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Electrical fields and transport properties of nonuniform plasmas in high magnetic fields
Journal of Magnetism and Magnetic Materials 11 (1979) 383-386 © North-Holland Publishing Company
ELECTRICAL FIELDS AND TRANSPORT PROPERTIES OF NONUNIFORM PLASMAS IN HIGH MAGNETIC FIELDS
Y.P. EMETS, L.V. GORODGA, V.F. REZTSOV and Y.M. VASETSKY Institute o f Electrodynamics Academy o f Sciences, Ukrainian SSR, Kiev, USSR Received 31 July 1978; in revised form 9 November 1978
The electrical fields and their statistical moments are defined in high magnetic fields. A plasma is considered as the conductive medium with random fluctuations and a double periodic structure of heterogeneities. The anomalous minimum of the GaAs Hall factor is explained on the basis of the obtained effective conductivity tensor.
parameter/3. The general task consists of determining the tensor be, which connects spatial average fields
The effective parameters and characteristics of electrical fields in heteogeneous media, which are typical for plasmas and semiconductors, suffer strong changes in high magnetic fields (/3 = mr > 1; 6o = cyclotron frequency of electrons and r = collision time of electrons.) [ 1 - 5 ] . In this paper we shall consider some applications of the theory o f statistical functions and the theory of boundary-value problems for the investigation of the characteristics of heterogeneous media in high magnetic fields. We shall pay more attention to the not well-studied question of the local electrical fields in such media and their statistical characteristics. (1) Let us consider a conductive two-dimensional medium with statistically homogeneous and isotropic distribution o f micro regions of different conductivity which are oriented along the magnetic field. We suppose that the dimensions o f heterogeneities are considerably larger than a free-path length. The distribution of current and electrical field intensity in such a medium are described by equations div j = 0,
rot • = 0,
j = be,
(q,
o-
(j:~ = Oe(e)
(2)
and in defining statistical moments of electrical fields. An application of the methods of statistical functions theory to the fields e a n d L which have axial s y m metry [7] permits a determination of the relation between tensor components Pe and qe. One of the possible relations in the case of a two-dimensional, twophase medium with conductivities ol, tr2 and relative volume concentrations cl, c2, respectively can be written as 2/3(1 + 0 2) (c2 - cl)Pe A + 2(1 + f32)qe =/~(1 - A2)(ox + 0 2 ) , = (al - a2)/(al + a2).
(3)
Eq. (3) was obtained from Ohm's law, eq. (2) after determination of statistical moments. For this purpose we used the standard procedure of the calcula. tion of moments with the aid of Green's function. This relation together with additional expression obtained by Dykhne [2]
,
(1)
4/3(1 +/32)(pe2 +qe2) - 4(1 +/32)(o I + o2)qe
where p = o/(1 +/~2), q = o/3/(1 +/32) _ are components of the conductivity tensor defined by conductivity = o in the absence of a magnetic field and
+/3(1 - A2)(Ol + o2) 2 = 0 permits a definition of the effective parameters for 383
(4)
Y.P. Emets et al. /Electrical fields of nonuniform plasmas
384
arbitrary concentrations of phases and conductivity fluctuations. Expressions (3) and (4) correspond to the solution obtained in [6] by a self-consistent determination of the local fields. For the average electrical fields values in the heterogeneous medium phases (e) 1 and (e>2 we obtained formulae
2.0
I,l I<~'>1
L5
C1 = 0.35
(ei)l = {E1 + P e - ( P ) + 13(qe_-(q >)']'-I
cl(ol a2)
_Sik
-
_~qe - (q~)s 13(.Pe-_-(P)) CI(OI -- 02)
/ 6ik) ( e k ) ' -LO
(ei)2= {I 1
-0.5
0
0.5
to
Pe-(P)+13(qe-(q)7~ Fig. 2. Relative average electrical field in one phase vs. conductivity fluctuations at c 1 = 0.35.
qe -- (q) -- 13(Pe -- (P))eik} (ek>
(5)
:
/
where 5ik = unit tensor, eik = unit antisymmetrical tensor. Analogous expressions can be obtained for the average current dencities (J)l and (J>2. In order to find the average values (e)l, (e)2 and < J ) l , (J)2 we need to know only the effective parameters Pe and qe. Expression (5) was obtained with the aid of determined statistical moments for f-distribution function of inhomogeneities.
