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Frequency dependent galvanomagnetic effects in multivalley semiconductors
Volume 31A, number 10
FREQUENCY
PHYSICS
DEPENDENT MULTIVALLEY
LETTERS
18 May 1970
GALVANOMAGNETIC SEMICONDUCTORS
EFFECTS
IN
J. M. BRETTELL Department
of Physics,
University
of New England,
Armidale,
N.S.W.,
Australia
Received 10 March 1970
Explicit expressions are presented for the imaginary parts of the high frequency galvanomaguetic coefficients for a cubically symmetric multivalley semiconductor in the limit of extreme degeneracy.
Calculations by Champlin [l] and Roy [2] indicate that a study of the real and imaginary parts of the high frequency galvanomagnetic coefficients of a many-valley semiconductor can provide useful information on the relaxation time anisotropy. In this letter we extend the calculations to include explicit expressions for the imaginary parts of the inverse Seitz coefficients [3,4] for the particular case of a cubically symmetric degenerate semiconductor. Quadratic energy surfaces are assumed with respectively 6, 12 and 4 ellipsoids for each of the (loo), (1lO)and (1ll)orientations. We consider only one type of carrier with two sets of effective masses for each class of ellipsoid, one m ,, along the axis and one m, perpendicular to it. Each effective mass is weighted by the reciprocal of the corresponding relaxation time so that the high frequency mass and scattering time anisotropy parameter is given by
where w is the angular frequency of the electric field. If WT << 1 then K’ =KCl + iw(‘r,, -TV)] where K = Re (K’). It is assumed that T,, and TV have the sample simple energy dependence. In the limit of weak magnetic fields (i.e. wcr << 1 where wc is the cyclotron frequency) the electric field E is related to the current density J according to E = P,[J+aJx
H + bH2J+c(J.
where p. is the zero-field resistivity, H the magnetic field T a diagonal tensor with elements H$ Hi, $$ an d a, b, c and d the inverse Seitz coefficients. Since the real parts of the coefficient6 have been worked out in detail [5-‘I] we present only the imaginary parts which, using the com-
Table 1 The magneto-resistance functions I’i(K). Orientation of ellipsoids Function
100
5K3-3K-2 r1
K(K+ 2)3 6K2(1 -K)
r2
K(K+ 2)3 -(11K3-6K2-3K-2)
r3 _
558
K(K+ 2)3
H)H+dTJ]
111
110
3(3K3 + 2KZ - 3K - 2) 4K(K+
2)3
- (K3 - 6K2 + 3K+ 2) 2K(K+
2)3
llK3-6K2-3K-2 4K(K+
2)3
2(ZK2 -K - 1) 3K(K+
2)2
2(2K2-K-2) 3K(K + 2) 2 2(11K3 - SK2 - 3K - 2) 3K(K+
2)2
Volume
PHYSICS
31A, number 10
putational method of Abeles and Meiboom assuming kT << EF, become
various ellipsoid orientations. As expected the function l?i(K) obey relations similar to the Shibuya symmetry relations [8] for the real parts of the magnetoresistance coefficients i.e.
[5] and
Im (R) = Im (ape) = WlK, w(r,, - 71) (2K+ 1)3ne Im (b’) =
rl(mw(T,, - ~~1 +
18 May 1970
LETTERS
Im(w2H) Re (b’)
rl(K)
- l?2(K) +l?g(K)
=0
(100)
Reip2$
q(K)
- ram
=0
(110)
=0
(111)
quo -
Imb2H)
Im (cl) =
r2(m~(7,~ - TV) +
Im (d’) =
r3(kq t0(7,, - TV)+~
Re (c’)
Re(p2d
r,(m
although in the present case there is no stipulation on the sign of F3(K) except in as far as we specify the shape of the ellipsoids. For example in the case of a (100) orientation we have F3(K) > 0 if the ellipsoids are oblate and F3(K) < 0 if they are prolate.
Re(W2,) Im(p2H)
- r3(K)
Re (d’)
where n
14
P‘H)
Re(C12$
References 1. K.S. Champlin,
4[K2 +K+ l] = (K+2)(2K+ 1) w(7;i - ‘l)
Phys. Rev. 130 (1963) 137. 2. S. K.Roy, J. Phys. C. (solid state) 2 (1969) 1487. 3. F.Seitz, Phys. Rev. 79 (1950) 372.
