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Plasma domains in extrinsic GaAs and InP
.L Phys. Chem. Solids Vol. 46, No. 1, pp. 103-105, 1985 Printed in the U.S.A.
0022-3697/85 $3.00 + .00 © 1985 Pergamon Press Ltd.
P L A S M A D O M A I N S IN EXTRINSIC GaAs A N D InP RICHARD L. LIBOFF Schools of Electrical Engineering and Applied Physics, Cornell University, Ithaca, NY 14853, U.S.A. (Received 24 February 1984; accepted 21 June 1984)
Abstract--Various parameters are introduced relevant to criteria for physical domains in solid-state plasmas. Application is made to extrinsicGaAs and InP at 300 K and varyingcharge-carder concentrations. At concentrations less than ~ 1015 cm -3, charge-carder plasmas for both p- and n-type semiconductors, respectively, are classical and weakly coupled. At a concentration of 1016 cm-I the plasmas grow degenerate. At a concentration of l0 ~7 cm-3 both p-type materials approach a degenerate state whereas both n-type materials are weakly-coupleddegenerate.
1. INTRODUCFION
I ~ <~ 1
In a previous work by the author criteria relevant to various physical domains important to solid-state plasmas were obtained [ 1]. Application of these criteria were made to the compound semiconductors GaAs and InP in the intrinsic domain. It was found that at 300 K charge-carder plasmas in both these materials remain classical and weakly coupled. At 1000 K, it was found that the charge-carder plasmas grow degenerate [2-4] but otherwise stay removed from the so-called weakly-coupled degenerate domain [5-7]. In the present study these criteria are employed to examine physical domains for extrinsic samples of these materials. The work examines properties due to donor and acceptor impurities separately at 300 K where typical impurity concentrations exceed intrinsic carrier densities [8, 9]. Values of impurity charge-carder concentrations are examined from 10s to 1017 c m -3. At lower concentrations the chargecarrier plasmas for p- and n-type conduction, in both semiconducting materials, respectively, remain in the weakly-coupled, classical domain. At carrier densities of 1016 cm -3, charge-carrier plasmas in both 13- and n-type GaAs and InP, respectively, grow degenerate. At a concentration of 1017 cm -3, charge-carrier plasmas in n-type materials for both GaAs and InP, respectively, are weakly-coupled degenerate, whereas for both p-type materials at this same concentration charge-carrier plasmas approach degeneracy.
weakly-coupled plasma
l~ ~ 1 strongly-coupled plasma.
(2)
In the strong-coupling domain, the mean interparticle potential is of the same order or greater than the mean kinetic energy. Consequently, the dynamics of a strongly-coupled plasma involves no less than the two-particle distribution function [12-14]. Classical and degenerate plasma domains are separated according to the quantum degeneracy parameter
27rh 2 n 2/3 A2 - - -m*
(3)
kBT
where m* represents the effective mass of electrons or holes. The parameter A represents the ratio of the thermal deBroglie wavelength [15, 16] to the mean interparticle spacing. Related criteria are given by A >~ 1 degenerate plasma A ,~ 1 classical plasma.
(4)
A degenerate plasma may further collapse into what is commonly referred to as a weakly-coupled degenerate plasma [5-7]. This domain is characterized in terms of the quantum compression parameter ( 3 ) 1/3 m*e 2 rs = \47r1
~h2n 1/3
(5)
2. PARAMETERS AND PHYSICAL CRITERIA The division between weakly and strongly-coupled classical plasmas is given in terms of the plasma parameter [10, l 1], which in cgs units is given by
which represents the ratio between the mean interparticle spacing and the effective Bohr radius ~h 2
a~ = m . e 2 .
47me 6
F2
(~kBT)3 .
(6)
( 1)
In this relation n is the charge-carrier concentration, e the electronic charge, e the dielectric constant and T the temperature. Criteria involving r are given by
The condition for a weakly-coupled degenerate plasma is given by r, ~ 1 weakly-coupled degenerate. 103
(7)
104
R. L. LIBOFF Table 1. Parameter characteristics Parameter
Independence
Dependence
F (plasma) A (quantum degeneracy) r~ (quantum compression)
m
F 2 ~ n/(~T) 3
~ T
A 2 ~ n2/3/m*T r~ ~ m*/~n 1/3
A2 -
This inequality corresponds, for the m o s t part, to high concentrations. A recapitulation o f these criteria is given below
Classical: A
1
, P "~ 1 weakly coupled
\
F > 1 strongly coupled.
