- Email: [email protected]
Hot electron conduction in lead telluride
Volume 56A, number 4
PHYSICS LETTERS
5 April 1976
HOT ELECTRON CONDUCTION IN LEAD TELLURIDE D.K. FERRY Office of NavalResearch, Arlington, Virgina 22217, USA. Received 26 January 1976 The velocity of hot electrons in PbTe at 77 K is calculated by an iterative-integral technique. Scattering by polaroptical phonons, acoustic phonons, and first-order-coupled intervalley phonons is included.
An experimental investigation of the high electric field transport of electrons in lead telluride was first carried out by St.-Onge and co-workers [1, 2]. As expected for polar semiconductors, the mobility decreased at high electric fields because of the effectiveness of polar-optical-phonon scattering as an energy loss mechanism for the hot electrons. Recently, Hemrich et al [3] have reported measurements of electron transport in epitaxial thin films and have observed values of drift velocity well in excess of those reported in bulk material by St.-Onge et al. However, subsequent measurements by Carver et al. [4] on epitaxial thin films yield velocity values that are essentially in agreement with the bulk values, The theoretical understanding of these curves is not well in hand. Simple calculations based on a drifted Maxwellian distribution function were carried out by St.-Onge and co-workers, but it is known that this is not a good approximation for strongly polar materials [5, 6]. In recent years, solutions of the semiclassical Boltzmann equation, which retain the full form of the collision integral, have been obtained numerically using the Monte Carlo [7—9] and iterative techniques [10—12].These methods are exact in the sense that no approximations to the physics, other than in the material model itself, are introduced and arbitrarily high accuracy may be obtained. In particular, the iterative techniques build upon a path-variable formalism and offer a rapid computational result. Moreover, the iterative techniques exploit the stability of a steady-state distribution function. In the present work, Rees’ iterative integral technique [12] is utilized to calculate the high electric field transport properties of electrons in PbTe at 77 K. The low-field scattering properties of the lead salts were extensively reviewed by Ravich et al. [13], and it
is reasonably well established that polar-optical-phonon scattering is the dominant scattering mechanism. Acoustic scattering plays a contributive role, however. Coupling to the polar phonon is characterized by the unscreened low- and high-frequency dielectric constants, e~= 400 and e,,, = 37, while a value of 24 eV for the acoustic deformation potential has been deduced from various transport measurements [13]. The conduction band of PbTe is characterized by multiple ellipsoids located at the L-point in the Brillouin zone. Because of the multi-valley nature of the conduction band, it would normally be expected that equivalentintervalley scattering would be important [14]. However, because of the symmetry of the conduction band minima, the zero-order interaction via the X-point phonons that would be involved is forbidden by selection rules [15, 16]. Intervalley scattering can occur, however, by a first-order interaction [14] and the scattering rates for this process have been worked out previously [17]. In the present calculation, this first-order intervalley interaction has been included along with acoustic-phonon scattering and polar-optical-phonon scattering. The calculated electron.velocity for an electric field oriented along a (100) direction is shown in fig. 1. The coupling constant (in eV) for the first-ordercoupled intervalley scattering process is shown as a parameter. For reference, the experimental data is also shown. As expected, the dominant scattering mechanism is the interaction of the electrons with the polar-optical phonons. Velocity saturation does not occur per se, but a definite knee in the velocityfield curve is apparent with the velocity limited by emission of LO phonons at high fields. The major effects of the isotropic scattering by acoustic and equivalent-intervalley phonons is to randomize the momen317
Volume 56A, number 4 I
-
on the velocity arises because of this effect which serves to diminish the streaming of the carriers alo~igthe electric field direction. This also leads to the result that this process has little effect when the electron distribution is nearly Maxwellian. This has been confirmed by simple drifted Maxwellian calculations. As a consequence, one is essentially
x
-
2
D1”O
-
....—--
~ 0
X
-_
of
~
a I
-
0
.
