the longitudinal acoustic (LA) phonons,correspondingly, B is the diffusion coefficient of theabsolute value of the quasi-momentum of LA-phonons inthe momentum space at q ¼ q0; k1, q0 and qm are constantscited in the Ref. [6], Gs

q is the symmetric component of thefluctuation of the DF of LA-phonons ~g0

q, which originate atthe initial moment of time t ¼ 0 as a result of randomphonon–phonon scattering (Gq ¼ ~g0

qðt ¼ 0Þ, for more de-tails see Ref. [6]).The numerical estimations show, that the condition

k14kT generally is satisfied in the temperature rangeT�300K and below for macroscopic semiconductors withdimensions L410 mm. In that case, it is possible to make asimplification in the calculations by replacing the k1 byinfinity. Then, for the coefficients eb and x we obtaineb ¼ 3kBT/2, x ¼ 9p3n

�q30. Along with that usually x51, in

the non-degenerate semiconductors. So, we can ignore x incomparison with unity.The first and second items in the nominators of the right-

hand side of Eq. (9) describe the mobility fluctuationsconditioned by the fluctuations of electrons’ distribution byenergy and ‘‘local mobility’’ fluctuations, correspondingly.For simplification of the mathematical calculations theinput of the fluctuations of the ‘‘local mobility’’ is notanalyzed here. We confine ourselves by consideration of

the item containing ~f0;s

k component. Thus, neglecting the

second term in the nominator of Eq. (9) containing ~tsk;e�ph,and using Eq. (10) and the well-known dependencefor tk;e�ph [5], from Eq. (9) for the Fourier-component of

ARTICLE IN PRESSS.V. Melkonyan et al. / Physica B 382 (2006) 65–7068

the phonon fluctuations of the lattice mobility of theelectrons ~ml;ph�phðoÞ, we obtain

~ml;ph�phðoÞ ¼ml

8pkBTq20

ffiffiffiffiffiffiffiffiffiffiffiffiffiffii2poBp tanh

� qm

ffiffiffiffiffiioB

r !Zqj j�q0

dq_oqGsq. ð11Þ

This expression describes how the frequency dependence ofthe ~ml;ph�phðoÞ fluctuation is determined by the source Gs

q.Such single fluctuations originate not only at the initialmoment of time t ¼ 0 but at other moments of time as well.As is known [7], in such cases it is necessary to construct anew random quasi-continuous process Gs

qðtÞ as a super-position of the single events. So, for generalizing Eq. (11) itis necessary to replace the source Gs

q by new a functionGs

qðoÞ.In the further calculations the well-known simplified

expression for the spectral density of the phonon–phononfluctuations sources Gs

q is used SG ¼ 4ð2pÞ3g0qð1þ g0

qÞdðq�

q1Þ.tph�phðqÞV [8], where V is the volume of the sample, g0

q

is the equilibrium DF of LA-phonons.Using Eq. (11) for the spectrum of the phonon

fluctuations of the electron mobility we obtain

Sm;ph�phðoÞ ¼pðmlÞ

2

ð‘ þ 3ÞoBVq0tph�phðq0Þtanh qm

ffiffiffiffiffiioB

r !2

.

(12)

Here the exponent ‘ in the relation tph�ph�q�‘ charac-terizes the mechanism of phonon–phonon scattering [6,9].

As it is seen from Eq. (12) the frequency dependence ofthe spectrum Sm,ph–ph(o) is determined by the tangent-hyperbolic function with characteristic frequency o0 ¼

B=q2m. The spectrum is white (Sm,ph–ph ¼ const.) in the

frequency range ooo0. In the frequency range o4o0,where |tanh(y)|2E1, the spectrum Sm,ph–ph(o)N1/o. It isworth to mention that the law 1/o (or 1/f) applies in arestricted frequency range o0oooo1. The value of theupper frequency limit is o1 ¼ 1/t0, where t0 is the time ofestablishment of the dynamic equilibrium between thelonger-wavelength LA-phonons and the electrons (formore details see Ref. [6]). In the frequency rangeo0oooo1, we obtain from expression (12), for Hooge’scoefficient

al;ph�ph ¼n

2ð‘ þ 3ÞBq0tph�phðq0Þ. (13)

4. Comparison of the theoretical results with experimental

data

Use of Eq. (13) for the numerical determination of thecoefficient al,ph–ph and for determining the dependence ofal,ph–ph on the concentration of electrons n and temperatureT requires that we know which mechanism of phonon-

phonon scattering (i.e. value of tph�ph) and whichmechanism of phonon diffusion (i.e. value of B) operates.The main processes of relaxation of phonons are studied

in detail, e.g. in Ref. [9], where Eq. (14) is obtained (in adifferent notation). Eq. (14) holds, in particular, in thetemperature range lower than the Debije temperature, forthe relaxation time of the LA-phonons with normalphonon–phonon scattering. In case of a simple cubic lattice

t�1ph�phðqÞ ¼192

5

kBTg2Gprv0

q4. (14)

