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Transition to paramagnetism in semiconductor plasmas
Volume
27A, number
9
PHYSICS
LETTERS
Dans les hypotheses &on&es, la mesure du P. T. E. d’alliages binaires P basse temperature (1) et de sa variation en concentration a temperature Blevee (2) permettent d’evaluer St et St. Ainsi on obtient experimentalement les valeurs de St et S: pour les alliages de nickel a partir des mesures de Greig [4] et Koster [5] (voir la figure et le tableau). Ces valeurs s’accordent avec celles des resistivites [2] et sont compatibles avec l’existence d’un &at lie virtue1 de caractbre ST- dt et d’etat resonant d: - di. Cependant les hypotheses utilisees (p? = p?“, S? = ST) sont sujettes a caution dans le nickel [6] et les resultats obtenus ne peuvent etre que semi-quantitatif s . Une etude a basse temperature sur les alliages ternaires evite de faire de telles hypotheses. En effet, de m&me qu’il a ete montre que les resistivites residuelles d’alliages ternaires verifient bien la loi: &I,
c2) = Clfl(XI)
23 September
1968
centrations ((dS/dAl)kl+, , (dS/dX2)X =c sont des combinaisons lineaires des P. T. E. Blo et S2a associes a chaque impurete dans la bande u) et les P. T. E. des alliages binaires correspondants permettent la separation des contributions S t et S: pour chaque impurete. Cette separation actuellement entreprise pour les alliages de nickel et cobalt [6] ainsi que celle des resistivites fournissent un nouveau type d’information sur la structure Blectronique des impure& dans les ferromagnetiques. Nous remercions vivement D. Greig d’avoir Porte a notre connaissance ses mesures de resistivites et pouvoir thermoelectrique avant publication.
R&fhences 1. I.A.Campbell,
= c2f202)
A.Fert et A.R. Pomeroy, Phil. Mag. 15 (1967) 977. P.Leonard. M.C.Cadeville, J.Durand et F.Gautier Phys. Rev. Letters. a paratire. D. K. C. Mac Donald. Thermoelectricity: An introduction to the principles (John Wiley and Sons. Inc., New York. 1962) p. 114. D.Greig, Proc. Phys. Sot.. a parilrtre. W. Koster et W. Gmohling. 2. Metalkkde 52 (1961) 713. M. C, Cadeville et J. Dursnd. communication privee.
2.
oti cl et c2 sont les concentrations d’impuretes 1 et 2 et Al = l/X2 = c2/cl [2], le P. T. E. de ces alliages peut s’ecrire a basse temperature dans un modele I deux bandes:
3.
4. 5.
S(c1, c2) = g1@1) = 82(X2) . Ses variations dans la limite des faibles con-
6.
*****
TRANSITION
TO
PARAMAGNETISM
IN SEMICONDUCTOR
PLASMAS
8. HOLTER Institute
of Physics.
University
of Oslo,
Oslo,
Norway
and R. R . JOHNSON Boeing
Scientific
Research
Laboratories.
Received
Seattle,
Washington,
USA
19 June 1968
The internal axial magnetic field is calculated as a function of the amplitude of the helical It is shown that for large enough amplitudes the internal field changes direction.
oscillations.