X
0~ a
-
I
0
0.5 1.0 .5 ELECTRIC FIELD (ky/cm) Fig. 1. The curves are the calculated velocity of electrons in PbTe at 77 K for an electric field oriented along a (100) axis. The parameter is the deformation potential coupling constant (in eV) for the first-order coupled equivalent intervalley scattering process. The experimental data are from the following sources: The crosses are from Heinrich et at. [3], the squares are from St.-Onge et al. [1,2], and the circles from Carver et at. [4].
tum. Scattering by the polar LO phonons tends to funnel the electrons into a distribution function that has a large spike along the electric field direction, representing a streaming of the carriers along the field. The isotropic scatterers randomize the electrons in this spike and reduce its size and effect. While the thinfilm data of Heinrich et al. [3] is fit nicely by D1 = 0 eV, both the bulk data of St.-Onge et al. [1, 2] and the thin film data of Carver et al. [4] is best fit by a value ofD1 = 90 eV. Although little is known about the expected strength of this process, this value is higher than one would normally expect by extrapolation from other material. In silicon, for example, the value ofD1 is somewhat less than twice the value of the acoustic deformation potential. Here, the value of D1 = 90 eV necessary to fit the bulk data is almost four times the acoustic deformation potential. In these calculations, the first-order intervalley interaction is found to be mainly operative at a high electron energy and thus primarily serves to randomize 318
the momentum and energy in the electron distribution. Its large effect
-
~-
5 April 1976
PHYSICS LETTERS
temperature. The fit to the data obtained by using 90 eV would seem to justify its use, at least until other experiments are performed which can directly measure this parameter. The difference in the experimental data obtained by Heinrich et al. [3] and Carver et al. [4] is puzzling, since in both cases similar substrates and preparation procedures were used. In the latter measurements, several film thicknesses, geometries, and orientations were measured and the high field velocities were independent of these factors. In addition, samples with a range of low field mobilities were utilized and the .
.
.
high field velocity was relatively insensitive to the variation in low field mobility as well. From these calculations and the results shown in fig. 1, it is possible to suggest one possibility for the experimental discrepancy. It has recently been observed that these films .
are strained, and that at 4.2 K the strain is sufficient to raise the degeneracy of the equivalent ellipsoids of the conduction band [18]. From magnetophonon resonance and Shubnikov—de Haas measurements, it was ascertained that at 4.2 K all of the carriers are in a single ellipsoid. If the strain were large enough and the conduction band in the samples of Heinrich et al. were still strain split at 77 K, the equivalent intervalley scattering would be reduced, a result in line with the present calculations. However, this is only a conjecture, and the resolution of the difference must await further experimental investigations. The author would like to express his appreciation to Mrs. Carver, Burke and Houston for furnishing the results of their investigations prior to publication, and to the Naval Surface Weapons Center, White Oak Laboratory, where these calculations were performed.
Volume 56A, number 4
PHYSICS LETTERS
References [1] H. St.-Onge, J.N. Walpole and R.H. Rediker, Proc. 10th mt. Conf. Phys. Semiconductors (1970 Cambridge, Mass.) p. 391. [2] H. St.-Onge and J.N. Walpole, Phys. Rev. B6 (1972) 2337. [31 H. Heinrich, W. Jantsch and J. Rozenbergs, Solid Cornmun. 17 (1975) 1145. [4] G.P. Carver, B.B. Houston and J.R. Burke, to be published. [5] E.M. Conwell, High field transport in semiconductors (Academic, New York, 1967). [6] P.J. Price, IBM J. Res. Develop. 14 (1970) 12. [7J T. Kurosawa, J. Phys. Soc. Jpn. Suppl. 21(1966) 424. [81 A.D. Boardman, W. Fawcett and H.D. Rees, Solid State Commun. 6 (1968) 305.
.
5 April 1976
[9] W. Fawcett, A.D. BGardman and S. Swain, J. Phys. Chem. Solids 31(1970)1963. [101 H.F. Budd,Phys. Rev. 158 (1967) 798. [11] P.J. Price, in Proc. 9th tnt. Conf. Phys. Semiconductors, Moscow (Nauka, Leningrad, 1968) p. 753. [12] H.D. Rees, Phys. Lett. A26 (1968) 416;J. Phys. Chem. Solids 30(1969) 643; J. Phys. CS (1972) 641. [13] Yu.I. Ravich, B.A. Efimova and V.1. Tam~archenko, Phys. Stat. Sol. (b) 43 (1971) 11. [141 W.A. Harrison, Phys. Rev. 104 (1956) 1281. [15] D.M. Bercha and M.V. Tarnavs’ka, Ukrayin. Fiz. Zh. 9 (1964) 642. [16] J.O. Dimmock, private communication. [17] D.K. Ferry, to be published. [18] J.R. Burke and G.P. Carver, to be published.
