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Inelastic electron-polar optical phonon scattering in the solid solution CdxHg1−xTe
Journal of Alloys and Compounds 379 (2004) 60–63
Inelastic electron-polar optical phonon scattering in the solid solution Cdx Hg1−x Te O.P. Malyk∗ Semiconductor Electronics Department, Lviv Polytechnic National University, Bandera Street 12, 79013 Lviv, Ukraine Received 16 January 2004; received in revised form 17 February 2004; accepted 17 February 2004
Abstract A model of inelastic electron-polar optical phonon scattering in the solid solution Cdx Hg1−x Te is proposed in which the scattering probability does not depend on a macroscopic parameter, the crystal permittivity. The given model yields good agreement between theory and experiment in temperature range 120–300 K. © 2004 Elsevier B.V. All rights reserved. PACS: 72.10.-d Keywords: Semiconductors; Electronic transport
1. Introduction The electron-polar optical phonon scattering in the solid solution Cdx Hg1−x Te was considered in relaxation time approximation in [1,2]. In [3,4], this scattering mechanism was considered for the case x = 0 taking into account the inelastic character of the scattering within the framework of a precise solution of the stationary Boltzmann equation. It was shown that the usage of a standard model of electron-polar optical phonon scattering leads to a disagreement between the theory and experiment in temperature range T > 100 K that is in that temperature range where this mechanism is dominating. According to our opinion, this model has the following shortcomings: (1) the use of a macroscopic parameter, the permittivity, is not reasonable in a microscopic processes and (2) the interaction potential of an electron with the optical oscillations of a crystal lattice is long-range which contradicts the special relativity. The purpose of the present work is the build-up of such a model of scattering which firstly would well match with experiment and secondly would not have the above mentioned shortcomings.
∗
Tel.: +380-322-398627. E-mail address: [email protected] (O.P. Malyk).
0925-8388/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jallcom.2004.02.039
2. The model of the electron-polar optical phonon scattering Let us consider a displacement of the jth (j = 1 and 2) atom in the unit cell of a crystal with zinc-blende structure under the influence of optical oscillations [5] 1/2 h ¯ 1 Qj = √ [ξj (q, ν)bq,ν eiqρ G q,ν 2Mων (q) ∗ +ξj∗ (q, ν)bq,ν e−iqρ ],
(1)
where G is the number of unit cells in a crystal volume; M = xMCd + (1 − x)MHg + MTe is the mass of the unit cell; q and ωv (q) are wave vector and angular frequency of the vth branch of optical oscillations in the crystal (v = 4, 5, 6); ξ j are polarization vectors of crystal oscillations; ∗ are phonon annihilation and creation operators bq,v and bq,ν respectively of the vth branch with wave vector q; ρ = i(n2 + n3 )(a0 /2) + j(n1 + n3 )(a0 /2) + k(n2 + n1 )(a0 /2), (n1 , n2 , n3 = 1, 2, . . . ), a0 is the lattice constant; i, j, k are unit vectors along the principal crystal axes. Under optical oscillations in a unit cell a polarization vector arises: P=
e(Q1 − Q2 ) , V0
(2)
O.P. Malyk / Journal of Alloys and Compounds 379 (2004) 60–63
where V0 = a0 /4 is the volume of the unit cell; e is the elementary charge. Using Eq. (1) and taking into account only of long wave length (q → 0) oscillations one can obtain
P =
4e a03
q,ν
h ¯ 2GMων (q)
12 (ξ 1 (q, ν) − ξ 2 (q, ν))[bq,ν eiqρ
∗ + bq,ν e−iqρ ].