Quantum: A > 1 r~ ~ 1 weakly-coupled degenerate.
(8)
3. PLASMA CHARACTERISTICS Two immediate relations between these parameters are evident. The first o f these appears as A2 =
( ! ] '/3 k s T F 4 / 3 \16]
(9)
~
where R* represents the effective Rydberg constant R* - m*27r2e4
~2h~
m*/m
-
t~
m*/m
R -
~
13.6 eV.
(10)
One can note in particular that e 2
R* -
(11)
2~a~ "
The second relation between A and F involves r~ and is given by (37r/4)1/3 rs
F 2/3.
(12)
Particular physical properties o f these parameters are as follows. First one can note that F as given by eqn (1) is i n d e p e n d e n t o f mass. This property follows from the fact that P is a measure o f the m e a n twoparticle potential energy to thermal kinetic energy. Both these energies are mass independent (with kinetic energy given in terms o f temperature) so that P as well, is i n d e p e n d e n t o f mass. The parameter A on the other hand, is independent o f the dielectric constant e. This parameter is a measure o f the thermal deBroglie wavelength to m e a n interparticle spacing. The first o f these entities depends only on dynamics, whereas the second is determined in terms o f charge concentration n. Thus both parameters do not contain the C o u l o m b interaction and A is i n d e p e n d e n t o f ~. Lastly one can note that rs is not d e p e n d e n t on temperature. This results from the fact that rs is a ratio between two lengths which are likewise temperature independent: the Bohr radius, eqn (6), a n d the mean interparticle spacing. These properties are listed in Table 1. The first o f these properties suggests that classical plasma characteristics o f extrinsic semiconductors, at the same temperature, will be the same for p- and ntype doping o f equal concentrations. Thus, for example, if the extrinsic charge-carrier density in ntype GaAs is weakly coupled a n d classical, the same will be true o f p-type GaAs at the same charge-carrier concentration and temperature.
Table 2. Domain parameters in extrinsic GaAs and lnP GaAs, T = 300 K,~ = 13.1 Acceptor, p-type rh~' = 0.45
Donor, n-type th~ = 0.067
l0 s 1010 10 TM 1016 1017
A
F
r~
A
F
r~
7.7 X 10 -4 3.6 X 10-3 0.08 0.36 0.77
9.8 X 10 6 9.8 X 10_5 9.8 × 10 -3 9.8 X 10 2 0.310
1300 280 13 2.8 1.3
3.0 X 10-4 1.4X 10 3 0.03 0.14 0.30
9.8 × 10-6 9.8 X 10 s 9.8 x 10-3 9.8 × 10 -2 0.310
8700 1900 87 18 8.7
InP, T = 300 K, ~ = 12.4 Acceptor, p-type rh~' = 0.64
Donor, n-type the* = 0.077
l0 s 10 ~° 1014 1016
1017
A
F
r~
A
F
7.2 X 10-4 3.4 × 10-3 0.070 0.34 0.72
1.1 X 10-s 1.1 X 10-4 1.1 × 10-2 0.11 0.34
1600 340 16 3.4 1.6
2.5 X 10 4 1.2 X 10 -3 2.5 × 10 z 0.12 0.25
1.1 X 10-5 1.1 X 10 -4 1.1 X 10-2 0.11 0.34
r~ 1.3 X 1 0 4 2800 130 28 13
Plasma domains in extrinsic GaAs and InP One can also note the following. Due to the inverse e-dependence of I" and relatively large values c of the semiconductors presently considered [ 17], the plasma has a propensity to be weakly coupled. Furthermore, the q u a n t u m compression parameter rs is also inversely proportional to e as well as directly proportional to m*. These properties give rise to the interesting result that charge-carrier plasmas in semiconductors with relatively large values of ~ and small values of rn* may effect a weakly-coupled degenerate plasma [4]. It should be noted, however, that static values of ~ were used to tabulate terms in Table 2. More generally ~ is frequency dependent [18] and these results should change in the non-static case. Note that one has set rh* -= m*/me in Table 2. Effective masses were obtained from Ref. [17].