319
PHYSICS LETTERS
5 April 1976
HOT ELECTRON CONDUCTION IN LEAD TELLURIDE D.K. FERRY Office of NavalResearch, Arlington, Virgina 22217, USA. Received 26 January 1976 The velocity of hot electrons in PbTe at 77 K is calculated by an iterative-integral technique. Scattering by polaroptical phonons, acoustic phonons, and first-order-coupled intervalley phonons is included.
An experimental investigation of the high electric field transport of electrons in lead telluride was first carried out by St.-Onge and co-workers [1, 2]. As expected for polar semiconductors, the mobility decreased at high electric fields because of the effectiveness of polar-optical-phonon scattering as an energy loss mechanism for the hot electrons. Recently, Hemrich et al [3] have reported measurements of electron transport in epitaxial thin films and have observed values of drift velocity well in excess of those reported in bulk material by St.-Onge et al. However, subsequent measurements by Carver et al. [4] on epitaxial thin films yield velocity values that are essentially in agreement with the bulk values, The theoretical understanding of these curves is not well in hand. Simple calculations based on a drifted Maxwellian distribution function were carried out by St.-Onge and co-workers, but it is known that this is not a good approximation for strongly polar materials [5, 6]. In recent years, solutions of the semiclassical Boltzmann equation, which retain the full form of the collision integral, have been obtained numerically using the Monte Carlo [7—9] and iterative techniques [10—12].These methods are exact in the sense that no approximations to the physics, other than in the material model itself, are introduced and arbitrarily high accuracy may be obtained. In particular, the iterative techniques build upon a path-variable formalism and offer a rapid computational result. Moreover, the iterative techniques exploit the stability of a steady-state distribution function. In the present work, Rees’ iterative integral technique [12] is utilized to calculate the high electric field transport properties of electrons in PbTe at 77 K. The low-field scattering properties of the lead salts were extensively reviewed by Ravich et al. [13], and it
is reasonably well established that polar-optical-phonon scattering is the dominant scattering mechanism. Acoustic scattering plays a contributive role, however. Coupling to the polar phonon is characterized by the unscreened low- and high-frequency dielectric constants, e~= 400 and e,,, = 37, while a value of 24 eV for the acoustic deformation potential has been deduced from various transport measurements [13]. The conduction band of PbTe is characterized by multiple ellipsoids located at the L-point in the Brillouin zone. Because of the multi-valley nature of the conduction band, it would normally be expected that equivalentintervalley scattering would be important [14]. However, because of the symmetry of the conduction band minima, the zero-order interaction via the X-point phonons that would be involved is forbidden by selection rules [15, 16]. Intervalley scattering can occur, however, by a first-order interaction [14] and the scattering rates for this process have been worked out previously [17]. In the present calculation, this first-order intervalley interaction has been included along with acoustic-phonon scattering and polar-optical-phonon scattering. The calculated electron.velocity for an electric field oriented along a (100) direction is shown in fig. 1. The coupling constant (in eV) for the first-ordercoupled intervalley scattering process is shown as a parameter. For reference, the experimental data is also shown. As expected, the dominant scattering mechanism is the interaction of the electrons with the polar-optical phonons. Velocity saturation does not occur per se, but a definite knee in the velocityfield curve is apparent with the velocity limited by emission of LO phonons at high fields. The major effects of the isotropic scattering by acoustic and equivalent-intervalley phonons is to randomize the momen317
Volume 56A, number 4 I
-
on the velocity arises because of this effect which serves to diminish the streaming of the carriers alo~igthe electric field direction. This also leads to the result that this process has little effect when the electron distribution is nearly Maxwellian. This has been confirmed by simple drifted Maxwellian calculations. As a consequence, one is essentially
x
-
2
D1”O
-
....—--
~ 0
X
-_
of
~
a I
-
0
.