(3)
It must be noticed that the polarization vector is a function of discrete variables P = P(n1 , n2 , n3 ). To calculate the bound charge ρ = −divP let us use the following expression of a partial derivative of the polarization vector ∂Px Px (n1 + 1, n2 , n3 ) − Px (n1 , n2 , n3 ) → ∂x x Px (n1 , n2 + 1, n3 ) − Px (n1 , n2 , n3 ) + x Px (n1 , n2 , n3 + 1) − Px (n1 , n2 , n3 ) + , x
(4)
1/2 ρ h ¯ 8ie ∇ ϕ =− = 3 ε0 2Gων (q) a 0 ε0 q xMCd + (1 − x)MHg + MTe 1/2 × {xMCd + (1 − x)MHg }MTe
ρ 2ε0
(5)
1 R2 − r 2 , 3
0≤r≤R
(qx + qy + qz )2 [bq eiqρ + bq∗ e−iqρ ]. q
(7)
It should be borne in mind that the potential Eq. (7) is short-range as it takes into account the interaction of an electron only with one unit cell. To calculate the transition probability connected with electron–phonon interaction let us write the wave function of the electron–phonon system in the form 1 Ψ = √ exp(ikr)Φ(x1 , x2 , . . . , xn ), (8) V
Nq , k |U|Nq , k 12 4ie2 h ¯ 1 = 3 exp(isr) R2 − r 2 dr 3 2Gων (q) a 0 ε0 V q xMCd + (1 − x)MHg + MTe 1/2 (qx + qy + qz )2 × (xMCd + (1 − x)MHg )MTe q × Φ∗ (x1 , x2 , . . . , xn )[bq eiqρ + bq∗ e−iqρ ] s = k − k (9)
where the relation qi (a0 /2) 1(i = x, y, z) is used and only optical longitudinal vibrations are taken into account, ε0 is the dielectric constant. To solve Eq. (5), let us substitute the unit cell by a sphere of effective radius R = γa0 the magnitude of which lies within the limits of half of the smaller diagonal up √ to half of the larger diagonal of a unit cell (0.5 < γ < 3/2). The magnitude of γ is chosen so to adjust the theory with experiment. The spherically symmetric solution of a Poisson equation looks like ϕ=
×
2
× Φ(x1 , x2 , . . . , xn ) dx1 , dx2 , . . . , dxn ,
2
(qx + qy + qz )2 [bq eiqρ + bq∗ e−iqρ ], q
12 h ¯ 1 2 R − r U = −eϕ = 3 3 2Gων (q) a 0 ε0 q 12 xMCd + (1 − x)MHg + MTe × (xMCd + (1 − x)MHg )MTe 4ie2
where V is the crystal volume; Φ (x1 , x2 , . . . , xn ) is the wave function of the system of independent harmonic oscillators. Then a transition matrix element from interaction energy looks like
where x = a0 /2 for a unit cell of the zinc-blende structure. A similar relation can be written for the partial derivatives (∂Py /∂y) and (∂Pz /∂z) with y = z = a0 /2. Then Poisson equation for a scalar potential associated with crystal oscillations becomes
×
61
(6)
Then the interaction energy of an electron with polar optical oscillations of the lattice is obtained from the expression
The integration over the electron coordinates is carried out within the limits of the unit cell and leads to 1 2 2 I(s) = exp(isr) R − r dr 3 π(8 sinRs − 8Rs cos Rs − 8/3R3 s3 cos Rs) (10) s5 The calculation shows that the electron wave vector (and s together with it) varies within the limits from 0 up to 109 m−1 at energy changing from 0 up to 10kB T (kB is the Boltzmann constant) at the temperature range 4.2–300 K. In Fig. 1, the dependence of the value I(s)/I(0) on s is shown. It is seen that for the indicated limits of the wave vector variation the following relation is well fulfilled =
I(s) ≈ I(0) =
5 16 15 πR
=
5 5 16 15 πa0 γ
The integration over oscillators coordinates the harmonic gives the factors Nq and Nq + 1 (Nq is the number of phonons with a frequency ω(q) = ω0 at q → 0) for phonon annihilation and creation operators respectively. To calculate the sum over the vector q let us make the following simplifications (1). We take into consideration a quasicontinuous
62
O.P. Malyk / Journal of Alloys and Compounds 379 (2004) 60–63
sorption and radiation W(k, k ) =
64π7 γ 10 e4 xMCd + (1 − x)MHg + MTe 225ε20 a04 Gω0 (xMCd + (1 − x)MHg )MTe
× [Nq δ(ε − ε − h ¯ ω0 ) + (Nq + 1) ¯ ω0 )], × δ(ε − ε + h
(13) nm Kβαab
from a prewhere ε is the electron energy.The values cise solution of the stationary Boltzmann equation for intraand inter-band electron transitions can now be obtained [3,4] nm Kβα11 =−
Fig. 1. The dependence of function I(s)/I(0) on s = |k − k |.