In the positive column [l] and in semiconductor plasmas [2], experiments show that the originally diamagnetic plasma columns become paramagnetic as helical instabilities attain finite amplitudes. In this paper we derive an expression for the internal magnetic field created by finite amplitude helical oscillations in semiconductor plasmas. We investigate a long cylindrical semiconductor bar of radius R with uniformly distributed thermal 642
PHYSICS
Volume 27A. number 9
LETTERS
23 September 1968
carriers of density no and to for electrons and holes respectively, and with an injected plasma density ni. The axial electric field 1s E,, and the axial applied magnetic field is B. We consider finite amplitude helical oscillations in the density ni and the potential U [3]. The equation for the zero order radial density distribution is
h,(d+y h,(y) - 1 + cvO1n (-~+r
) + ($,Y)~2
0
+/3;j
Y’ ln
6
_f
‘y’
(1
+ -&)
fW)W)
h,(r')
dr’ +
(1)
y1 -
(;)
0
dr’
-
J/v0
,I
y
Art)
mdr’
dg(r’) dr’ = 0
,
0
where we have assumed fl0) = g(0) = 0, and put h,(O) = 1, and where $ = $(b+l)m PZ = (l+y)(b+l)
sin6
NI VI N,(.V,+Vh) ;
5 pe(ve+Vh)
(1 -Y)tlo]
;
PO-no ‘lo = (b+!;;+y)
No
*
Ve and Vh are the temperatures, b = pe/ph the mobility ratio, and y = 1-1/FOBS. No is the injected plasma density on the axis, N1 and U1 are amplitude constants of the oscl*flations, f(r) and g(r) represent the radial dependencies of the density and potential oscillations, respectively. The axial internal magnetic field is created by the time averaged azimuthal part of the helically shaped current. For the internal magnetic field Bi on the axis we obtain Bi
=
,ep,N,( Ve + vh)/..lh
F
L
-I-qOln(l+l,y)
+$lR (1 -g);gdr],
(2)
0
where p, is the permeability of the material. For sufficiently large amplitudes of the helical mode this term may become large enough to make the plasma column paramagnetic. To compute Bi numerically we have to solve eq. (1) for h . To do this we specify b, y, No/p,, No ino, 6, rc/, and the functions Ar) and g(r). The quantity 8 can not be selected independent of the others since it is related to $ through 6. We can determine 6 as a function of @ from the condition for marginal stability of the finite amplitude oscillations [4]. With b >> 1 an magnetic fields for which 1 >> y ,> 10w3, we have + << 6. Further, since the magnetic field does not depend explicitly on rc/, we shall put Q = 0 in eq. (1). We approximate the functions f(r) and g(r) by [4,5] f(r) = J1(/31~), g(r) = = J1@ly)/(h,(r) +Y), where PlR is the first zero of the first order Bessel function Jl(pl~-). For small 4 the density profile h,(r) is shifted below its 4 = 0 value. For larger @I it changes in a manner similar to that of the positive column [3]. This will then for fixed injection level result in a change in the conductance R u = neRa
CchPo+ pen0 + (pe + ~h)N,2Rm2 _f y’hodY’
C
0
I9
when $Jincreases. We have computed the internal field for situations corr sponding to experiments in p-type InSb. We have used Ve = Vh = 6.6 X low3 eV, p = 4 X 1014 part/cm 8 R -- 4 X 10-S cm. We have selected values The first value of Y and b according to the relation yPb = (phB)2 for (GhB) = 6 X 10m4and 3 X 10-4. corresponds to the zero field value used in Ancher-Johnsons [2] measurements of the internal field. The results are shown in fig. 1. The paramagnetic field exceeds the diamagnetic field by several orders of magnitude for moderate $. This is in accordance with the experimental measurements which, however, yields internal fields about an order of magnitude higher than those calculated here. To obtain 643
Volume
2’7A, number
9
PHYSICS
LETTERS
23 September
1968
2.5 i 2 ‘3 m 2.0
1.5
1.0
0.5
0.1 0 - 0.1
I
1.0
I
I
1
2.0
3.0
4.0.10 O(MHOS)
The injection level No/PO = 1.0. The last column of Fig. 1. The internal magnetic field as a function of conductance. the inserted table gives the diamagnetic field in gauss for $ = 0. The broken lines connects points on the curves with the same @-values.
quantitative comparisons the value of the zero field electron mobility in p-type InSb is needed, together with the variation of the temperatures and mobilities with the applied electric field.
References 1. R.R.Johnson. Proc. Sixth Intern. Conf. on Ionization phenomena p. 413. 2. B.Ancker-Johnson, Phys. Rev. 134 (1964) A1465. 3. 0.Holter and R.R. Johnson, Phys. Fluids 8 (1965) 333. 4. 0.Holter and R.R.Johnson, to be published. 5. 0.Holter, Phys. Rev. 129 (1963) 2548.