4. CONCLUSIONS Extrinsic properties of GaAs and InP at varying charge-carrier concentrations at 300 K have been examined. Values of characteristic parameters listed in Table 2 indicate the following. At the lower chargecarrier concentrations, the charge-carrier plasmas remain classical and weakly coupled. At a concentration of 1016 cm -3 these plasmas grow degenerate. This is more pronounced for the n-type samples than for the p-type samples. At a concentration of 1017 cm -3 both the n-type materials remain degenerate and grow weakly coupled whereas the p-type materials approach a degenerate state. In degenerate domains classical plasma physics becomes inappropriate and must be replaced by the theory of the Fermi liquid [2-4, 19].
105
Acknowledgments--I am grateful to Brian Jones for his critical reading of the final manuscript of this paper. Thanks are also due to Kenneth Gardner for his assistance with numerical calculations for this paper. This research was supported in part by a contract with Battelle Columbus Lab. REFERENCES
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.
Liboff R. L., J. Appl. Phys. 56, 2532 (1984). Landau L., Soy. Phys. JETP3, (1957); 5, 101 (1957). Galitskii V., Soy. Phys. JETP 7, 104 (1958). Platzman P. M. and Wolf P. A., Waves and Interactions in Solid State Plasmas. Academic Press, New York (1973). Wigner E. P., Phys. Rev. 46, 1002 (1934); Trans. Faraday Soc. 34, 678 (1938). Gel-Mann M. and Brueckner K., Phys. Rev. 106, 364 (1957). LiboffR. L., Phys. Lett. 74A, 323 (1979). Thurmond C. D., J. Electrochem. Soc. 122, 1133 (1975). Hall R. N. and Racette J. H., J. Appl. Phys. 35, 379 (1964). Kunkel W. B., Plasma Physics in Theory and Application. McGraw-Hill, New York (1966). Caroff L. J. and Liboff R. L., J. Plasma Phys. 4, 83 (1970). Guillen M. A. and Liboff R. L., J. Plasma Phys. (1984), in press. Brush S. G., Sahlin H. L. and Teller E., J. Chem. Phys. 45, 2102 (1966). Ichimaru S., Rev. Mod. Phys. 54, 1017 (1982). Pathria R. K., Statistical Mechanics. Pergamon Press, New York (1972). Lifshitz E. M. and Pitaevskii, Statistical Physics. Pergamon Press, New York (1980). Sze S. M., Physics of Semiconductor Devices (2nd edn.). Wiley, New York (1981). Blakemore J. S., J. Appl. Phys. 53, 123 (1982). Lie T. J. and Liboff R. L., Ann. Phys. 67, 349 (1971).
0022-3697/85 $3.00 + .00 © 1985 Pergamon Press Ltd.
P L A S M A D O M A I N S IN EXTRINSIC GaAs A N D InP RICHARD L. LIBOFF Schools of Electrical Engineering and Applied Physics, Cornell University, Ithaca, NY 14853, U.S.A. (Received 24 February 1984; accepted 21 June 1984)
Abstract--Various parameters are introduced relevant to criteria for physical domains in solid-state plasmas. Application is made to extrinsicGaAs and InP at 300 K and varyingcharge-carder concentrations. At concentrations less than ~ 1015 cm -3, charge-carder plasmas for both p- and n-type semiconductors, respectively, are classical and weakly coupled. At a concentration of 1016 cm-I the plasmas grow degenerate. At a concentration of l0 ~7 cm-3 both p-type materials approach a degenerate state whereas both n-type materials are weakly-coupleddegenerate.
1. INTRODUCFION
I ~ <~ 1
In a previous work by the author criteria relevant to various physical domains important to solid-state plasmas were obtained [ 1]. Application of these criteria were made to the compound semiconductors GaAs and InP in the intrinsic domain. It was found that at 300 K charge-carder plasmas in both these materials remain classical and weakly coupled. At 1000 K, it was found that the charge-carder plasmas grow degenerate [2-4] but otherwise stay removed from the so-called weakly-coupled degenerate domain [5-7]. In the present study these criteria are employed to examine physical domains for extrinsic samples of these materials. The work examines properties due to donor and acceptor impurities separately at 300 K where typical impurity concentrations exceed intrinsic carrier densities [8, 9]. Values of impurity charge-carder concentrations are examined from 10s to 1017 c m -3. At lower concentrations the chargecarrier plasmas for p- and n-type conduction, in both semiconducting materials, respectively, remain in the weakly-coupled, classical domain. At carrier densities of 1016 cm -3, charge-carrier plasmas in both 13- and n-type GaAs and InP, respectively, grow degenerate. At a concentration of 1017 cm -3, charge-carrier plasmas in n-type materials for both GaAs and InP, respectively, are weakly-coupled degenerate, whereas for both p-type materials at this same concentration charge-carrier plasmas approach degeneracy.
weakly-coupled plasma
l~ ~ 1 strongly-coupled plasma.
