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Phonon emission in a degenerate semiconductor at low lattice temperatures
Physica B ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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Phonon emission in a degenerate semiconductor at low lattice temperatures S. Midday, S. Nag, D.P. Bhattacharya n Faculty of Science, Department of Physics, Jadavpur University, Kolkata 700 032, India
art ic l e i nf o
a b s t r a c t
Article history: Received 6 August 2014 Received in revised form 27 October 2014 Accepted 30 October 2014
The characteristics of phonon growth in a degenerate semiconductor at low lattice temperatures have been studied for inelastic interaction of non-equilibrium electrons with the intravalley acoustic phonons. The energy of the phonon and the full form of the phonon distribution are taken into account. The results reveal significant changes in the growth characteristics compared to the same for a non-degenerate material. & 2014 Published by Elsevier B.V.
Keywords: Degenerate semiconductor Phonon growth Low temperature
1. Introduction At low lattice temperatures TL ≤ 20 K , the free electrons in a high purity covalent semiconductor interact dominantly only with intravalley deformation acoustic phonons. Under this condition, the electron ensemble may be significantly perturbed from the state of thermodynamic equilibrium even for a field of only a few V/cm or less [1–3]. The electrons then attain an effective temperature Te which is higher than the lattice temperature TL, and so emit more phonons per unit time compared to how much they absorb. This leads to a growth in the number of phonons. In the theoretical analysis for the same interaction one traditionally neglects the phonon energy εph compared to the carrier energy εk and thus assumes electron–phonon collisions to be elastic and also approximates the phonon distribution by the equipartition law. Considering the long wavelength acoustic phonon of wave vector q, the phonon energy εph may be taken to be ħu lq, where ħ¼h/2π, h being Planck's constant and ul is the velocity of longitudinal acoustic mode in the material. Thus εph/εk turns out to be of the order of the ratio of the acoustic velocity ul to the average thermal velocity of the carriers. Hence, such traditional simplifications can be made only at higher temperatures. At lower temperatures, whenever the average thermal velocity becomes comparable with the acoustic velocity, the electron–phonon interaction can neither be taken to be elastic, nor the phonon distribution be approximated by the equipartition law. In an earlier theoretical analysis made by one of the present authors and n
Corresponding author. E-mail address: [email protected] (D.P. Bhattacharya).
another, it has been seen how do these traditional approximations like the elastic interaction and the equipartition law, lead to significant errors in determining the phonon growth characteristics of a non-degenerate semiconductor material at low lattice temperatures [4]. With more and more doping, when the electron concentration of an n-type material exceeds the effective density of states, the Fermi level εF moves into the conduction band and the material can no longer be treated as a non-degenerate one. At low lattice temperatures, when εF is not much lower than kBTL of the band edge (kB being the Boltzmann constant), and the electron densities are beyond the insulator-to-metal transition, the free electron ensemble in the material may be considered as degenerate [5]. A rough estimate of the critical concentration of the donor density ND for the onset of the degeneracy may be obtained from
ε F = (ℏ2/2m⁎)(3π 2ND )2/3 > Ed where m* is the effective mass of an electron and Ed is the donor binding energy [6]. The degeneracy is said to be extreme when εF largely exceeds kBTL [5]. The purpose of this communication is to make an analysis for the phonon growth for the case of inelastic interaction of an electron with intravalley acoustic phonons in a degenerate semiconductor at low lattice temperatures. The analysis is made here under the condition when neither the phonon energy can be neglected in comparison to the carrier energy, nor can the phonon distribution Nq be represented by the equipartition law. The net rate of increase of the number of phonons (∂Nq/∂t) with wave vector q due to emission and absorption by all the carriers in the
http://dx.doi.org/10.1016/j.physb.2014.10.027 0921-4526/& 2014 Published by Elsevier B.V.
Please cite this article as: S. Midday, et al., Physica B (2014), http://dx.doi.org/10.1016/j.physb.2014.10.027i
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degenerate material is calculated. The numerical results which are obtained from the present analysis for some degenerate samples of Si are then compared with the rates which have been reported earlier for the non-degenerate materials in the same framework satisfying the low temperature conditions [4].