xMCd + (1 − x)MHg + MTe δαβ (xMCd + (1 − x)MHg )MTe × Nq f0 (ε) × [1 − f0 (ε + h ¯ ω0 )]k2 (ε + h ¯ ω0 )
×
character of wave vector variation and pass from summation to integration over q and (2) We change from integration of a cube with diameter 2π/a0 to integration of a sphere with effective radius π/a0 π/a0 π/a0 π/a0 V ··· → · · · dqx dqy dqz (2π)3 −π/a0 −π/a0 −π/a0 q 2π π π/a0 → . . . q2 sin θ dq dθ dϕ (11) 0
0
q
8 cos ρQ + 8ρQ sin ρQ − 4ρ2 Q2 cos ρQ − 8 ρ4 Nq − absorption, × . Nq + 1 − radiation, Q = π/a0
=π
(12)
∂k(ε + h ¯ ω0 ) ¯ ω0 )f0 (ε) + (Nq + 1)θ(ε − h ∂ε
∂k(ε − h ¯ ω0 ) 2 × [1 − f0 (ε− h ¯ ω0 )] × k (ε − h ¯ ω0 ) ∂ε ∂k(ε) εn+m dε; × k4 (ε) ∂ε ¯ 2 γ 10 2V 32π6 e4h 2mhh 3/2 =− (2π)3 675ε20 a0 ω0 kB T h ¯2 xMCd + (1 − x)MHg + MTe × δαβ (xMCd + (1 − x)MHg )MTe × (Nq + 1) × f0 (ε)[1 − f0 (ε − h ¯ ω0 )] ×
0
Then we obtain for the sum the following expression · · · = F(ρ)
¯ 2 γ 10 2V 64π6 e4h (2π)3 675ε20 a0 ω0 kB T
nm Kβα12
¯ ω0 )1/2 k4 (ε) × (−ε − εg + h
∂k(ε) n+m ε dε, ∂ε
In Fig. 2, the dependence of the value F(ρ)/F(0) on ρ is shown. It follows that the function F(ρ) can be approximated by the expression Nq 4 F(0) = πQ . (12a) Nq + 1
where δαβ is the Kronecker delta; f0 (ε) is the Fermi–Dirac function; θ(x) is a step function.
After some calculations we can obtain the expression for electron transition probability connected with phonon ab-
A comparison of the theoretical temperature dependences of the electron mobility was made with the experimental data presented in [2] for Cdx Hg1−x Te crystals with compositions x = 0.08; 0.095; 0.12; 0.135 and 0.17. The corresponding parameter values for these compositions are γ = 0.6708, 0.6952, 0.7103, 0.7125 and 0.7125. The calculation of the theoretical electron mobilities µ(T) was made on the basis of electroneutrality equation: n − p = Nd , where Nd = n4.2 is the electron concentration at 4.2 K. In these calculations, the same scattering mechanisms and crystal parameters as in [2,3,6,7] were taken into consideration. The theoretical µ(T) curves are presented in Fig. 3a–e. The dashed lines represent the curves calculated on the basis of the long-range model within the framework of the precise solution of the Boltzmann equation. It is seen that the theoretical curve well agree with the experimental data (deviation about ∼10–20%) in
Fig. 2. The dependence of function F(ρ)/F(0) on ρ.
