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Thermal radiation law for a small semiconductor body
J. Quan~. Specbosc. Radiat. Transfer Vol. 49, No. 3, Printed in Great Britain. All rights reserved
THERMAL
pp. 259-261,
1993 Copyright
0022-4073/93 $6.00 + 0.00 0 1993 Pergamon Press Ltd
RADIATION LAW FOR A SMALL SEMICONDUCTOR BODY S.
HAvAt and M. AUSLENDER
Department of Electrical and Computer Engineering, Ben-Gurion University of the Negev, P.O.B. 653, Beer-Sheva 84105, Israel (Received 20 April 1992)
Abstract-The temperature dependence of the thermal radiant flux from a small semiconductor body is considered. The modification of the thermal radiation law for this case is discussed.
It is shown that for a wide interval of temperature, the thermal radiant flux is proportional to T’.’ rather than T4.
INTRODUCTION The thermal radiation law for a partially transparent body of dimensions greater than the emitted wavelength is well known. ‘9’The thermal radiation for blackbody microcavities of dimensions comparable to the wavelength was considered in Ref. 3. The purpose of the present paper is to discuss the intermediate case when the dimensions remain greater than the wavelength but are smaller than the inverse absorption coefficient. This situation can be realized for weakly doped semiconductors. The general form of the spectral thermal radiation law is’
& Z(w, T, Cl) = ~(a,
T, Cl) x
ho’ (2m)*(ewKT - 1) ’
where Z(o, T, i2) is the directional spectral radiant flux, o the frequency of the emitted wave, T the temperature of the body, R a unit vector in the direction of propagation, and t(~, T, 0) the directional spectral emittance of the body. Using Kirchoffs law,’ c=cI
(2)
reduces the calculation of t(~, T, Q) to that of absorptance o1(0, T, Q) of the incident nonpolarized plane wave with the frequency o and wave vector k opposite to R, i.e. k = - (co/c)R. Thus, by means of Eq. (2), one returns to a purely electrodynamical problem which yields a. The solution of this problem requires knowledge of the material dielectric constant c(o) = 6, (0) + iQ(O) (3) and of the geometry. Much attention has been paid to the calculation of a for non-opaque bodies with given geometries.* These calculations predicted various reductions of the emittance c as compared to the opaque limit value. However, no result has been given in this connection concerning possible modifications of the temperature dependence of the total radiant flux. This dependence is the well known Stefan-Boltzmann law and is obtained by integration of Eq. (1) with respect to o supposing the emittance to be independent on w. This is justified for opaque materials or for rather large transparent bodies. Small transparent bodies require special consideration. ABSORPTANCE
OF A SMALL
BODY
We now clarify the term small more quantitatively. If the material is transparent, then E, > 0 and 6, % .5*. Therefore, the refractive index n(o) and extinction coefficient k(o) are given by fl(~)&XG tTo whom all correspondence
should be addressed. 259
(4)
S. HAVA and M. ALELENDER
260
k(w) = cz(CO)i2n(o). The transparency
property
of the small body means
(5)
that
[cok(w)lc]d < 1, where values defines Baltes We
(6)
d is a typical body dimension in the emission direction. It is clear from Eqs. (4)-(6) that all of the ratio d/i are possible, where 1 = 2nc/c1 is the wavelength. The condition d/l - 1 the microbody whose thermal radiation has peculiarities discussed in a series of papers by et al.’ will now consider the case when the condition given by Eq. (6) coexists with the condition d/E. $ I.
It is clear that the absorption of electromagnetic ~~(0). In this case, the absorptance is
(7) radiation
of such a body is proportional
3( = Y,{h\IP,,, . where Pabs and P,, are the absorbed contains cl(w) as a proportionality
(8)
and incident powers respectively. factor.’ i.e.