2. Development We consider a degenerate ensemble of non-equilibrium electrons in a volume V of a semiconductor. As a result of interaction with intravalley acoustic phonons let there be electronic transition → → → between the two wave vector states k and k + q , with attendant emission and absorption of a phonon. On applying the time-dependent perturbation theory to the electron–phonon system one can see that the rate of increase in the number of phonons is given by [1]
∂Nq ∂t
=
→ → 2π ∑ M k, k + → q ℏ →
(
k
2
) δ (ε
→ ⎤ ⎡ 1−f k ⎥− ⎣⎢ ⎦ → ⎡ → ⎢⎣1 − f k + q
→, k
2
→ → ( ) M ( k + →q , k ) δ (ε ⎤ ( )⎥⎦
⎛→ → →⎞ where M ⎜ k , k + q ⎟ ⎝ ⎠ given by
→ → → M k, k + q
(
)
→ → → → N→ q + 1 − ε k +→ q , Nq f k + q
2
=
→ →, k +q
)(
)
→ → → N→ q − 1 − εk , Nq f k
)( ) (1)
2
is the square of the matrix element and is
⎡ → ⎤ E12ℏq ⎢ N q ⎥ → 2Vρul ⎢⎣ N q + 1⎥⎦
(2)
E1 is the deformation potential constant, ρ is the density of the ⎛→⎞ material and f ⎜ k ⎟ is the distribution function for the non-equili⎝ ⎠ brium electrons. It is well known that the non-equilibrium distribution function ⎛→⎞ ⎛→⎞ f ⎜ k ⎟ can be taken as a sum of a spherically symmetric term f0 ⎜ k ⎟ ⎝ ⎠ ⎝ ⎠ and a small additional term f1 which gives the asymmetry in the field direction. In the analysis what follows, the contribution of ⎛→⎞ this small asymmetry may be neglected and f0 ⎜ k ⎟ is taken to be ⎝ ⎠ Fermi Dirac function with an effective electron temperature Te [1,5]. → In performing the summation over k in (1) it is now required to have a knowledge about the energy band structure of the semiconductor material and the form of the phonon distribution function N→ q. Bulk Si is actually a many valley semiconductor, having more than one conduction band minimum. But, although the complex form of the band structure is known to lead to some non-trivial results, in many cases of practical interest however, the band anisotropy and the many valley effects give rise to only some changes in the numerical coefficients of the transport characteristics. Moreover, in the region of low temperature of our consideration here, the electrons are mostly confined to the lowest valley. The energy band for Si is not parabolic either. But in the region of low lattice temperatures of interest here, when the electrons are scattered by acoustic phonons, the electron temperature may assume a value of the order of 100 times the lattice temperature for an electric field of only a few V/cm. Under these circumstances, the energy which an electron may gain or lose in a single scattering event amounts to of the order of 0.01 eV. Thus the electrons will be mainly confined to a short segment of the energy dispersion
curve where the nature of the curve can ordinarily be regarded as parabolic. Thus, the calculations here may be carried out taking into account only a single, parabolic, spherically symmetric conduction band without any serious loss of accuracy [1,2]. For normal collisions of the electron–phonon system, the total crystal momentum is conserved. In the umklapp process on the other hand, an electron on absorbing a phonon arrives in a state at the boundary of the Brillouin zone, whereupon it suffers a reflection, and the initial and final crystal momenta differ by a nonzero reciprocal lattice vector. But it is well known that at the low temperatures the scattering processes that can occur at an appreciable rate are usually the normal processes, the umklapp processes are frozen out [7]. Moreover, the umklapp processes are hardly known to disturb the equilibrium phonon population at low temperatures and the rather strong anharmonicity of the lattice forces maintains the phonon equilibrium in high purity materials of interest here [1,8]. Hence the tendency of the phonon accumulation around the Brillouin zone states and their nonequilibrium population eventually influencing the generation rate may be set aside for the purpose of the present analysis. In the presence of a relatively high electric field, when the electron ensemble is perturbed from the state of thermodynamic equilibrium, the power supplied to the electrons must be transmitted to the phonons, and this should reasonably lead to a perturbation of the equilibrium phonon distribution. However, the deviation of the phonon distribution from its equilibrium value as a result of such perturbation of the electron ensemble is actually known to be an effect of the second order only [1,2,9]. Again, be it due to the phonon–phonon interaction or to the boundary scattering, the normal collision process yields a relaxation time which is usually the shortest among the relevant relaxation times. Thus under the condition of low temperature, even when the electron ensemble gets heated up for a field of just a few V/cm, one can assume that the phonon ensemble still remains unperturbed and thus follow the equilibrium Bose Einstein distribution function [1,2]. → The summation over k in (1) may be converted to an in→ tegration of the spherical polar coordinates k, θ and ϕ with q direction taken along the Z-axis. Considering the spin degeneracy into account and integrating over the polar and azimuthal angles one may obtain
∂Nq ∂t
=
E12 m⁎ 2πρℏ2ul ⎡ → → ⎢ (Nq + 1) f0 k + q k ⎣
→
→
){1 − f ( k )} − N f ( k )
(
∫
q 0
0
⎤
→
{1 − f ( k + →q )}⎥⎦k dk 0
(3)
At the low temperatures of interest here, the phonon energy is to be taken into account for the inelastic interaction which is being considered here. The lower and upper limits for the integration over k may be obtained from the energy and momentum balance equation as (q /2 − m⁎ul /ℏ) and ∞ respectively. Now carrying out the integration over k one can obtain
⎛ ∂Nq ⎞ E 2 m⁎ 2 k B Te [Ι1 − Ι2 − Ι3 ] ⎜ ⎟ = 1 ⎝ ∂t ⎠deg 2πρℏ4ul
(4)
where I1 = (Nq + 1) ln [1 + exp (β − (x/Tn ))]; I2 = Nq ln λ ; I3 = ((1/λ)
(
)
+ ln λ − 1) + (x/Tn ) (1/λ) − (1/2λ2) − (1/2) ; Nq = (e x − 1)−1; the full phonon distribution without the truncation to equipartition approximation, x = ℏul q /kB TL ; β = η − a (x − b)2; η = εF /kB Te ; a = kB TL /8m⁎ul2 Tn ; b = 2m⁎ul2 /kB TL ; Tn = Te/TL ; λ = 1 + exp (β)
Please cite this article as: S. Midday, et al., Physica B (2014), http://dx.doi.org/10.1016/j.physb.2014.10.027i
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3
It may be noted that when the Fermi level is below the conduction band edge and lies at a level much exceeding kB Te i.e. the material is non-degenerate [5], result (4) which is obtained here for a degenerate ensemble reduces to what has been reported earlier for the non-degenerate ensemble [4]. The result for the non-degenerate ensemble may be quoted here for a ready reference.