2.0 I':K>I 1.,5"
1.0
o.s /
~oj~
zx=o.3
# 0
0.25"
O.S
0.75
¢, 1.0
Fig. 1. Relative average electrical field in one phase vs. volume concentration of this phase at A = 0.3.
To illustrate the obtained results, the modulus of the average field in one phase is plotted against the concentration of inhomogeneities (fig. 1), for comparatively small (A = 0.3) conductivity fluctuations. As can be seen from the curves, the average electrical field in the phase increases if cl > 0 . 5 and decreases if c~ < 0.5. This means that the electrical field is "dislodged" from the phase with concentration c < 0.5 into another phase with a larger one. The sharpest change in I(e)l I occurs near c 1 = 0.5. The characteristic changes by jump from zero to 2 I(e>l when 13~ (thick curve). The effect of conductivity fluctuations in different magnetic fields is shown in fig. 2. If the magnetic field is absent, the magnitude of the average electrical field I(e)ll in phase with the smaller concentration (cl = 0.35) will be smoothly changed from 21(e)l to zero when conductivity Ol is changed from zero to o 1 ~ oo. The characteristic shape is sharply changed in the high magnetic field. The average electrical field intensity is smaller everywhere than 2 I(e)l. If13 ~ oo the average electrical field intensity goes to zero for any conductivity fluctuations except A ~ 0. All values I(e)l I lie between curves for/3 = 0 and 13-+ oo (thick lines). Accordingly for any conductivity fluctuations the average electrical field I(e)21 goes up with increasing t3. (2) It follows from the above analysis that large electric field fluctuations occur near the percolation
Y.P. Emets et al. /Electrical fields o f nonuniform plasmas
385
threshold (c 1 = c2 = 0.5). To explain such a state of the nonuniform system we shall consider its electrical characteristics in detail for the inhomogeneous medium with double periodical distribution of inhomogeneities. Such a medium also allows the analysis of the local structure of fields. Using for boundary values the existing ohmic contacts of cells having different conductivities it is possible to obtain exact expressions for current densities. The magnitudes of local current densities in two adjacent cells are connected by the relations of symmetry ~2('Z) = (0.2/0.1) 1]2 (--1)(--1/2--6)]1(g)
,
g =
(7)
(]')2 = (02/0"1)112(--1)c+(])1 '
where 5 = lr -1 arc tan [(o231 - 01fl2)/(0.1 - 02)], 12/= if--1 arc t a n [(02/31 - 01fl2)/(01 ÷ 0"2)]. Using the available symmetry we can determine the average values of the electrical field in the cells and the effective parameters. For any one phase we have formulae which connect the average field values in this phase with the average field in the whole system (/)1 = 2 [1 + ( o 2 / o l ) l / 2 e i + m ] ( / ) ,
(8)
= 2(t72/01) 1/2 [ ( 0 2 / O l ) 1/2 + e i n a ] ( e ) .
Analogous expressions can be obtained for the other phases. The exact values of eq. (8) coincide with (e)l, (e)2 given by formulae (5) when cl = c2 = 0.5. It can been seen from the formulae, that if the conductivity of any phase decreases, the current dislodges and hence the field intensity increases accordingly; however, its value will not exceed twice the value of the average field. The magnitude of field fluctuations may be derived from a consideration of the field dispersion in separate cells and in the system as a whole. When/3 = const the field dispersion in the system is determined from the expression
D =
(le12) - I ( e ) l 2 I(e)l 2
~ - x / 1 - A
fl=l°z
p=+o
x + iy. (6)
Here,T(f) is the conjugate value of the complex current in conjugate points z. The same type of relations of symmetry are varied for averaged values in every cells
(e)l
.