/Em,(r,
to
0,
We will use the formula
sZ)l’d3r.,
which
(9)
where Eint is the electric field of the electromagnetic wave inside the body. When finding this field, we may neglect t*(w) because of the condition given by Eq. (6) and take the dielectric constant to be real: c(w) = n(w)‘. Using the familiar expression P!, = cEf,S/4n and Eqs. (8), (9), we obtain (!M ,
(0) )
(10) (11)
where S is the body surface area and e,“, is E,,, provided that the amplitude of incident wave electric field E,, is chosen as unity. The quantity rl(o, Q) has the meaning of an effective linear dimension of the body and depends essentially on its geometry. CALCULATION
OF THE
THERMAL
RADIANT
Substituting Eq. (10) in Eq. (2) and Eq. (2) in Eq. (1) and then integrating to o, we obtain for the total directional flux
FLUX both parts with respect
(12) For conventional semiconductors in the transparency window, the refractive index does not depend strongly on w. For this case and under the condition given by Eq. (7) d(w, Q) is a strongly oscillating function of o. Hence, when integrating, we may replace this function by its o-average, d(R), over the interval (0, zo). It can be shown for simple geometries that this procedure is equivalent to the well known multiple-reflection approximation. We are forced to perform such averaging also for an additional reason. The emitted wave is fully incoherent. Due to this, the absorption c( in Eq. (2) has to be averaged with respect to the random phase of the wave. Since this phase will enter GIadditively with the regular phase, which is proportional to w, we may replace the random-phase average by an o average. In our case, c*(w) does not vanish due to intra-band electron scattering processes. For sufficiently high temperatures, the intra-band scattering is dominated by electron-photon scattering and is almost elastic. Also, the frequencies of interest satisfy the condition
Thermal radiation law for a small semiconductor body
261
where r is the transport relaxation time. Under these conditions, c*(w) can be calculated explicitly by using the transition probability method,s i.e. (13) F(x) = x sinh(x)&(x),
(14)
where or is the plasma frequency and K,(x) is the McDonald function. After substitution of Eqs. (13), (14) into Eq. (12) replacing the integration variable by o = 2kT x x, and using the formula6 m emXx2K2(x)dx = S/S, s0 we arrive at the final result Z(T, LI) =
d(Q) 0.0444 x aT=T=P' c-c
(15)
where 0 is the Stefan-Boltzmann constant and Tp = hop/k. Comparing Eq. (15) with the Stefan-Boltzmann law,’ we find that the small semiconductor body is described by an effective emittance that is strongly dependent on the temperature as follows: e&T,
f2) =
d(R)
0.0444 x -
CT
T=
x2
T2 ’
For a weakly-doped semiconductor, there is a rather broad temperature interval for which the carrier concentration N and hence TiaN are constant.’ Taking into account that l/r a T’.‘.’ we obtain t&T, Q)cc T-o.5. (17) REFERENCES 1. R. Siegel and J. R. Howell, Thermal Radiation Heat Transfer, McGraw-Hill, New York, NY (1981). 2. Y. S. Toulukian ed., Thermophysical Properties of Matter, Vol. 8, Thermal Radiative Properties, IFI/Plenum, New York, NY (1972). 3. H. P. Baltes and F. K. Kneubuehl, Helv. Phys. Acta 45, 481 (1972); H. P. Baltes, Infrared Phys. 16, 1 (1976); B. Steinle, H. P. Baltes, and M. Pabst, ibid. 16, 25 (1976). 4. L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 8, Electrodynamics of Continuous Media, Pergamon Press, New York, NY (1984). 5. K. Seeger, Semiconductor Physics, Springer Series in Solid State Science 40, Springer, Heidelberg (1982). 6. G. N. Watson, A Treatise on the Theory of Bessel Functions, University Press, Cambridge (1966).
THERMAL
pp. 259-261,
1993 Copyright
0022-4073/93 $6.00 + 0.00 0 1993 Pergamon Press Ltd
RADIATION LAW FOR A SMALL SEMICONDUCTOR BODY S.