⎛ ⎛ ∂Nq ⎞ E 2 m⁎ 2 k B Te e η ⎜⎜1 + (Nq + 1) ⎜ ⎟ = 1 ⎝ ∂t ⎠non − deg 2πρℏ4ul ⎝
∞
∑ j=1
exp ⎡⎣−a (x − b)2⎤⎦
⎞ (−x/Tn ) j ⎟ ⎟ j! ⎠ (5)
When the material is highly degenerate the Fermi level lies at a level much higher than kB Te in the conduction band [5]. Under this condition, the rate of phonon growth (4) reduces to a simple form
⎛ ∂Nq ⎞ ⎤ E 2 m⁎ 2k B Te ⎡ x ⎜ ⎟ = 1 1 + 2Nq ) ⎥ ⎢1 − ( 4 2Tn ⎦ ⎝ ∂t ⎠high − deg 2πρℏ ul ⎣
(6)
3. Results and discussion Comparing (4) with (5) and (6) it may be noted that the effect of degeneracy on the phonon growth rate is quite complex and the degeneracy makes the dependence of (∂Nq/∂t) upon the normalized phonon wave vector x significantly different from what transpires had the degeneracy factor not been taken into account. The numerical results are calculated for samples of Si with the following parameter values: E1 = 9.0 eV ; ρ = 2.329 × 103 kg m−3; ul = 9.037 × 103 m s−1; m⁎ = 0.32m0 , m0 being the free electron mass. The results obtained for the dependence of the phonon growth rate (∂Nq/∂t) upon x at lattice temperatures of 1 and 20 K for values of Tn ¼ 2.5 and 10 and of the degeneracy parameter εF /kBTL ¼ 5 and 15, are represented in Figs. 1 and 2. Again, for a quick reference to the changes that are effected in the phonon growth characteristics of the material due to degeneracy, relative to the same for the non-degenerate material under
Fig. 1. For the degeneracy parameter εF/kBTL ¼ 5 the dependence of the phonon growth rate (∂Nq/∂t) upon the phonon wave vector x in a sample of Si for different values of TL, the lattice temperature and Tn = (Te/TL ) , the effective electron temperature normalized to TL. The curves 1 and 2 are for Tn ¼2.5 and 10 respectively. The curves marked a or c and b or d correspond to the lattice temperature of 1 and 20 K respectively. The solid curves represent the results one obtains considering the degeneracy of the ensemble and the dashed ones follow on neglecting the same.
Fig. 2. For the degeneracy parameter εF/kBTL ¼ 15 the dependence of the phonon growth rate (∂Nq/∂t) upon the phonon wave vector x in a sample of Si for different values of TL, the lattice temperature and Tn = (Te/TL ) , the effective electron temperature normalized to TL. The curves 1 and 2 are for Tn ¼2.5 and 10 respectively. The curves marked a or c and b or d correspond to the lattice temperature of 1 and 20 K respectively. The solid curves represent the results one obtains considering the degeneracy of the ensemble and the dashed ones follow on neglecting the same.
the condition of low temperature, one can normalize expression (4) to expression (5) and thus obtains (∂Nq/∂t)norm . The dependence of such normalized rate upon x at the lattice temperatures of 1, 4 and 20 K, for Tn ¼ 2.5, 4 and 20, and for εF/kBTL ¼5 and 15 are thus shown in Figs. 3 and 4. The figures reveal that the degeneracy factor brings in significant qualitative as well as quantitative changes in the phonon growth characteristics under the condition of low lattice temperature. The consideration of the degeneracy of the material makes the rate of phonon growth fall significantly below that of the non-degenerate material for lower values of the phonon wave vector x. The fall is again greater the greater is the value of the degeneracy parameter εF/kBTL. The discrepancy between (∂Nq/∂t)deg and (∂Nq/∂t)non − deg characteristics at any x decreases
Fig. 3. For the degeneracy parameter εF/kBTL ¼ 5 the dependence of the phonon growth rate (∂Nq/∂t)deg as normalized to the rate (∂Nq/∂t)non − deg , upon the phonon wave vector x, in a sample of Si for different values of the lattice temperature TL and the normalized electron temperatureTn = (Te/TL ) . The curves 1, 2 and 3 are for Tn ¼2.5, 5, and 10 respectively, and those marked a, b, and c correspond to the lattice temperature of 1, 4, and 20 K respectively.