2
(9)
- l.o
-0.5
o
A I.o
0.5
Fig. 3. Square root from relative dispersion of electrical field in one phase vs. conductivity fluctuations at Cl = c2 = 0.5.
The formula for the dispersion in a separate phase is D I - (lelt2)-I(e)tl2 = I(e)l 2 / ~ 1 / 1 = v,
+
- A i- +
+,,, -
2 ( 1 - A)
]
_
(10) Values for D 1 and D2 in strong magnetic fields change proportionally with/3A, which is not small even for small conductivity fluctuations. The dispersion growth in the limiting cases, A + +1, as can be seen from fig. 3, is due to one cell while the fluctua. tions of parameters in the second cell can be negligible. (3) Up to here, we have considered media with twodimensional inhomogeneities which are typical for films, composite materials and for plasma media in some cases. In the three-dimensional case, the conductivity along the magnetic field depends on the parameter/3. For example, the longitudinal conductivity value in a strong magnetic field for the case of nonconductive spherical inhomogeneities is given by Oeaa = o1(1 - 2c2/31//r),
c2/31 ~ 0 . 2 - 0 . 3 .
(11)
In the case of circular cylindrical inhomogeneities oriented along the magnetic field we have 0e33 = Ol(1 -- C 2 ) .
(12)
Here, ol,/31 are the conductivity and Hall parameter for the matrix.
Y.P. Emets et aL/ Electricalfields of nonuniform plasmas
t.t6.~
Results of comparison of theory and experiment are presented on fig. 4, where the dotted line corresponds to experimental data of [8]; curve 1 is the dependence of the Hall factor rn(H) for a homogeneous semiconductor; curves 2 - 4 are theoretical dependences of the Hall factor rile (/4) for inhomogeneous semiconductor calculated according to formula (13) when concentrations are 1.5% (curve 2), 2;2% (curve 3), 3.5% (curve 4). For comparison, the calculated results of the theoretical model for a film with clusters of cylindrical shape, carried out in [8] (curve 5), are shown in this figure. Comparing these results one can come to the conclusion that the observed effect of an anomalous minimum in the GaAs Hall factor in a finite magnetic field is apparently connected with the presence of a conductive Ga spherical cluster with a volume concentration of metallic phase of about 2.2%.
the
1.12"~
386
I
1.08" ~
Fig. 4. GaAs Hall factor vs. intensity of magnetic field.
References From the expression obtained for the effective conductivity tensor components one can easily derived the analytical formulae for mobility, Hall constant and other electrophysical characteristics of semiconductors. The inmonotonous character of the Hall factor behavior discovered recently in inhomogeneous GaAs films [8] can be explained by using formula ( 2c2 ) rile = r u 1 -n(X,)( 1 +/31:) , where
n (x') is the polarization factor for ellipsoid
x 2 +y2+(l+/3~)z 2=1 along axis x.
(13)
[1] C.J. Herring, J. Appl. Phys. 31 (1961) 1939. [2] A.M. Dykhne, Zh. Ehksp. Teor. Fiz. 59 (1970) 641. [3] M. Martinez-Sanchez, R. De Saro and J.F. Louis, in: Sixth Int. Conf. MHD Power Generation, Vol. 4 (Washington, 1975) p. 247. [4] N.R. Hower and M. Mitchner, in: Sixth Int. Conf. MHD Power Generation, Vol. 4 (Washington, 1975) p. 217. [5] Y.P. Emets, Zh. Techn. Fiz. 44 (1974) 916. [6] S.E. Shama, M. Martinez-Sanchez and J.F. Louis, in: 16th Symp. Engineering Aspects of MHD (Pittsburg, 1977) p. VII, 1.1. [ 7] G.K. Batchelor, The Theory of Homogeneous Turbulence, (Cambridge, 1953). [8] C.M. Wolf, C.E. Stillman, D.L. Spears, D.E. Hill and F.V. Williams, J. Appl. Phys. 44 (1973) 732.