(2)
In the strong-coupling domain, the mean interparticle potential is of the same order or greater than the mean kinetic energy. Consequently, the dynamics of a strongly-coupled plasma involves no less than the two-particle distribution function [12-14]. Classical and degenerate plasma domains are separated according to the quantum degeneracy parameter
27rh 2 n 2/3 A2 - - -m*
(3)
kBT
where m* represents the effective mass of electrons or holes. The parameter A represents the ratio of the thermal deBroglie wavelength [15, 16] to the mean interparticle spacing. Related criteria are given by A >~ 1 degenerate plasma A ,~ 1 classical plasma.
(4)
A degenerate plasma may further collapse into what is commonly referred to as a weakly-coupled degenerate plasma [5-7]. This domain is characterized in terms of the quantum compression parameter ( 3 ) 1/3 m*e 2 rs = \47r1
~h2n 1/3
(5)
2. PARAMETERS AND PHYSICAL CRITERIA The division between weakly and strongly-coupled classical plasmas is given in terms of the plasma parameter [10, l 1], which in cgs units is given by
which represents the ratio between the mean interparticle spacing and the effective Bohr radius ~h 2
a~ = m . e 2 .
47me 6
F2
(~kBT)3 .
(6)
( 1)
In this relation n is the charge-carrier concentration, e the electronic charge, e the dielectric constant and T the temperature. Criteria involving r are given by
The condition for a weakly-coupled degenerate plasma is given by r, ~ 1 weakly-coupled degenerate. 103
(7)
104
R. L. LIBOFF Table 1. Parameter characteristics Parameter
Independence
Dependence
F (plasma) A (quantum degeneracy) r~ (quantum compression)
m
F 2 ~ n/(~T) 3
~ T
A 2 ~ n2/3/m*T r~ ~ m*/~n 1/3
A2 -
This inequality corresponds, for the m o s t part, to high concentrations. A recapitulation o f these criteria is given below
Classical: A
1
, P "~ 1 weakly coupled
\
F > 1 strongly coupled.
Quantum: A > 1 r~ ~ 1 weakly-coupled degenerate.
(8)
3. PLASMA CHARACTERISTICS Two immediate relations between these parameters are evident. The first o f these appears as A2 =
( ! ] '/3 k s T F 4 / 3 \16]
(9)
~
where R* represents the effective Rydberg constant R* - m*27r2e4
~2h~
m*/m
-
t~
m*/m
R -
~
13.6 eV.
(10)
One can note in particular that e 2
R* -
(11)
2~a~ "
The second relation between A and F involves r~ and is given by (37r/4)1/3 rs
F 2/3.
(12)
Particular physical properties o f these parameters are as follows. First one can note that F as given by eqn (1) is i n d e p e n d e n t o f mass. This property follows from the fact that P is a measure o f the m e a n twoparticle potential energy to thermal kinetic energy. Both these energies are mass independent (with kinetic energy given in terms o f temperature) so that P as well, is i n d e p e n d e n t o f mass. The parameter A on the other hand, is independent o f the dielectric constant e. This parameter is a measure o f the thermal deBroglie wavelength to m e a n interparticle spacing. The first o f these entities depends only on dynamics, whereas the second is determined in terms o f charge concentration n. Thus both parameters do not contain the C o u l o m b interaction and A is i n d e p e n d e n t o f ~. Lastly one can note that rs is not d e p e n d e n t on temperature. This results from the fact that rs is a ratio between two lengths which are likewise temperature independent: the Bohr radius, eqn (6), a n d the mean interparticle spacing. These properties are listed in Table 1. The first o f these properties suggests that classical plasma characteristics o f extrinsic semiconductors, at the same temperature, will be the same for p- and ntype doping o f equal concentrations. Thus, for example, if the extrinsic charge-carrier density in ntype GaAs is weakly coupled a n d classical, the same will be true o f p-type GaAs at the same charge-carrier concentration and temperature.