Please cite this article as: S. Midday, et al., Physica B (2014), http://dx.doi.org/10.1016/j.physb.2014.10.027i
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much that the high degeneracy condition is hardly satisfied. If εF /kBTL is small, such negative value of (∂Nq/∂t) can hardly occur. Again, in the range for lower values of x, when the equipartition approximation holds good, (∂Nq/∂t) seems to be almost always independent of x. The above theory may be used for subsequent calculation of the non-ohmic transport characteristics of degenerate semiconductors at low lattice temperatures. The theory that has been developed so far in this communication is based on a number of simplifying assumptions which may be easily validated in suitably designed experiments. But there being a dearth of experimental data, the phonon growth characteristics that are obtained here could not be compared with the experimental findings. However, the results seem to be encouraging and thus prompt further development of the low temperature transport theory in degenerate materials.
4. Conclusion Fig. 4. For the degeneracy parameter εF/kBTL ¼15 the dependence of the phonon growth rate (∂Nq/∂t)deg as normalized to the rate (∂Nq/∂t)non − deg , upon the phonon wave vector x in a sample of Si for different values of the lattice temperature TL and the normalized electron temperatureTn = (Te/TL ) . The curves 1, 2 and 3 are for Tn ¼2.5, 5, and 10 respectively, and those marked a, b, and c correspond to the lattice temperature of 1, 4, and 20 K respectively.
with the increase in x and eventually, as expected, the characteristics of the phonon growth rate for the degenerate material tend to be the same as that for the non-degenerate material for higher values of x where the effects of degeneracy are hardly manifested. The discrepancy is vividly greater the lower is the value of the normalized electron temperature Tn and continues for higher values of x the lower is the lattice temperature. The discrepancy at any Tn is quite significantly susceptible to any change in the degeneracy factor, and assumes greater and greater values the greater is the factor. It may be noted also that the relative changes in the (∂Nq/∂t) characteristics that are effected at the lower values of x are mainly due to the degeneracy factor, whereas those at higher values of x are mainly due to inelasticity of the collisions and the true phonon distribution. In the lower range of the normalized phonon wave vector x, so long as the degeneracy parameter εF/kBTL remains greater than x, the material seems to be degenerate and the effect of degeneracy on both the qualitative and quantitative aspects of the growth characteristics is significantly manifested. Beyond that, as x increases, the material gradually turns out to be more and more non-degenerate and the growth characteristics starting from a relatively low value, asymptotically approaches the same for the non-degenerate material. For higher values of the degeneracy parameter and at low lattice temperatures the growth curves display rather sharp variation around the transition region. Over this region as x increases the effect of degeneracy gradually becomes weaker and the growth rate tends to increase. But the finite value of the phonon energy and the true phonon distribution now begin to influence the growth rate more and more and thus make it decrease. As a result the growth rate here may exhibit some transients in the form extrema, particularly when the temperature is quite low. For highly degenerate materials it can be seen from (6) that (∂Nq/∂t) may turn out to be negative if x > 2Tn . It may be noted that for the high values of the degeneracy parameter like 15 or so, the growth rate for Tn ¼ 2.5 drops quite fast with increase in x and, indeed tends to be negative for x 45. For Tn 42.5, η decreases so
Following the standard perturbation theory, the phonon emission characteristics in a degenerate material have been analyzed here. The lattice temperature has been assumed to be low enough, so that none of the traditional assumptions, like elastic collisions with intravalley acoustic phonons and equipartition law for the phonon distribution could be made. The explicit expressions obtained for the phonon growth rate describe how does the degeneracy of the material effect changes in the growth characteristics for various values of the lattice temperature TL and the effective electron temperature Te. Numerical results have been obtained for samples of bulk Si. The results which thus follow show that under the condition of low temperature, the degeneracy factor of the material beings in significant changes in the growth characteristics compared to what follows for the non-degenerate materials under the same prevalent conditions of low temperature. The results seem to be interesting and thus stimulate further studies on the non-ohmic transport in degenerate materials under the low temperature conditions.
Acknowledgments S. Midday is indebted to the University Grants Commission, New Delhi, India, for the award of a Teacher fellowship under the faculty improvement program. D.P. Bhattacharya is grateful to the same commission for their support on being awarded Emeritus Fellowship. The authors acknowledge the help rendered by B. Das and A. Basu.
References [1] E.M. Conwell, High Field Transport in Semiconductors, Academic Press, New York, 1967. [2] Z.S. Kachlishvili, Phys. Status Solidi (a) 88 (1976) 15. [3] S. Midday, D.P. Bhattacharya, J. Phys. Chem. Solids 72 (2011) 1343. [4] N. Chakraborty, D.P. Bhattacharya, J. Phys. Chem. Solids 57 (1995) 653. [5] B.R. Nag, Electron Transport in Compound Semiconductors, Springer-Verlag, Berlin, 1980. [6] S.M. Sze, Physics of Semiconductor Devices, Wiley Eastern Limited, New Delhi, 1983. [7] N.W. Aschroft, N.D. Mermin, Solid State Physics, CBS Publishing Japan, Ltd., Tokyo, 1981. [8] P.G. Klemens, Proc. Phys. Soc. (Lond.) A 64 (1951) 1030. [9] A. Anselm, Introduction to Semiconductor Theory, Mir Publisher, Moscow, 1981.