ELECTRICAL FIELDS AND TRANSPORT PROPERTIES OF NONUNIFORM PLASMAS IN HIGH MAGNETIC FIELDS
Y.P. EMETS, L.V. GORODGA, V.F. REZTSOV and Y.M. VASETSKY Institute o f Electrodynamics Academy o f Sciences, Ukrainian SSR, Kiev, USSR Received 31 July 1978; in revised form 9 November 1978
The electrical fields and their statistical moments are defined in high magnetic fields. A plasma is considered as the conductive medium with random fluctuations and a double periodic structure of heterogeneities. The anomalous minimum of the GaAs Hall factor is explained on the basis of the obtained effective conductivity tensor.
parameter/3. The general task consists of determining the tensor be, which connects spatial average fields
The effective parameters and characteristics of electrical fields in heteogeneous media, which are typical for plasmas and semiconductors, suffer strong changes in high magnetic fields (/3 = mr > 1; 6o = cyclotron frequency of electrons and r = collision time of electrons.) [ 1 - 5 ] . In this paper we shall consider some applications of the theory o f statistical functions and the theory of boundary-value problems for the investigation of the characteristics of heterogeneous media in high magnetic fields. We shall pay more attention to the not well-studied question of the local electrical fields in such media and their statistical characteristics. (1) Let us consider a conductive two-dimensional medium with statistically homogeneous and isotropic distribution o f micro regions of different conductivity which are oriented along the magnetic field. We suppose that the dimensions o f heterogeneities are considerably larger than a free-path length. The distribution of current and electrical field intensity in such a medium are described by equations div j = 0,
rot • = 0,
j = be,
(q,
o-
(j:~ = Oe(e)
(2)
and in defining statistical moments of electrical fields. An application of the methods of statistical functions theory to the fields e a n d L which have axial s y m metry [7] permits a determination of the relation between tensor components Pe and qe. One of the possible relations in the case of a two-dimensional, twophase medium with conductivities ol, tr2 and relative volume concentrations cl, c2, respectively can be written as 2/3(1 + 0 2) (c2 - cl)Pe A + 2(1 + f32)qe =/~(1 - A2)(ox + 0 2 ) , = (al - a2)/(al + a2).
(3)
Eq. (3) was obtained from Ohm's law, eq. (2) after determination of statistical moments. For this purpose we used the standard procedure of the calcula. tion of moments with the aid of Green's function. This relation together with additional expression obtained by Dykhne [2]
,
(1)
4/3(1 +/32)(pe2 +qe2) - 4(1 +/32)(o I + o2)qe
where p = o/(1 +/~2), q = o/3/(1 +/32) _ are components of the conductivity tensor defined by conductivity = o in the absence of a magnetic field and
+/3(1 - A2)(Ol + o2) 2 = 0 permits a definition of the effective parameters for 383
(4)
Y.P. Emets et al. /Electrical fields of nonuniform plasmas
384
arbitrary concentrations of phases and conductivity fluctuations. Expressions (3) and (4) correspond to the solution obtained in [6] by a self-consistent determination of the local fields. For the average electrical fields values in the heterogeneous medium phases (e) 1 and (e>2 we obtained formulae
2.0
I
L5
C1 = 0.35
(ei)l = {E1 + P e - ( P ) + 13(qe_-(q >)']'-I
cl(ol a2)
_Sik
-
_~qe - (q~)s 13(.Pe-_-(P)) CI(OI -- 02)
/ 6ik) ( e k ) ' -LO
(ei)2= {I 1
-0.5
0
0.5
to
Pe-(P)+13(qe-(q)7~ Fig. 2. Relative average electrical field in one phase vs. conductivity fluctuations at c 1 = 0.35.
qe -- (q) -- 13(Pe -- (P))eik} (ek>
(5)
:
/
where 5ik = unit tensor, eik = unit antisymmetrical tensor. Analogous expressions can be obtained for the average current dencities (J)l and (J>2. In order to find the average values (e)l, (e)2 and < J ) l , (J)2 we need to know only the effective parameters Pe and qe. Expression (5) was obtained with the aid of determined statistical moments for f-distribution function of inhomogeneities.