*****
644
in gases
(S.E.R.M.A.,
Paris
1964) Vol. 1,
27A, number
9
PHYSICS
LETTERS
Dans les hypotheses &on&es, la mesure du P. T. E. d’alliages binaires P basse temperature (1) et de sa variation en concentration a temperature Blevee (2) permettent d’evaluer St et St. Ainsi on obtient experimentalement les valeurs de St et S: pour les alliages de nickel a partir des mesures de Greig [4] et Koster [5] (voir la figure et le tableau). Ces valeurs s’accordent avec celles des resistivites [2] et sont compatibles avec l’existence d’un &at lie virtue1 de caractbre ST- dt et d’etat resonant d: - di. Cependant les hypotheses utilisees (p? = p?“, S? = ST) sont sujettes a caution dans le nickel [6] et les resultats obtenus ne peuvent etre que semi-quantitatif s . Une etude a basse temperature sur les alliages ternaires evite de faire de telles hypotheses. En effet, de m&me qu’il a ete montre que les resistivites residuelles d’alliages ternaires verifient bien la loi: &I,
c2) = Clfl(XI)
23 September
1968
centrations ((dS/dAl)kl+, , (dS/dX2)X =c sont des combinaisons lineaires des P. T. E. Blo et S2a associes a chaque impurete dans la bande u) et les P. T. E. des alliages binaires correspondants permettent la separation des contributions S t et S: pour chaque impurete. Cette separation actuellement entreprise pour les alliages de nickel et cobalt [6] ainsi que celle des resistivites fournissent un nouveau type d’information sur la structure Blectronique des impure& dans les ferromagnetiques. Nous remercions vivement D. Greig d’avoir Porte a notre connaissance ses mesures de resistivites et pouvoir thermoelectrique avant publication.
R&fhences 1. I.A.Campbell,
= c2f202)
A.Fert et A.R. Pomeroy, Phil. Mag. 15 (1967) 977. P.Leonard. M.C.Cadeville, J.Durand et F.Gautier Phys. Rev. Letters. a paratire. D. K. C. Mac Donald. Thermoelectricity: An introduction to the principles (John Wiley and Sons. Inc., New York. 1962) p. 114. D.Greig, Proc. Phys. Sot.. a parilrtre. W. Koster et W. Gmohling. 2. Metalkkde 52 (1961) 713. M. C, Cadeville et J. Dursnd. communication privee.
2.
oti cl et c2 sont les concentrations d’impuretes 1 et 2 et Al = l/X2 = c2/cl [2], le P. T. E. de ces alliages peut s’ecrire a basse temperature dans un modele I deux bandes:
3.
4. 5.
S(c1, c2) = g1@1) = 82(X2) . Ses variations dans la limite des faibles con-
6.
*****
TRANSITION
TO
PARAMAGNETISM
IN SEMICONDUCTOR
PLASMAS
8. HOLTER Institute
of Physics.
University
of Oslo,
Oslo,
Norway
and R. R . JOHNSON Boeing
Scientific
Research
Laboratories.
Received
Seattle,
Washington,
USA
19 June 1968
The internal axial magnetic field is calculated as a function of the amplitude of the helical It is shown that for large enough amplitudes the internal field changes direction.
oscillations.