Please cite this article as: S. Midday, et al., Physica B (2014), http://dx.doi.org/10.1016/j.physb.2014.10.027i
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Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Phonon emission in a degenerate semiconductor at low lattice temperatures S. Midday, S. Nag, D.P. Bhattacharya n Faculty of Science, Department of Physics, Jadavpur University, Kolkata 700 032, India
art ic l e i nf o
a b s t r a c t
Article history: Received 6 August 2014 Received in revised form 27 October 2014 Accepted 30 October 2014
The characteristics of phonon growth in a degenerate semiconductor at low lattice temperatures have been studied for inelastic interaction of non-equilibrium electrons with the intravalley acoustic phonons. The energy of the phonon and the full form of the phonon distribution are taken into account. The results reveal significant changes in the growth characteristics compared to the same for a non-degenerate material. & 2014 Published by Elsevier B.V.
Keywords: Degenerate semiconductor Phonon growth Low temperature
1. Introduction At low lattice temperatures TL ≤ 20 K , the free electrons in a high purity covalent semiconductor interact dominantly only with intravalley deformation acoustic phonons. Under this condition, the electron ensemble may be significantly perturbed from the state of thermodynamic equilibrium even for a field of only a few V/cm or less [1–3]. The electrons then attain an effective temperature Te which is higher than the lattice temperature TL, and so emit more phonons per unit time compared to how much they absorb. This leads to a growth in the number of phonons. In the theoretical analysis for the same interaction one traditionally neglects the phonon energy εph compared to the carrier energy εk and thus assumes electron–phonon collisions to be elastic and also approximates the phonon distribution by the equipartition law. Considering the long wavelength acoustic phonon of wave vector q, the phonon energy εph may be taken to be ħu lq, where ħ¼h/2π, h being Planck's constant and ul is the velocity of longitudinal acoustic mode in the material. Thus εph/εk turns out to be of the order of the ratio of the acoustic velocity ul to the average thermal velocity of the carriers. Hence, such traditional simplifications can be made only at higher temperatures. At lower temperatures, whenever the average thermal velocity becomes comparable with the acoustic velocity, the electron–phonon interaction can neither be taken to be elastic, nor the phonon distribution be approximated by the equipartition law. In an earlier theoretical analysis made by one of the present authors and n
Corresponding author. E-mail address: [email protected] (D.P. Bhattacharya).
another, it has been seen how do these traditional approximations like the elastic interaction and the equipartition law, lead to significant errors in determining the phonon growth characteristics of a non-degenerate semiconductor material at low lattice temperatures [4]. With more and more doping, when the electron concentration of an n-type material exceeds the effective density of states, the Fermi level εF moves into the conduction band and the material can no longer be treated as a non-degenerate one. At low lattice temperatures, when εF is not much lower than kBTL of the band edge (kB being the Boltzmann constant), and the electron densities are beyond the insulator-to-metal transition, the free electron ensemble in the material may be considered as degenerate [5]. A rough estimate of the critical concentration of the donor density ND for the onset of the degeneracy may be obtained from
ε F = (ℏ2/2m⁎)(3π 2ND )2/3 > Ed where m* is the effective mass of an electron and Ed is the donor binding energy [6]. The degeneracy is said to be extreme when εF largely exceeds kBTL [5]. The purpose of this communication is to make an analysis for the phonon growth for the case of inelastic interaction of an electron with intravalley acoustic phonons in a degenerate semiconductor at low lattice temperatures. The analysis is made here under the condition when neither the phonon energy can be neglected in comparison to the carrier energy, nor can the phonon distribution Nq be represented by the equipartition law. The net rate of increase of the number of phonons (∂Nq/∂t) with wave vector q due to emission and absorption by all the carriers in the
http://dx.doi.org/10.1016/j.physb.2014.10.027 0921-4526/& 2014 Published by Elsevier B.V.
Please cite this article as: S. Midday, et al., Physica B (2014), http://dx.doi.org/10.1016/j.physb.2014.10.027i
67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102
S. Midday et al. / Physica B ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
degenerate material is calculated. The numerical results which are obtained from the present analysis for some degenerate samples of Si are then compared with the rates which have been reported earlier for the non-degenerate materials in the same framework satisfying the low temperature conditions [4].