4. G. L. Pearson and H. Suhl, Phys. Rev. 83 (1951) 768. 5. B. Abeles and S. Meiboom, Phys. Rev. 95 (1954) 31.
Here R is the Hall coefficient, PH the complex Hall mobility and b’, c’ and d’ the dimensionless magnetoresistance coefficients defined as the inverse Seitz coefficients divided by Re (p2~). The coefficients ri(K) are listed in table 1 for the
6. R.S.Allgaier, Phys. Rev. 112 (1958) 828. 7. R. S.Allgaier, Phys. Rev. 115 (1959) 1185. 8. M.Shibuya, Phys. Rev. 95 (1954) 1385.
*****
MESURE DES ECARTS DANS LES TRANSITIONS
ISOTOPIQUES LASER DU XENON
R. VETTER Laboratoire
AIME
II- 91 Orsay,
COTTON
- C.N.R.S.
Received
21 March 1970
This letter gives the shift between the isotopes lines of xenon.
Nous avons mesure les &carts isotopiques dans les transitions laser infrarouges du xenon, a l’aide d’une experience [l] dont nous rappelons ici brievement le principe: Les deux isotopes purs dont on veut mesurer l’ecart sont contenus dans deux tubes a decharge align&s suivant l’axe d’une cavite laser oscillant en mode unique: ils sont excites alternativement pendant des intervalles de temps egaux. Le balayage continu en frequence de la cavite permet
France
136, 134, 132, 130 and 128 for fifteen infrared
laser
l’enregistrement “en pointille” des profils relatifs a chacun des isotopes. La precision des mesures varie selon les transitions, elle peut atteindre + 0.01 mK dans les meilleurs cas; pour certaines transitions, elle est limit&e par des effets de pression, en particulier lorsque la decharge contient un melange Xe - He. Les echantillons de xenon isotopique ont et& fournis par le laboratoire de Spectrometrie de Masse du C.N.R.S. a Orsay. 559
FREQUENCY
PHYSICS
DEPENDENT MULTIVALLEY
LETTERS
18 May 1970
GALVANOMAGNETIC SEMICONDUCTORS
EFFECTS
IN
J. M. BRETTELL Department
of Physics,
University
of New England,
Armidale,
N.S.W.,
Australia
Received 10 March 1970
Explicit expressions are presented for the imaginary parts of the high frequency galvanomaguetic coefficients for a cubically symmetric multivalley semiconductor in the limit of extreme degeneracy.
Calculations by Champlin [l] and Roy [2] indicate that a study of the real and imaginary parts of the high frequency galvanomagnetic coefficients of a many-valley semiconductor can provide useful information on the relaxation time anisotropy. In this letter we extend the calculations to include explicit expressions for the imaginary parts of the inverse Seitz coefficients [3,4] for the particular case of a cubically symmetric degenerate semiconductor. Quadratic energy surfaces are assumed with respectively 6, 12 and 4 ellipsoids for each of the (loo), (1lO)and (1ll)orientations. We consider only one type of carrier with two sets of effective masses for each class of ellipsoid, one m ,, along the axis and one m, perpendicular to it. Each effective mass is weighted by the reciprocal of the corresponding relaxation time so that the high frequency mass and scattering time anisotropy parameter is given by
where w is the angular frequency of the electric field. If WT << 1 then K’ =KCl + iw(‘r,, -TV)] where K = Re (K’). It is assumed that T,, and TV have the sample simple energy dependence. In the limit of weak magnetic fields (i.e. wcr << 1 where wc is the cyclotron frequency) the electric field E is related to the current density J according to E = P,[J+aJx
H + bH2J+c(J.