Here gG is the Gruneisen parameter of thermal expansion,r is the crystal density.To determine the diffusion coefficient of the absolute

value of the phonon quasi-momentum in q-space, it ispossible to make a simplification and relay on theanalogous case of electrons (Ref. [5]). It is possible toshow that in the case when the phonon diffusion isdetermined by the phonon–electron scattering

B ¼ q2�2tph�eðqÞ (15)

where tph�e is the relaxation time of the single phonon–electron scattering.Expression (13) can be presented in the following way:

al;ph�ph ¼6

35p6kBT

edefgG

� �2ffiffiffiffiffiffiffiffiffiffiffiffiffi2kBT

pmnv20

s(16)

where edef is the deformation potential constant [4,8]. Herewe used Eqs. (14), (15), the definition of the parameter q0from Ref. [6] and the well-known expression of tph�eðqÞ[4,5].For the comparison of the theoretical results with

experimental data, the numerical computations of thecoefficients al,ph–ph were carried out on the base of Eq. (16)(at averaged value edefffi5 eV, m0 ¼ 9.1� 10�28 g is the freeelectron mass, kB ¼ 8.63� 10�5 eV/K, deg is the Boltz-mann constant) for several semiconductors: Ge, Si, InSband GaAs with the following parameters:

Semiconductors

gG[10]

r (g/cm3)[11]

M0/m0,[10]

v0(103m/s),

[11]a

Ge

0.76 5.32 0.12 4.87 Si 0.45 2.33 0.26 8.43 InSb 0.55 5.77 0.014 [16] 3.4 GaAs 1 5.32 0.043 4.73

aHere we used data for /100S crystalline direction.

In Table 1, the calculated results are presented togetherwith experimental values of the Hooge’s coefficient. Herewe used data from reviews of L. Vandamme and Sh.Kogan only for mono-crystalline n-type semiconductors[12,13]. For additional reference on alpha-values for otherparticular cases, for example, see also Refs. [14,15].As it is seen from Table 1 the maximal deviation of the

theoretical values from the experimental data does not

ARTICLE IN PRESS

Table 1

a-values and characteristic frequency f0 for single crystal semiconductors

Semiconductors T (K) n (cm�3) or r0 (O cm) V (cm3) Experimental aH Theoretical al,ph–ph Theoretical f0 (HZ)

n-Ge 300 8� 1014 2.1� 10�3 1� 10�3 [13]

1.5� 1014 1.8� 10�3 3.5� 10�3 [13] 1.8� 10�3 10�3–10�2

1.3� 1014 1.8� 10�3 1.8� 10�4 [13]

5� 1013 1.8� 10�5 6� 10�4 [13]

0.1 — 2� 10�3 [12]

3.5 — 4� 10�3 [12]

n-Si [12] 77–400 1013 — 3� 10�5–4� 10�4

77–300 2� 103 — 2� 10�4–10�3 2.6� 10�3 (T ¼ 300K) 10�5–10�4 (T ¼ 300K)

300 7 — 10�5–3� 10�3

n-InSb 76–77 1� 1014 2.1� 10�4 4� 10�4 [12,13]

1� 1014 1.1� 10�3 3.4� 10�5 [13] 1.3� 10�4 10�8–10�7

1.6� 1014 3.2� 10�5 1� 10�3 [12,13]

1.6� 1014 1.1� 10�4 1.6� 10�3 [13]]

n-GaAs 295 2.3� 1016 8� 10�4 6� 10�3 [12,13] 5.4� 10�3 10�11–10�10

1000 / T, K

310-4

α H

Sample - 1

Sample - 2

5 7

10-3

10-2

9 11 13

theory

Fig. 1. Dependence of Hooge’s coefficient on temperature T.

S.V. Melkonyan et al. / Physica B 382 (2006) 65–70 69

exceed one order of the magnitude for 4 types ofsemiconductor materials, (al,ph–ph/aH)max�10. We shallnot discuss and analyze the causes of the deviations eachseparate case. We shall mention the most significant ofthem, which were not considered in the derivation ofEq. (16): that is the anisotropy of the semiconductor crystalparameters (gG, edef, v0), the input of semiconductor surfacescattering processes, and the presence of shallow and/ordeep impurity scattering centers. On the other hand, it iswell known [12,13], that in samples with the same electro-physical parameters deviations among the Hooge’s coeffi-cients can be up to one order of magnitude. Therefore, itcan be concluded that the theoretical results do agree withthe experimental data quite well.

It is worth to mention, that Eq. (16) shows that thecoefficient al,ph–ph does not depend on the electronconcentration. This property is connected with the use ofEq. (14), when tph�ph�q�‘ with ‘ ¼ 4. If another (acoustic)phonon-phonon scattering mechanism works with ‘a4(e.g. when ‘ ¼ 2; 3, etc., [9]), then the coefficient al,ph–ph nolonger is constant, but it becomes a quantity weaklydependent on the electron concentration.