2. Development We consider a degenerate ensemble of non-equilibrium electrons in a volume V of a semiconductor. As a result of interaction with intravalley acoustic phonons let there be electronic transition → → → between the two wave vector states k and k + q , with attendant emission and absorption of a phonon. On applying the time-dependent perturbation theory to the electron–phonon system one can see that the rate of increase in the number of phonons is given by [1]
∂Nq ∂t
=
→ → 2π ∑ M k, k + → q ℏ →
(
k
2
) δ (ε
→ ⎤ ⎡ 1−f k ⎥− ⎣⎢ ⎦ → ⎡ → ⎢⎣1 − f k + q
→, k
2
→ → ( ) M ( k + →q , k ) δ (ε ⎤ ( )⎥⎦
⎛→ → →⎞ where M ⎜ k , k + q ⎟ ⎝ ⎠ given by
→ → → M k, k + q
(
)
→ → → → N→ q + 1 − ε k +→ q , Nq f k + q
2
=
→ →, k +q
)(
)
→ → → N→ q − 1 − εk , Nq f k
)( ) (1)
2
is the square of the matrix element and is
⎡ → ⎤ E12ℏq ⎢ N q ⎥ → 2Vρul ⎢⎣ N q + 1⎥⎦
(2)
E1 is the deformation potential constant, ρ is the density of the ⎛→⎞ material and f ⎜ k ⎟ is the distribution function for the non-equili⎝ ⎠ brium electrons. It is well known that the non-equilibrium distribution function ⎛→⎞ ⎛→⎞ f ⎜ k ⎟ can be taken as a sum of a spherically symmetric term f0 ⎜ k ⎟ ⎝ ⎠ ⎝ ⎠ and a small additional term f1 which gives the asymmetry in the field direction. In the analysis what follows, the contribution of ⎛→⎞ this small asymmetry may be neglected and f0 ⎜ k ⎟ is taken to be ⎝ ⎠ Fermi Dirac function with an effective electron temperature Te [1,5]. → In performing the summation over k in (1) it is now required to have a knowledge about the energy band structure of the semiconductor material and the form of the phonon distribution function N→ q. Bulk Si is actually a many valley semiconductor, having more than one conduction band minimum. But, although the complex form of the band structure is known to lead to some non-trivial results, in many cases of practical interest however, the band anisotropy and the many valley effects give rise to only some changes in the numerical coefficients of the transport characteristics. Moreover, in the region of low temperature of our consideration here, the electrons are mostly confined to the lowest valley. The energy band for Si is not parabolic either. But in the region of low lattice temperatures of interest here, when the electrons are scattered by acoustic phonons, the electron temperature may assume a value of the order of 100 times the lattice temperature for an electric field of only a few V/cm. Under these circumstances, the energy which an electron may gain or lose in a single scattering event amounts to of the order of 0.01 eV. Thus the electrons will be mainly confined to a short segment of the energy dispersion
curve where the nature of the curve can ordinarily be regarded as parabolic. Thus, the calculations here may be carried out taking into account only a single, parabolic, spherically symmetric conduction band without any serious loss of accuracy [1,2]. For normal collisions of the electron–phonon system, the total crystal momentum is conserved. In the umklapp process on the other hand, an electron on absorbing a phonon arrives in a state at the boundary of the Brillouin zone, whereupon it suffers a reflection, and the initial and final crystal momenta differ by a nonzero reciprocal lattice vector. But it is well known that at the low temperatures the scattering processes that can occur at an appreciable rate are usually the normal processes, the umklapp processes are frozen out [7]. Moreover, the umklapp processes are hardly known to disturb the equilibrium phonon population at low temperatures and the rather strong anharmonicity of the lattice forces maintains the phonon equilibrium in high purity materials of interest here [1,8]. Hence the tendency of the phonon accumulation around the Brillouin zone states and their nonequilibrium population eventually influencing the generation rate may be set aside for the purpose of the present analysis. In the presence of a relatively high electric field, when the electron ensemble is perturbed from the state of thermodynamic equilibrium, the power supplied to the electrons must be transmitted to the phonons, and this should reasonably lead to a perturbation of the equilibrium phonon distribution. However, the deviation of the phonon distribution from its equilibrium value as a result of such perturbation of the electron ensemble is actually known to be an effect of the second order only [1,2,9]. Again, be it due to the phonon–phonon interaction or to the boundary scattering, the normal collision process yields a relaxation time which is usually the shortest among the relevant relaxation times. Thus under the condition of low temperature, even when the electron ensemble gets heated up for a field of just a few V/cm, one can assume that the phonon ensemble still remains unperturbed and thus follow the equilibrium Bose Einstein distribution function [1,2]. → The summation over k in (1) may be converted to an in→ tegration of the spherical polar coordinates k, θ and ϕ with q direction taken along the Z-axis. Considering the spin degeneracy into account and integrating over the polar and azimuthal angles one may obtain
∂Nq ∂t
=
E12 m⁎ 2πρℏ2ul ⎡ → → ⎢ (Nq + 1) f0 k + q k ⎣
→
→
){1 − f ( k )} − N f ( k )
(
∫
q 0
0
⎤
→
{1 − f ( k + →q )}⎥⎦k dk 0
(3)
At the low temperatures of interest here, the phonon energy is to be taken into account for the inelastic interaction which is being considered here. The lower and upper limits for the integration over k may be obtained from the energy and momentum balance equation as (q /2 − m⁎ul /ℏ) and ∞ respectively. Now carrying out the integration over k one can obtain
⎛ ∂Nq ⎞ E 2 m⁎ 2 k B Te [Ι1 − Ι2 − Ι3 ] ⎜ ⎟ = 1 ⎝ ∂t ⎠deg 2πρℏ4ul
(4)
where I1 = (Nq + 1) ln [1 + exp (β − (x/Tn ))]; I2 = Nq ln λ ; I3 = ((1/λ)
(
)
+ ln λ − 1) + (x/Tn ) (1/λ) − (1/2λ2) − (1/2) ; Nq = (e x − 1)−1; the full phonon distribution without the truncation to equipartition approximation, x = ℏul q /kB TL ; β = η − a (x − b)2; η = εF /kB Te ; a = kB TL /8m⁎ul2 Tn ; b = 2m⁎ul2 /kB TL ; Tn = Te/TL ; λ = 1 + exp (β)
Please cite this article as: S. Midday, et al., Physica B (2014), http://dx.doi.org/10.1016/j.physb.2014.10.027i
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3
It may be noted that when the Fermi level is below the conduction band edge and lies at a level much exceeding kB Te i.e. the material is non-degenerate [5], result (4) which is obtained here for a degenerate ensemble reduces to what has been reported earlier for the non-degenerate ensemble [4]. The result for the non-degenerate ensemble may be quoted here for a ready reference.