2.0 I':K>I 1.,5"
1.0
o.s /
~oj~
zx=o.3
# 0
0.25"
O.S
0.75
¢, 1.0
Fig. 1. Relative average electrical field in one phase vs. volume concentration of this phase at A = 0.3.
To illustrate the obtained results, the modulus of the average field in one phase is plotted against the concentration of inhomogeneities (fig. 1), for comparatively small (A = 0.3) conductivity fluctuations. As can be seen from the curves, the average electrical field in the phase increases if cl > 0 . 5 and decreases if c~ < 0.5. This means that the electrical field is "dislodged" from the phase with concentration c < 0.5 into another phase with a larger one. The sharpest change in I(e)l I occurs near c 1 = 0.5. The characteristic changes by jump from zero to 2 I(e>l when 13~ (thick curve). The effect of conductivity fluctuations in different magnetic fields is shown in fig. 2. If the magnetic field is absent, the magnitude of the average electrical field I(e)ll in phase with the smaller concentration (cl = 0.35) will be smoothly changed from 21(e)l to zero when conductivity Ol is changed from zero to o 1 ~ oo. The characteristic shape is sharply changed in the high magnetic field. The average electrical field intensity is smaller everywhere than 2 I(e)l. If13 ~ oo the average electrical field intensity goes to zero for any conductivity fluctuations except A ~ 0. All values I(e)l I lie between curves for/3 = 0 and 13-+ oo (thick lines). Accordingly for any conductivity fluctuations the average electrical field I(e)21 goes up with increasing t3. (2) It follows from the above analysis that large electric field fluctuations occur near the percolation
Y.P. Emets et al. /Electrical fields o f nonuniform plasmas
385
threshold (c 1 = c2 = 0.5). To explain such a state of the nonuniform system we shall consider its electrical characteristics in detail for the inhomogeneous medium with double periodical distribution of inhomogeneities. Such a medium also allows the analysis of the local structure of fields. Using for boundary values the existing ohmic contacts of cells having different conductivities it is possible to obtain exact expressions for current densities. The magnitudes of local current densities in two adjacent cells are connected by the relations of symmetry ~2('Z) = (0.2/0.1) 1]2 (--1)(--1/2--6)]1(g)
,
g =
(7)
(]')2 = (02/0"1)112(--1)c+(])1 '
where 5 = lr -1 arc tan [(o231 - 01fl2)/(0.1 - 02)], 12/= if--1 arc t a n [(02/31 - 01fl2)/(01 ÷ 0"2)]. Using the available symmetry we can determine the average values of the electrical field in the cells and the effective parameters. For any one phase we have formulae which connect the average field values in this phase with the average field in the whole system (/)1 = 2 [1 + ( o 2 / o l ) l / 2 e i + m ] ( / ) ,
(8)
= 2(t72/01) 1/2 [ ( 0 2 / O l ) 1/2 + e i n a ] ( e ) .
Analogous expressions can be obtained for the other phases. The exact values of eq. (8) coincide with (e)l, (e)2 given by formulae (5) when cl = c2 = 0.5. It can been seen from the formulae, that if the conductivity of any phase decreases, the current dislodges and hence the field intensity increases accordingly; however, its value will not exceed twice the value of the average field. The magnitude of field fluctuations may be derived from a consideration of the field dispersion in separate cells and in the system as a whole. When/3 = const the field dispersion in the system is determined from the expression
D =
(le12) - I ( e ) l 2 I(e)l 2
~ - x / 1 - A
fl=l°z
p=+o
x + iy. (6)
Here,T(f) is the conjugate value of the complex current in conjugate points z. The same type of relations of symmetry are varied for averaged values in every cells
(e)l
.