Table 2. Domain parameters in extrinsic GaAs and lnP GaAs, T = 300 K,~ = 13.1 Acceptor, p-type rh~' = 0.45
Donor, n-type th~ = 0.067
l0 s 1010 10 TM 1016 1017
A
F
r~
A
F
r~
7.7 X 10 -4 3.6 X 10-3 0.08 0.36 0.77
9.8 X 10 6 9.8 X 10_5 9.8 × 10 -3 9.8 X 10 2 0.310
1300 280 13 2.8 1.3
3.0 X 10-4 1.4X 10 3 0.03 0.14 0.30
9.8 × 10-6 9.8 X 10 s 9.8 x 10-3 9.8 × 10 -2 0.310
8700 1900 87 18 8.7
InP, T = 300 K, ~ = 12.4 Acceptor, p-type rh~' = 0.64
Donor, n-type the* = 0.077
l0 s 10 ~° 1014 1016
1017
A
F
r~
A
F
7.2 X 10-4 3.4 × 10-3 0.070 0.34 0.72
1.1 X 10-s 1.1 X 10-4 1.1 × 10-2 0.11 0.34
1600 340 16 3.4 1.6
2.5 X 10 4 1.2 X 10 -3 2.5 × 10 z 0.12 0.25
1.1 X 10-5 1.1 X 10 -4 1.1 X 10-2 0.11 0.34
r~ 1.3 X 1 0 4 2800 130 28 13
Plasma domains in extrinsic GaAs and InP One can also note the following. Due to the inverse e-dependence of I" and relatively large values c of the semiconductors presently considered [ 17], the plasma has a propensity to be weakly coupled. Furthermore, the q u a n t u m compression parameter rs is also inversely proportional to e as well as directly proportional to m*. These properties give rise to the interesting result that charge-carrier plasmas in semiconductors with relatively large values of ~ and small values of rn* may effect a weakly-coupled degenerate plasma [4]. It should be noted, however, that static values of ~ were used to tabulate terms in Table 2. More generally ~ is frequency dependent [18] and these results should change in the non-static case. Note that one has set rh* -= m*/me in Table 2. Effective masses were obtained from Ref. [17].
4. CONCLUSIONS Extrinsic properties of GaAs and InP at varying charge-carrier concentrations at 300 K have been examined. Values of characteristic parameters listed in Table 2 indicate the following. At the lower chargecarrier concentrations, the charge-carrier plasmas remain classical and weakly coupled. At a concentration of 1016 cm -3 these plasmas grow degenerate. This is more pronounced for the n-type samples than for the p-type samples. At a concentration of 1017 cm -3 both the n-type materials remain degenerate and grow weakly coupled whereas the p-type materials approach a degenerate state. In degenerate domains classical plasma physics becomes inappropriate and must be replaced by the theory of the Fermi liquid [2-4, 19].
105
Acknowledgments--I am grateful to Brian Jones for his critical reading of the final manuscript of this paper. Thanks are also due to Kenneth Gardner for his assistance with numerical calculations for this paper. This research was supported in part by a contract with Battelle Columbus Lab. REFERENCES
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.
Liboff R. L., J. Appl. Phys. 56, 2532 (1984). Landau L., Soy. Phys. JETP3, (1957); 5, 101 (1957). Galitskii V., Soy. Phys. JETP 7, 104 (1958). Platzman P. M. and Wolf P. A., Waves and Interactions in Solid State Plasmas. Academic Press, New York (1973). Wigner E. P., Phys. Rev. 46, 1002 (1934); Trans. Faraday Soc. 34, 678 (1938). Gel-Mann M. and Brueckner K., Phys. Rev. 106, 364 (1957). LiboffR. L., Phys. Lett. 74A, 323 (1979). Thurmond C. D., J. Electrochem. Soc. 122, 1133 (1975). Hall R. N. and Racette J. H., J. Appl. Phys. 35, 379 (1964). Kunkel W. B., Plasma Physics in Theory and Application. McGraw-Hill, New York (1966). Caroff L. J. and Liboff R. L., J. Plasma Phys. 4, 83 (1970). Guillen M. A. and Liboff R. L., J. Plasma Phys. (1984), in press. Brush S. G., Sahlin H. L. and Teller E., J. Chem. Phys. 45, 2102 (1966). Ichimaru S., Rev. Mod. Phys. 54, 1017 (1982). Pathria R. K., Statistical Mechanics. Pergamon Press, New York (1972). Lifshitz E. M. and Pitaevskii, Statistical Physics. Pergamon Press, New York (1980). Sze S. M., Physics of Semiconductor Devices (2nd edn.). Wiley, New York (1981). Blakemore J. S., J. Appl. Phys. 53, 123 (1982). Lie T. J. and Liboff R. L., Ann. Phys. 67, 349 (1971).