⎛ ⎛ ∂Nq ⎞ E 2 m⁎ 2 k B Te e η ⎜⎜1 + (Nq + 1) ⎜ ⎟ = 1 ⎝ ∂t ⎠non − deg 2πρℏ4ul ⎝
∞
∑ j=1
exp ⎡⎣−a (x − b)2⎤⎦
⎞ (−x/Tn ) j ⎟ ⎟ j! ⎠ (5)
When the material is highly degenerate the Fermi level lies at a level much higher than kB Te in the conduction band [5]. Under this condition, the rate of phonon growth (4) reduces to a simple form
⎛ ∂Nq ⎞ ⎤ E 2 m⁎ 2k B Te ⎡ x ⎜ ⎟ = 1 1 + 2Nq ) ⎥ ⎢1 − ( 4 2Tn ⎦ ⎝ ∂t ⎠high − deg 2πρℏ ul ⎣
(6)
3. Results and discussion Comparing (4) with (5) and (6) it may be noted that the effect of degeneracy on the phonon growth rate is quite complex and the degeneracy makes the dependence of (∂Nq/∂t) upon the normalized phonon wave vector x significantly different from what transpires had the degeneracy factor not been taken into account. The numerical results are calculated for samples of Si with the following parameter values: E1 = 9.0 eV ; ρ = 2.329 × 103 kg m−3; ul = 9.037 × 103 m s−1; m⁎ = 0.32m0 , m0 being the free electron mass. The results obtained for the dependence of the phonon growth rate (∂Nq/∂t) upon x at lattice temperatures of 1 and 20 K for values of Tn ¼ 2.5 and 10 and of the degeneracy parameter εF /kBTL ¼ 5 and 15, are represented in Figs. 1 and 2. Again, for a quick reference to the changes that are effected in the phonon growth characteristics of the material due to degeneracy, relative to the same for the non-degenerate material under
Fig. 1. For the degeneracy parameter εF/kBTL ¼ 5 the dependence of the phonon growth rate (∂Nq/∂t) upon the phonon wave vector x in a sample of Si for different values of TL, the lattice temperature and Tn = (Te/TL ) , the effective electron temperature normalized to TL. The curves 1 and 2 are for Tn ¼2.5 and 10 respectively. The curves marked a or c and b or d correspond to the lattice temperature of 1 and 20 K respectively. The solid curves represent the results one obtains considering the degeneracy of the ensemble and the dashed ones follow on neglecting the same.
Fig. 2. For the degeneracy parameter εF/kBTL ¼ 15 the dependence of the phonon growth rate (∂Nq/∂t) upon the phonon wave vector x in a sample of Si for different values of TL, the lattice temperature and Tn = (Te/TL ) , the effective electron temperature normalized to TL. The curves 1 and 2 are for Tn ¼2.5 and 10 respectively. The curves marked a or c and b or d correspond to the lattice temperature of 1 and 20 K respectively. The solid curves represent the results one obtains considering the degeneracy of the ensemble and the dashed ones follow on neglecting the same.
the condition of low temperature, one can normalize expression (4) to expression (5) and thus obtains (∂Nq/∂t)norm . The dependence of such normalized rate upon x at the lattice temperatures of 1, 4 and 20 K, for Tn ¼ 2.5, 4 and 20, and for εF/kBTL ¼5 and 15 are thus shown in Figs. 3 and 4. The figures reveal that the degeneracy factor brings in significant qualitative as well as quantitative changes in the phonon growth characteristics under the condition of low lattice temperature. The consideration of the degeneracy of the material makes the rate of phonon growth fall significantly below that of the non-degenerate material for lower values of the phonon wave vector x. The fall is again greater the greater is the value of the degeneracy parameter εF/kBTL. The discrepancy between (∂Nq/∂t)deg and (∂Nq/∂t)non − deg characteristics at any x decreases
Fig. 3. For the degeneracy parameter εF/kBTL ¼ 5 the dependence of the phonon growth rate (∂Nq/∂t)deg as normalized to the rate (∂Nq/∂t)non − deg , upon the phonon wave vector x, in a sample of Si for different values of the lattice temperature TL and the normalized electron temperatureTn = (Te/TL ) . The curves 1, 2 and 3 are for Tn ¼2.5, 5, and 10 respectively, and those marked a, b, and c correspond to the lattice temperature of 1, 4, and 20 K respectively.