3. Discussion
O.P. Malyk / Journal of Alloys and Compounds 379 (2004) 60–63
63
Fig. 3. Temperature dependence of the electron mobility in Cdx Hg1−x Te crystals. (Solid line) Mixed scattering mechanism (short-range model); (dashed line) mixed scattering mechanism (long-range model); (curve 1) nonpolar; (curve 2) acoustic; (curve 3) piezo-acoustic; (curve 4) disorder; (curve 5) intraband polar; and (curve 6) ionized impurity scattering mechanisms.
the temperature range T = 120–300 K whereas the curves calculated on the basis of the long-range model differ from experiment in 1.9–3.8 times. To estimate the role of the different scattering mechanisms in Fig. 3e the dotted lines represent the appropriate dependences for a composition x = 0.17. It is seen that in all investigated temperature intervals the main scattering mechanism is intraband scattering on polar optical phonons. Other scattering mechanisms such as acoustic scattering, piezo-acoustic scattering, nonpolar optical phonon scattering, disorder scattering, ionized impurity scattering, give negligibly small contributions. A similar situation exists for crystals of other compositions.
4. Conclusion A short-range model of inelastic electron scattering on polar optical phonons in Cdx Hg1−x Te is developed within
the framework of precise solution of the Boltzmann equation. There is good agreement with experimental data in the temperature range T = 120–300 K.
References [1] W. Szymanska, T. Dietl, J. Phys. Chem. Solids 39 (1978) 1025– 1040. [2] J. Dubowski, T. Dietl, W. Szymanska, J. Phys. Chem. Solids 42 (1981) 351–362. [3] O.P. Malyk, Ukr. Phys. Zhur. 47 (2002) 842–845. [4] O.P. Malyk, J. Alloys Compd. 371 (2003) 146–149. [5] V.L. Bonch-Bruevich, S.G. Kalashnikov, Physics of Semiconductors, Nauka, Moscow, 1977, pp. 391–394 (in Russian). [6] O.P. Malyk, Ukr. Phys. Zhur. 35 (1990) 1374–1376. [7] R.G. Mani, J.R. Anderson, J.B. Choi, D.A. Nelson, Phys. Rev. B: Condens. Matter. 36 (1987) 9146–9149.
Inelastic electron-polar optical phonon scattering in the solid solution Cdx Hg1−x Te O.P. Malyk∗ Semiconductor Electronics Department, Lviv Polytechnic National University, Bandera Street 12, 79013 Lviv, Ukraine Received 16 January 2004; received in revised form 17 February 2004; accepted 17 February 2004
Abstract A model of inelastic electron-polar optical phonon scattering in the solid solution Cdx Hg1−x Te is proposed in which the scattering probability does not depend on a macroscopic parameter, the crystal permittivity. The given model yields good agreement between theory and experiment in temperature range 120–300 K. © 2004 Elsevier B.V. All rights reserved. PACS: 72.10.-d Keywords: Semiconductors; Electronic transport
1. Introduction The electron-polar optical phonon scattering in the solid solution Cdx Hg1−x Te was considered in relaxation time approximation in [1,2]. In [3,4], this scattering mechanism was considered for the case x = 0 taking into account the inelastic character of the scattering within the framework of a precise solution of the stationary Boltzmann equation. It was shown that the usage of a standard model of electron-polar optical phonon scattering leads to a disagreement between the theory and experiment in temperature range T > 100 K that is in that temperature range where this mechanism is dominating. According to our opinion, this model has the following shortcomings: (1) the use of a macroscopic parameter, the permittivity, is not reasonable in a microscopic processes and (2) the interaction potential of an electron with the optical oscillations of a crystal lattice is long-range which contradicts the special relativity. The purpose of the present work is the build-up of such a model of scattering which firstly would well match with experiment and secondly would not have the above mentioned shortcomings.
∗
Tel.: +380-322-398627. E-mail address: [email protected] (O.P. Malyk).