where p. is the zero-field resistivity, H the magnetic field T a diagonal tensor with elements H$ Hi, $$ an d a, b, c and d the inverse Seitz coefficients. Since the real parts of the coefficient6 have been worked out in detail [5-‘I] we present only the imaginary parts which, using the com-
Table 1 The magneto-resistance functions I’i(K). Orientation of ellipsoids Function
100
5K3-3K-2 r1
K(K+ 2)3 6K2(1 -K)
r2
K(K+ 2)3 -(11K3-6K2-3K-2)
r3 _
558
K(K+ 2)3
H)H+dTJ]
111
110
3(3K3 + 2KZ - 3K - 2) 4K(K+
2)3
- (K3 - 6K2 + 3K+ 2) 2K(K+
2)3
llK3-6K2-3K-2 4K(K+
2)3
2(ZK2 -K - 1) 3K(K+
2)2
2(2K2-K-2) 3K(K + 2) 2 2(11K3 - SK2 - 3K - 2) 3K(K+
2)2
Volume
PHYSICS
31A, number 10
putational method of Abeles and Meiboom assuming kT << EF, become
various ellipsoid orientations. As expected the function l?i(K) obey relations similar to the Shibuya symmetry relations [8] for the real parts of the magnetoresistance coefficients i.e.
[5] and
Im (R) = Im (ape) = WlK, w(r,, - 71) (2K+ 1)3ne Im (b’) =
rl(mw(T,, - ~~1 +
18 May 1970
LETTERS
Im(w2H) Re (b’)
rl(K)
- l?2(K) +l?g(K)
=0
(100)
Reip2$
q(K)
- ram
=0
(110)
=0
(111)
quo -
Imb2H)
Im (cl) =
r2(m~(7,~ - TV) +
Im (d’) =
r3(kq t0(7,, - TV)+~
Re (c’)
Re(p2d
r,(m
although in the present case there is no stipulation on the sign of F3(K) except in as far as we specify the shape of the ellipsoids. For example in the case of a (100) orientation we have F3(K) > 0 if the ellipsoids are oblate and F3(K) < 0 if they are prolate.
Re(W2,) Im(p2H)
- r3(K)
Re (d’)
where n
14
P‘H)
Re(C12$
References 1. K.S. Champlin,
4[K2 +K+ l] = (K+2)(2K+ 1) w(7;i - ‘l)
Phys. Rev. 130 (1963) 137. 2. S. K.Roy, J. Phys. C. (solid state) 2 (1969) 1487. 3. F.Seitz, Phys. Rev. 79 (1950) 372.
4. G. L. Pearson and H. Suhl, Phys. Rev. 83 (1951) 768. 5. B. Abeles and S. Meiboom, Phys. Rev. 95 (1954) 31.
Here R is the Hall coefficient, PH the complex Hall mobility and b’, c’ and d’ the dimensionless magnetoresistance coefficients defined as the inverse Seitz coefficients divided by Re (p2~). The coefficients ri(K) are listed in table 1 for the
6. R.S.Allgaier, Phys. Rev. 112 (1958) 828. 7. R. S.Allgaier, Phys. Rev. 115 (1959) 1185. 8. M.Shibuya, Phys. Rev. 95 (1954) 1385.
*****
MESURE DES ECARTS DANS LES TRANSITIONS
ISOTOPIQUES LASER DU XENON
R. VETTER Laboratoire
AIME
II- 91 Orsay,
COTTON
- C.N.R.S.
Received
21 March 1970
This letter gives the shift between the isotopes lines of xenon.
Nous avons mesure les &carts isotopiques dans les transitions laser infrarouges du xenon, a l’aide d’une experience [l] dont nous rappelons ici brievement le principe: Les deux isotopes purs dont on veut mesurer l’ecart sont contenus dans deux tubes a decharge align&s suivant l’axe d’une cavite laser oscillant en mode unique: ils sont excites alternativement pendant des intervalles de temps egaux. Le balayage continu en frequence de la cavite permet
France
136, 134, 132, 130 and 128 for fifteen infrared
laser
l’enregistrement “en pointille” des profils relatifs a chacun des isotopes. La precision des mesures varie selon les transitions, elle peut atteindre + 0.01 mK dans les meilleurs cas; pour certaines transitions, elle est limit&e par des effets de pression, en particulier lorsque la decharge contient un melange Xe - He. Les echantillons de xenon isotopique ont et& fournis par le laboratoire de Spectrometrie de Masse du C.N.R.S. a Orsay. 559