Please cite this article as: S. Midday, et al., Physica B (2014), http://dx.doi.org/10.1016/j.physb.2014.10.027i
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S. Midday et al. / Physica B ∎ (∎∎∎∎) ∎∎∎–∎∎∎
much that the high degeneracy condition is hardly satisfied. If εF /kBTL is small, such negative value of (∂Nq/∂t) can hardly occur. Again, in the range for lower values of x, when the equipartition approximation holds good, (∂Nq/∂t) seems to be almost always independent of x. The above theory may be used for subsequent calculation of the non-ohmic transport characteristics of degenerate semiconductors at low lattice temperatures. The theory that has been developed so far in this communication is based on a number of simplifying assumptions which may be easily validated in suitably designed experiments. But there being a dearth of experimental data, the phonon growth characteristics that are obtained here could not be compared with the experimental findings. However, the results seem to be encouraging and thus prompt further development of the low temperature transport theory in degenerate materials.
4. Conclusion Fig. 4. For the degeneracy parameter εF/kBTL ¼15 the dependence of the phonon growth rate (∂Nq/∂t)deg as normalized to the rate (∂Nq/∂t)non − deg , upon the phonon wave vector x in a sample of Si for different values of the lattice temperature TL and the normalized electron temperatureTn = (Te/TL ) . The curves 1, 2 and 3 are for Tn ¼2.5, 5, and 10 respectively, and those marked a, b, and c correspond to the lattice temperature of 1, 4, and 20 K respectively.
with the increase in x and eventually, as expected, the characteristics of the phonon growth rate for the degenerate material tend to be the same as that for the non-degenerate material for higher values of x where the effects of degeneracy are hardly manifested. The discrepancy is vividly greater the lower is the value of the normalized electron temperature Tn and continues for higher values of x the lower is the lattice temperature. The discrepancy at any Tn is quite significantly susceptible to any change in the degeneracy factor, and assumes greater and greater values the greater is the factor. It may be noted also that the relative changes in the (∂Nq/∂t) characteristics that are effected at the lower values of x are mainly due to the degeneracy factor, whereas those at higher values of x are mainly due to inelasticity of the collisions and the true phonon distribution. In the lower range of the normalized phonon wave vector x, so long as the degeneracy parameter εF/kBTL remains greater than x, the material seems to be degenerate and the effect of degeneracy on both the qualitative and quantitative aspects of the growth characteristics is significantly manifested. Beyond that, as x increases, the material gradually turns out to be more and more non-degenerate and the growth characteristics starting from a relatively low value, asymptotically approaches the same for the non-degenerate material. For higher values of the degeneracy parameter and at low lattice temperatures the growth curves display rather sharp variation around the transition region. Over this region as x increases the effect of degeneracy gradually becomes weaker and the growth rate tends to increase. But the finite value of the phonon energy and the true phonon distribution now begin to influence the growth rate more and more and thus make it decrease. As a result the growth rate here may exhibit some transients in the form extrema, particularly when the temperature is quite low. For highly degenerate materials it can be seen from (6) that (∂Nq/∂t) may turn out to be negative if x > 2Tn . It may be noted that for the high values of the degeneracy parameter like 15 or so, the growth rate for Tn ¼ 2.5 drops quite fast with increase in x and, indeed tends to be negative for x 45. For Tn 42.5, η decreases so
Following the standard perturbation theory, the phonon emission characteristics in a degenerate material have been analyzed here. The lattice temperature has been assumed to be low enough, so that none of the traditional assumptions, like elastic collisions with intravalley acoustic phonons and equipartition law for the phonon distribution could be made. The explicit expressions obtained for the phonon growth rate describe how does the degeneracy of the material effect changes in the growth characteristics for various values of the lattice temperature TL and the effective electron temperature Te. Numerical results have been obtained for samples of bulk Si. The results which thus follow show that under the condition of low temperature, the degeneracy factor of the material beings in significant changes in the growth characteristics compared to what follows for the non-degenerate materials under the same prevalent conditions of low temperature. The results seem to be interesting and thus stimulate further studies on the non-ohmic transport in degenerate materials under the low temperature conditions.
Acknowledgments S. Midday is indebted to the University Grants Commission, New Delhi, India, for the award of a Teacher fellowship under the faculty improvement program. D.P. Bhattacharya is grateful to the same commission for their support on being awarded Emeritus Fellowship. The authors acknowledge the help rendered by B. Das and A. Basu.
References [1] E.M. Conwell, High Field Transport in Semiconductors, Academic Press, New York, 1967. [2] Z.S. Kachlishvili, Phys. Status Solidi (a) 88 (1976) 15. [3] S. Midday, D.P. Bhattacharya, J. Phys. Chem. Solids 72 (2011) 1343. [4] N. Chakraborty, D.P. Bhattacharya, J. Phys. Chem. Solids 57 (1995) 653. [5] B.R. Nag, Electron Transport in Compound Semiconductors, Springer-Verlag, Berlin, 1980. [6] S.M. Sze, Physics of Semiconductor Devices, Wiley Eastern Limited, New Delhi, 1983. [7] N.W. Aschroft, N.D. Mermin, Solid State Physics, CBS Publishing Japan, Ltd., Tokyo, 1981. [8] P.G. Klemens, Proc. Phys. Soc. (Lond.) A 64 (1951) 1030. [9] A. Anselm, Introduction to Semiconductor Theory, Mir Publisher, Moscow, 1981.
Please cite this article as: S. Midday, et al., Physica B (2014), http://dx.doi.org/10.1016/j.physb.2014.10.027i
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