0925-8388/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jallcom.2004.02.039
2. The model of the electron-polar optical phonon scattering Let us consider a displacement of the jth (j = 1 and 2) atom in the unit cell of a crystal with zinc-blende structure under the influence of optical oscillations [5] 1/2 h ¯ 1 Qj = √ [ξj (q, ν)bq,ν eiqρ G q,ν 2Mων (q) ∗ +ξj∗ (q, ν)bq,ν e−iqρ ],
(1)
where G is the number of unit cells in a crystal volume; M = xMCd + (1 − x)MHg + MTe is the mass of the unit cell; q and ωv (q) are wave vector and angular frequency of the vth branch of optical oscillations in the crystal (v = 4, 5, 6); ξ j are polarization vectors of crystal oscillations; ∗ are phonon annihilation and creation operators bq,v and bq,ν respectively of the vth branch with wave vector q; ρ = i(n2 + n3 )(a0 /2) + j(n1 + n3 )(a0 /2) + k(n2 + n1 )(a0 /2), (n1 , n2 , n3 = 1, 2, . . . ), a0 is the lattice constant; i, j, k are unit vectors along the principal crystal axes. Under optical oscillations in a unit cell a polarization vector arises: P=
e(Q1 − Q2 ) , V0
(2)
O.P. Malyk / Journal of Alloys and Compounds 379 (2004) 60–63
where V0 = a0 /4 is the volume of the unit cell; e is the elementary charge. Using Eq. (1) and taking into account only of long wave length (q → 0) oscillations one can obtain
P =
4e a03
q,ν
h ¯ 2GMων (q)
12 (ξ 1 (q, ν) − ξ 2 (q, ν))[bq,ν eiqρ
∗ + bq,ν e−iqρ ].
(3)
It must be noticed that the polarization vector is a function of discrete variables P = P(n1 , n2 , n3 ). To calculate the bound charge ρ = −divP let us use the following expression of a partial derivative of the polarization vector ∂Px Px (n1 + 1, n2 , n3 ) − Px (n1 , n2 , n3 ) → ∂x x Px (n1 , n2 + 1, n3 ) − Px (n1 , n2 , n3 ) + x Px (n1 , n2 , n3 + 1) − Px (n1 , n2 , n3 ) + , x
(4)
1/2 ρ h ¯ 8ie ∇ ϕ =− = 3 ε0 2Gων (q) a 0 ε0 q xMCd + (1 − x)MHg + MTe 1/2 × {xMCd + (1 − x)MHg }MTe
ρ 2ε0
(5)
1 R2 − r 2 , 3
0≤r≤R
(qx + qy + qz )2 [bq eiqρ + bq∗ e−iqρ ]. q
(7)
It should be borne in mind that the potential Eq. (7) is short-range as it takes into account the interaction of an electron only with one unit cell. To calculate the transition probability connected with electron–phonon interaction let us write the wave function of the electron–phonon system in the form 1 Ψ = √ exp(ikr)Φ(x1 , x2 , . . . , xn ), (8) V
Nq , k |U|Nq , k 12 4ie2 h ¯ 1 = 3 exp(isr) R2 − r 2 dr 3 2Gων (q) a 0 ε0 V q xMCd + (1 − x)MHg + MTe 1/2 (qx + qy + qz )2 × (xMCd + (1 − x)MHg )MTe q × Φ∗ (x1 , x2 , . . . , xn )[bq eiqρ + bq∗ e−iqρ ] s = k − k (9)
where the relation qi (a0 /2) 1(i = x, y, z) is used and only optical longitudinal vibrations are taken into account, ε0 is the dielectric constant. To solve Eq. (5), let us substitute the unit cell by a sphere of effective radius R = γa0 the magnitude of which lies within the limits of half of the smaller diagonal up √ to half of the larger diagonal of a unit cell (0.5 < γ < 3/2). The magnitude of γ is chosen so to adjust the theory with experiment. The spherically symmetric solution of a Poisson equation looks like ϕ=
×
2
× Φ(x1 , x2 , . . . , xn ) dx1 , dx2 , . . . , dxn ,
2
(qx + qy + qz )2 [bq eiqρ + bq∗ e−iqρ ], q
12 h ¯ 1 2 R − r U = −eϕ = 3 3 2Gων (q) a 0 ε0 q 12 xMCd + (1 − x)MHg + MTe × (xMCd + (1 − x)MHg )MTe 4ie2