In Fig. 1, the theoretical curve of al,ph–ph(T) andexperimental points of aH from Ref. [16] for n-Si arepresented. It is clear that the theoretical curve agrees wellwith the experimental data (especially with curve for thesample 1) in the temperature range from 150 to 330K. Inthe range of lower temperatures the deviation is only afactor two. Along with that it should be mentioned that inthe numerical calculations we did not consider thetemperature dependences of the Gruneisen parameter gG(Gruneisen approximation), phonon velocity v0, anddeformation potential edef; they all were supposed to beconstant.

One of the most important parameters in this theoryis the numerical value of the characteristic frequencyf0 ¼ o0/2p, since below that frequency a saturation of the1/f spectrum takes place. Following from the definition of

the characteristic quantity qm [6] and using Eqs. (14) and(15) for f0 we obtain

f 0 ¼e2def nq0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffimn=2p

p9rðkBTÞ3=2 102=3 � 1

h i2 . (17)

For carrying out numerical estimation of the frequencyf0 let us mention the following. Expressions (16) and (17) inprinciple are valid only for crystals with ideal structure,because we used here the results obtained in Ref. [6]. Formaximal simplification in Ref. [6] the case of an idealsemiconductor was considered, where the scattering isdetermined exclusively by electron–phonon, phonon–elec-tron and phonon–phonon scattering. Actually in Ref. [6],the case of an isotropic intrinsic semiconductor wasstudied. That is why for carrying out numerical estimationsof the frequency f0 instead of n we must use here theintrinsic concentration of electrons ni. The interestingresults of the numerical calculations are presented in theTable 1.

ARTICLE IN PRESSS.V. Melkonyan et al. / Physica B 382 (2006) 65–7070

It is important to note, that in contrast with f0 thequantity al,ph–ph does not depend on the electron concen-tration n. Thus, formula (16) in contrast with Eq. (17) isapplicable not only for intrinsic semiconductors but fordoped semiconductors as well.

The results of preliminary comparison of the theoreticaland experimental data (Table 1) show that in futureit is necessary, by developing the theory, to considerthe case of a semiconductor with different static defectsand analyze the input of the ‘‘local mobility’’ fluctua-tions.

We also want point out that Musha and Tacano [17]have studied interesting problems related with equilibriumfluctuations in oscillator system. Their work is focused onsemiconductors, and their experimental data likely can beused to evaluate the role of optical phonons in thegeneration of 1/f noise.

Acknowledgment

The authors are very thankful to Prof. F.N. Hooge forvery valuable scientific discussions, critical remarks andconsiderable help. The authors are also very thankful toProf. L. Vandamme for providing data of well-selectedexperimental alpha-values for comparison with the theory.

References

[1] F.N. Hooge, T.G.M. Kleinpenning, L.K.J. Vandamme, Rep. Prog.

Phys. 44 (1981) 479.

[2] F.N. Hooge, IEEE Trans. Electron Devices 41 (1994) 1926.

[3] F.N. Hooge, Phys. Lett. 29A (1969) 139.

[4] R.A. Smith, Semiconductors, Second ed, Cambridge University

Press, 1978.

[5] V.F. Gantmaher, I.B. Levinson, Current Carriers Scattering in

Metals and Semiconductors, Moscow, Nauka, 1986 (in Russian).

[6] S.V. Melkonyan, et al., Physica B 357 (2005) 398.

[7] A. Van der Ziel, Fluctuation Phenomena in Semiconductors, Butter-

worths Scientific Publications, London, 1959.

[8] I.M. Dykman, P.M. Tomchuk, Phenomena of Transport and Fluctua-

tions in Semiconductors, Kiev, Naukova Dumka, 1981 (in Russian).

[9] B.M. Magilevski, A.F. Chudnovski, Thermal Capacity of Semicon-

ductors, Moskow, Nauka, 1972 (in Russian).

[10] P.I. Baranski, V.P. Klochkov, I.V. Potikevich, Semiconductor

Electronics Handbook, Kiev, Naukova Dumka, 1975 (in Russian).

[11] M. Levishtein, S. Rumyantsev, M. Shur, Semiconductors Parameters,

World Scientific Publ., 1996.

[12] L.K.J. Vandamme, In: M. Savelli, G. Lecoy, J.-P. Nougier, (Eds.),

Proceedings of the Seventh International Conference on Noise in

Physical Systems and 1/f noise, 1983, p. 183.

[13] Sh. Kogan, Electronic Noise and Fluctuations in Solids, Cambridge

University Press, Cambridge, 1996.

[14] A. Mercha, L.K.J. Vandamme, L. Pichon, R. Carin, O. Bonnaud,

J. Appl. Phys. 90 (8) (2001) 4019.

[15] L.K.J. Vandamme, IEEE Trans. Electron Devices 36 (5) (1989) 987.

[16] L. Luo, W.F. Love, S.C. Miller, Appl. Phys. Lett. 60 (1986) 3196.

[17] T. Musha, M. Tacano, Physica A 346 (2005) 339.