2
(9)
- l.o
-0.5
o
A I.o
0.5
Fig. 3. Square root from relative dispersion of electrical field in one phase vs. conductivity fluctuations at Cl = c2 = 0.5.
The formula for the dispersion in a separate phase is D I - (lelt2)-I(e)tl2 = I(e)l 2 / ~ 1 / 1 = v,
+
- A i- +
+,,, -
2 ( 1 - A)
]
_
(10) Values for D 1 and D2 in strong magnetic fields change proportionally with/3A, which is not small even for small conductivity fluctuations. The dispersion growth in the limiting cases, A + +1, as can be seen from fig. 3, is due to one cell while the fluctua. tions of parameters in the second cell can be negligible. (3) Up to here, we have considered media with twodimensional inhomogeneities which are typical for films, composite materials and for plasma media in some cases. In the three-dimensional case, the conductivity along the magnetic field depends on the parameter/3. For example, the longitudinal conductivity value in a strong magnetic field for the case of nonconductive spherical inhomogeneities is given by Oeaa = o1(1 - 2c2/31//r),
c2/31 ~ 0 . 2 - 0 . 3 .
(11)
In the case of circular cylindrical inhomogeneities oriented along the magnetic field we have 0e33 = Ol(1 -- C 2 ) .
(12)
Here, ol,/31 are the conductivity and Hall parameter for the matrix.
Y.P. Emets et aL/ Electricalfields of nonuniform plasmas
t.t6.~
Results of comparison of theory and experiment are presented on fig. 4, where the dotted line corresponds to experimental data of [8]; curve 1 is the dependence of the Hall factor rn(H) for a homogeneous semiconductor; curves 2 - 4 are theoretical dependences of the Hall factor rile (/4) for inhomogeneous semiconductor calculated according to formula (13) when concentrations are 1.5% (curve 2), 2;2% (curve 3), 3.5% (curve 4). For comparison, the calculated results of the theoretical model for a film with clusters of cylindrical shape, carried out in [8] (curve 5), are shown in this figure. Comparing these results one can come to the conclusion that the observed effect of an anomalous minimum in the GaAs Hall factor in a finite magnetic field is apparently connected with the presence of a conductive Ga spherical cluster with a volume concentration of metallic phase of about 2.2%.
the
1.12"~
386
I
1.08" ~
Fig. 4. GaAs Hall factor vs. intensity of magnetic field.
References From the expression obtained for the effective conductivity tensor components one can easily derived the analytical formulae for mobility, Hall constant and other electrophysical characteristics of semiconductors. The inmonotonous character of the Hall factor behavior discovered recently in inhomogeneous GaAs films [8] can be explained by using formula ( 2c2 ) rile = r u 1 -n(X,)( 1 +/31:) , where
n (x') is the polarization factor for ellipsoid
x 2 +y2+(l+/3~)z 2=1 along axis x.
(13)
[1] C.J. Herring, J. Appl. Phys. 31 (1961) 1939. [2] A.M. Dykhne, Zh. Ehksp. Teor. Fiz. 59 (1970) 641. [3] M. Martinez-Sanchez, R. De Saro and J.F. Louis, in: Sixth Int. Conf. MHD Power Generation, Vol. 4 (Washington, 1975) p. 247. [4] N.R. Hower and M. Mitchner, in: Sixth Int. Conf. MHD Power Generation, Vol. 4 (Washington, 1975) p. 217. [5] Y.P. Emets, Zh. Techn. Fiz. 44 (1974) 916. [6] S.E. Shama, M. Martinez-Sanchez and J.F. Louis, in: 16th Symp. Engineering Aspects of MHD (Pittsburg, 1977) p. VII, 1.1. [ 7] G.K. Batchelor, The Theory of Homogeneous Turbulence, (Cambridge, 1953). [8] C.M. Wolf, C.E. Stillman, D.L. Spears, D.E. Hill and F.V. Williams, J. Appl. Phys. 44 (1973) 732.