In the positive column [l] and in semiconductor plasmas [2], experiments show that the originally diamagnetic plasma columns become paramagnetic as helical instabilities attain finite amplitudes. In this paper we derive an expression for the internal magnetic field created by finite amplitude helical oscillations in semiconductor plasmas. We investigate a long cylindrical semiconductor bar of radius R with uniformly distributed thermal 642
PHYSICS
Volume 27A. number 9
LETTERS
23 September 1968
carriers of density no and to for electrons and holes respectively, and with an injected plasma density ni. The axial electric field 1s E,, and the axial applied magnetic field is B. We consider finite amplitude helical oscillations in the density ni and the potential U [3]. The equation for the zero order radial density distribution is
h,(d+y h,(y) - 1 + cvO1n (-~+r
) + ($,Y)~2
0
+/3;j
Y’ ln
6
_f
‘y’
(1
+ -&)
fW)W)
h,(r')
dr’ +
(1)
y1 -
(;)
0
dr’
-
J/v0
,I
y
Art)
mdr’
dg(r’) dr’ = 0
,
0
where we have assumed fl0) = g(0) = 0, and put h,(O) = 1, and where $ = $(b+l)m PZ = (l+y)(b+l)
sin6
NI VI N,(.V,+Vh) ;
5 pe(ve+Vh)
(1 -Y)tlo]
;
PO-no ‘lo = (b+!;;+y)
No
*
Ve and Vh are the temperatures, b = pe/ph the mobility ratio, and y = 1-1/FOBS. No is the injected plasma density on the axis, N1 and U1 are amplitude constants of the oscl*flations, f(r) and g(r) represent the radial dependencies of the density and potential oscillations, respectively. The axial internal magnetic field is created by the time averaged azimuthal part of the helically shaped current. For the internal magnetic field Bi on the axis we obtain Bi
=
,ep,N,( Ve + vh)/..lh
F
L
-I-qOln(l+l,y)
+$lR (1 -g);gdr],
(2)
0
where p, is the permeability of the material. For sufficiently large amplitudes of the helical mode this term may become large enough to make the plasma column paramagnetic. To compute Bi numerically we have to solve eq. (1) for h . To do this we specify b, y, No/p,, No ino, 6, rc/, and the functions Ar) and g(r). The quantity 8 can not be selected independent of the others since it is related to $ through 6. We can determine 6 as a function of @ from the condition for marginal stability of the finite amplitude oscillations [4]. With b >> 1 an magnetic fields for which 1 >> y ,> 10w3, we have + << 6. Further, since the magnetic field does not depend explicitly on rc/, we shall put Q = 0 in eq. (1). We approximate the functions f(r) and g(r) by [4,5] f(r) = J1(/31~), g(r) = = J1@ly)/(h,(r) +Y), where PlR is the first zero of the first order Bessel function Jl(pl~-). For small 4 the density profile h,(r) is shifted below its 4 = 0 value. For larger @I it changes in a manner similar to that of the positive column [3]. This will then for fixed injection level result in a change in the conductance R u = neRa
CchPo+ pen0 + (pe + ~h)N,2Rm2 _f y’hodY’
C
0
I9
when $Jincreases. We have computed the internal field for situations corr sponding to experiments in p-type InSb. We have used Ve = Vh = 6.6 X low3 eV, p = 4 X 1014 part/cm 8 R -- 4 X 10-S cm. We have selected values The first value of Y and b according to the relation yPb = (phB)2 for (GhB) = 6 X 10m4and 3 X 10-4. corresponds to the zero field value used in Ancher-Johnsons [2] measurements of the internal field. The results are shown in fig. 1. The paramagnetic field exceeds the diamagnetic field by several orders of magnitude for moderate $. This is in accordance with the experimental measurements which, however, yields internal fields about an order of magnitude higher than those calculated here. To obtain 643
Volume
2’7A, number
9
PHYSICS
LETTERS
23 September
1968
2.5 i 2 ‘3 m 2.0
1.5
1.0
0.5
0.1 0 - 0.1
I
1.0
I
I
1
2.0
3.0
4.0.10 O(MHOS)
The injection level No/PO = 1.0. The last column of Fig. 1. The internal magnetic field as a function of conductance. the inserted table gives the diamagnetic field in gauss for $ = 0. The broken lines connects points on the curves with the same @-values.
quantitative comparisons the value of the zero field electron mobility in p-type InSb is needed, together with the variation of the temperatures and mobilities with the applied electric field.
References 1. R.R.Johnson. Proc. Sixth Intern. Conf. on Ionization phenomena p. 413. 2. B.Ancker-Johnson, Phys. Rev. 134 (1964) A1465. 3. 0.Holter and R.R. Johnson, Phys. Fluids 8 (1965) 333. 4. 0.Holter and R.R.Johnson, to be published. 5. 0.Holter, Phys. Rev. 129 (1963) 2548.
*****
644
in gases
(S.E.R.M.A.,
Paris
1964) Vol. 1,