where V is the crystal volume; Φ (x1 , x2 , . . . , xn ) is the wave function of the system of independent harmonic oscillators. Then a transition matrix element from interaction energy looks like
where x = a0 /2 for a unit cell of the zinc-blende structure. A similar relation can be written for the partial derivatives (∂Py /∂y) and (∂Pz /∂z) with y = z = a0 /2. Then Poisson equation for a scalar potential associated with crystal oscillations becomes
×
61
(6)
Then the interaction energy of an electron with polar optical oscillations of the lattice is obtained from the expression
The integration over the electron coordinates is carried out within the limits of the unit cell and leads to 1 2 2 I(s) = exp(isr) R − r dr 3 π(8 sinRs − 8Rs cos Rs − 8/3R3 s3 cos Rs) (10) s5 The calculation shows that the electron wave vector (and s together with it) varies within the limits from 0 up to 109 m−1 at energy changing from 0 up to 10kB T (kB is the Boltzmann constant) at the temperature range 4.2–300 K. In Fig. 1, the dependence of the value I(s)/I(0) on s is shown. It is seen that for the indicated limits of the wave vector variation the following relation is well fulfilled =
I(s) ≈ I(0) =
5 16 15 πR
=
5 5 16 15 πa0 γ
The integration over oscillators coordinates the harmonic gives the factors Nq and Nq + 1 (Nq is the number of phonons with a frequency ω(q) = ω0 at q → 0) for phonon annihilation and creation operators respectively. To calculate the sum over the vector q let us make the following simplifications (1). We take into consideration a quasicontinuous
62
O.P. Malyk / Journal of Alloys and Compounds 379 (2004) 60–63
sorption and radiation W(k, k ) =
64π7 γ 10 e4 xMCd + (1 − x)MHg + MTe 225ε20 a04 Gω0 (xMCd + (1 − x)MHg )MTe
× [Nq δ(ε − ε − h ¯ ω0 ) + (Nq + 1) ¯ ω0 )], × δ(ε − ε + h
(13) nm Kβαab
from a prewhere ε is the electron energy.The values cise solution of the stationary Boltzmann equation for intraand inter-band electron transitions can now be obtained [3,4] nm Kβα11 =−
Fig. 1. The dependence of function I(s)/I(0) on s = |k − k |.
xMCd + (1 − x)MHg + MTe δαβ (xMCd + (1 − x)MHg )MTe × Nq f0 (ε) × [1 − f0 (ε + h ¯ ω0 )]k2 (ε + h ¯ ω0 )
×
character of wave vector variation and pass from summation to integration over q and (2) We change from integration of a cube with diameter 2π/a0 to integration of a sphere with effective radius π/a0 π/a0 π/a0 π/a0 V ··· → · · · dqx dqy dqz (2π)3 −π/a0 −π/a0 −π/a0 q 2π π π/a0 → . . . q2 sin θ dq dθ dϕ (11) 0
0
q
8 cos ρQ + 8ρQ sin ρQ − 4ρ2 Q2 cos ρQ − 8 ρ4 Nq − absorption, × . Nq + 1 − radiation, Q = π/a0
=π
(12)
∂k(ε + h ¯ ω0 ) ¯ ω0 )f0 (ε) + (Nq + 1)θ(ε − h ∂ε
∂k(ε − h ¯ ω0 ) 2 × [1 − f0 (ε− h ¯ ω0 )] × k (ε − h ¯ ω0 ) ∂ε ∂k(ε) εn+m dε; × k4 (ε) ∂ε ¯ 2 γ 10 2V 32π6 e4h 2mhh 3/2 =− (2π)3 675ε20 a0 ω0 kB T h ¯2 xMCd + (1 − x)MHg + MTe × δαβ (xMCd + (1 − x)MHg )MTe × (Nq + 1) × f0 (ε)[1 − f0 (ε − h ¯ ω0 )] ×
0
Then we obtain for the sum the following expression · · · = F(ρ)
¯ 2 γ 10 2V 64π6 e4h (2π)3 675ε20 a0 ω0 kB T
nm Kβα12
¯ ω0 )1/2 k4 (ε) × (−ε − εg + h
∂k(ε) n+m ε dε, ∂ε
In Fig. 2, the dependence of the value F(ρ)/F(0) on ρ is shown. It follows that the function F(ρ) can be approximated by the expression Nq 4 F(0) = πQ . (12a) Nq + 1
where δαβ is the Kronecker delta; f0 (ε) is the Fermi–Dirac function; θ(x) is a step function.