X
0~ a
-
I
0
0.5 1.0 .5 ELECTRIC FIELD (ky/cm) Fig. 1. The curves are the calculated velocity of electrons in PbTe at 77 K for an electric field oriented along a (100) axis. The parameter is the deformation potential coupling constant (in eV) for the first-order coupled equivalent intervalley scattering process. The experimental data are from the following sources: The crosses are from Heinrich et at. [3], the squares are from St.-Onge et al. [1,2], and the circles from Carver et at. [4].
tum. Scattering by the polar LO phonons tends to funnel the electrons into a distribution function that has a large spike along the electric field direction, representing a streaming of the carriers along the field. The isotropic scatterers randomize the electrons in this spike and reduce its size and effect. While the thinfilm data of Heinrich et al. [3] is fit nicely by D1 = 0 eV, both the bulk data of St.-Onge et al. [1, 2] and the thin film data of Carver et al. [4] is best fit by a value ofD1 = 90 eV. Although little is known about the expected strength of this process, this value is higher than one would normally expect by extrapolation from other material. In silicon, for example, the value ofD1 is somewhat less than twice the value of the acoustic deformation potential. Here, the value of D1 = 90 eV necessary to fit the bulk data is almost four times the acoustic deformation potential. In these calculations, the first-order intervalley interaction is found to be mainly operative at a high electron energy and thus primarily serves to randomize 318
the momentum and energy in the electron distribution. Its large effect
-
~-
5 April 1976
PHYSICS LETTERS
temperature. The fit to the data obtained by using 90 eV would seem to justify its use, at least until other experiments are performed which can directly measure this parameter. The difference in the experimental data obtained by Heinrich et al. [3] and Carver et al. [4] is puzzling, since in both cases similar substrates and preparation procedures were used. In the latter measurements, several film thicknesses, geometries, and orientations were measured and the high field velocities were independent of these factors. In addition, samples with a range of low field mobilities were utilized and the .
.
.
high field velocity was relatively insensitive to the variation in low field mobility as well. From these calculations and the results shown in fig. 1, it is possible to suggest one possibility for the experimental discrepancy. It has recently been observed that these films .
are strained, and that at 4.2 K the strain is sufficient to raise the degeneracy of the equivalent ellipsoids of the conduction band [18]. From magnetophonon resonance and Shubnikov—de Haas measurements, it was ascertained that at 4.2 K all of the carriers are in a single ellipsoid. If the strain were large enough and the conduction band in the samples of Heinrich et al. were still strain split at 77 K, the equivalent intervalley scattering would be reduced, a result in line with the present calculations. However, this is only a conjecture, and the resolution of the difference must await further experimental investigations. The author would like to express his appreciation to Mrs. Carver, Burke and Houston for furnishing the results of their investigations prior to publication, and to the Naval Surface Weapons Center, White Oak Laboratory, where these calculations were performed.
Volume 56A, number 4
PHYSICS LETTERS
References [1] H. St.-Onge, J.N. Walpole and R.H. Rediker, Proc. 10th mt. Conf. Phys. Semiconductors (1970 Cambridge, Mass.) p. 391. [2] H. St.-Onge and J.N. Walpole, Phys. Rev. B6 (1972) 2337. [31 H. Heinrich, W. Jantsch and J. Rozenbergs, Solid Cornmun. 17 (1975) 1145. [4] G.P. Carver, B.B. Houston and J.R. Burke, to be published. [5] E.M. Conwell, High field transport in semiconductors (Academic, New York, 1967). [6] P.J. Price, IBM J. Res. Develop. 14 (1970) 12. [7J T. Kurosawa, J. Phys. Soc. Jpn. Suppl. 21(1966) 424. [81 A.D. Boardman, W. Fawcett and H.D. Rees, Solid State Commun. 6 (1968) 305.
.
5 April 1976
[9] W. Fawcett, A.D. BGardman and S. Swain, J. Phys. Chem. Solids 31(1970)1963. [101 H.F. Budd,Phys. Rev. 158 (1967) 798. [11] P.J. Price, in Proc. 9th tnt. Conf. Phys. Semiconductors, Moscow (Nauka, Leningrad, 1968) p. 753. [12] H.D. Rees, Phys. Lett. A26 (1968) 416;J. Phys. Chem. Solids 30(1969) 643; J. Phys. CS (1972) 641. [13] Yu.I. Ravich, B.A. Efimova and V.1. Tam~archenko, Phys. Stat. Sol. (b) 43 (1971) 11. [141 W.A. Harrison, Phys. Rev. 104 (1956) 1281. [15] D.M. Bercha and M.V. Tarnavs’ka, Ukrayin. Fiz. Zh. 9 (1964) 642. [16] J.O. Dimmock, private communication. [17] D.K. Ferry, to be published. [18] J.R. Burke and G.P. Carver, to be published.
319