After some calculations we can obtain the expression for electron transition probability connected with phonon ab-
A comparison of the theoretical temperature dependences of the electron mobility was made with the experimental data presented in [2] for Cdx Hg1−x Te crystals with compositions x = 0.08; 0.095; 0.12; 0.135 and 0.17. The corresponding parameter values for these compositions are γ = 0.6708, 0.6952, 0.7103, 0.7125 and 0.7125. The calculation of the theoretical electron mobilities µ(T) was made on the basis of electroneutrality equation: n − p = Nd , where Nd = n4.2 is the electron concentration at 4.2 K. In these calculations, the same scattering mechanisms and crystal parameters as in [2,3,6,7] were taken into consideration. The theoretical µ(T) curves are presented in Fig. 3a–e. The dashed lines represent the curves calculated on the basis of the long-range model within the framework of the precise solution of the Boltzmann equation. It is seen that the theoretical curve well agree with the experimental data (deviation about ∼10–20%) in
Fig. 2. The dependence of function F(ρ)/F(0) on ρ.
3. Discussion
O.P. Malyk / Journal of Alloys and Compounds 379 (2004) 60–63
63
Fig. 3. Temperature dependence of the electron mobility in Cdx Hg1−x Te crystals. (Solid line) Mixed scattering mechanism (short-range model); (dashed line) mixed scattering mechanism (long-range model); (curve 1) nonpolar; (curve 2) acoustic; (curve 3) piezo-acoustic; (curve 4) disorder; (curve 5) intraband polar; and (curve 6) ionized impurity scattering mechanisms.
the temperature range T = 120–300 K whereas the curves calculated on the basis of the long-range model differ from experiment in 1.9–3.8 times. To estimate the role of the different scattering mechanisms in Fig. 3e the dotted lines represent the appropriate dependences for a composition x = 0.17. It is seen that in all investigated temperature intervals the main scattering mechanism is intraband scattering on polar optical phonons. Other scattering mechanisms such as acoustic scattering, piezo-acoustic scattering, nonpolar optical phonon scattering, disorder scattering, ionized impurity scattering, give negligibly small contributions. A similar situation exists for crystals of other compositions.
4. Conclusion A short-range model of inelastic electron scattering on polar optical phonons in Cdx Hg1−x Te is developed within
the framework of precise solution of the Boltzmann equation. There is good agreement with experimental data in the temperature range T = 120–300 K.
References [1] W. Szymanska, T. Dietl, J. Phys. Chem. Solids 39 (1978) 1025– 1040. [2] J. Dubowski, T. Dietl, W. Szymanska, J. Phys. Chem. Solids 42 (1981) 351–362. [3] O.P. Malyk, Ukr. Phys. Zhur. 47 (2002) 842–845. [4] O.P. Malyk, J. Alloys Compd. 371 (2003) 146–149. [5] V.L. Bonch-Bruevich, S.G. Kalashnikov, Physics of Semiconductors, Nauka, Moscow, 1977, pp. 391–394 (in Russian). [6] O.P. Malyk, Ukr. Phys. Zhur. 35 (1990) 1374–1376. [7] R.G. Mani, J.R. Anderson, J.B. Choi, D.A. Nelson, Phys. Rev. B: Condens. Matter. 36 (1987) 9146–9149.