HAvAt and M. AUSLENDER
Department of Electrical and Computer Engineering, Ben-Gurion University of the Negev, P.O.B. 653, Beer-Sheva 84105, Israel (Received 20 April 1992)
Abstract-The temperature dependence of the thermal radiant flux from a small semiconductor body is considered. The modification of the thermal radiation law for this case is discussed.
It is shown that for a wide interval of temperature, the thermal radiant flux is proportional to T’.’ rather than T4.
INTRODUCTION The thermal radiation law for a partially transparent body of dimensions greater than the emitted wavelength is well known. ‘9’The thermal radiation for blackbody microcavities of dimensions comparable to the wavelength was considered in Ref. 3. The purpose of the present paper is to discuss the intermediate case when the dimensions remain greater than the wavelength but are smaller than the inverse absorption coefficient. This situation can be realized for weakly doped semiconductors. The general form of the spectral thermal radiation law is’
& Z(w, T, Cl) = ~(a,
T, Cl) x
ho’ (2m)*(ewKT - 1) ’
where Z(o, T, i2) is the directional spectral radiant flux, o the frequency of the emitted wave, T the temperature of the body, R a unit vector in the direction of propagation, and t(~, T, 0) the directional spectral emittance of the body. Using Kirchoffs law,’ c=cI
(2)
reduces the calculation of t(~, T, Q) to that of absorptance o1(0, T, Q) of the incident nonpolarized plane wave with the frequency o and wave vector k opposite to R, i.e. k = - (co/c)R. Thus, by means of Eq. (2), one returns to a purely electrodynamical problem which yields a. The solution of this problem requires knowledge of the material dielectric constant c(o) = 6, (0) + iQ(O) (3) and of the geometry. Much attention has been paid to the calculation of a for non-opaque bodies with given geometries.* These calculations predicted various reductions of the emittance c as compared to the opaque limit value. However, no result has been given in this connection concerning possible modifications of the temperature dependence of the total radiant flux. This dependence is the well known Stefan-Boltzmann law and is obtained by integration of Eq. (1) with respect to o supposing the emittance to be independent on w. This is justified for opaque materials or for rather large transparent bodies. Small transparent bodies require special consideration. ABSORPTANCE
OF A SMALL
BODY
We now clarify the term small more quantitatively. If the material is transparent, then E, > 0 and 6, % .5*. Therefore, the refractive index n(o) and extinction coefficient k(o) are given by fl(~)&XG tTo whom all correspondence
should be addressed. 259
(4)
S. HAVA and M. ALELENDER
260
k(w) = cz(CO)i2n(o). The transparency
property
of the small body means
(5)
that
[cok(w)lc]d < 1, where values defines Baltes We
(6)
d is a typical body dimension in the emission direction. It is clear from Eqs. (4)-(6) that all of the ratio d/i are possible, where 1 = 2nc/c1 is the wavelength. The condition d/l - 1 the microbody whose thermal radiation has peculiarities discussed in a series of papers by et al.’ will now consider the case when the condition given by Eq. (6) coexists with the condition d/E. $ I.
It is clear that the absorption of electromagnetic ~~(0). In this case, the absorptance is
(7) radiation
of such a body is proportional
3( = Y,{h\IP,,, . where Pabs and P,, are the absorbed contains cl(w) as a proportionality
(8)
and incident powers respectively. factor.’ i.e.
/Em,(r,
to
0,
We will use the formula
sZ)l’d3r.,
which
(9)
where Eint is the electric field of the electromagnetic wave inside the body. When finding this field, we may neglect t*(w) because of the condition given by Eq. (6) and take the dielectric constant to be real: c(w) = n(w)‘. Using the familiar expression P!, = cEf,S/4n and Eqs. (8), (9), we obtain (!M ,
(0) )
(10) (11)
where S is the body surface area and e,“, is E,,, provided that the amplitude of incident wave electric field E,, is chosen as unity. The quantity rl(o, Q) has the meaning of an effective linear dimension of the body and depends essentially on its geometry. CALCULATION
OF THE
THERMAL
RADIANT
Substituting Eq. (10) in Eq. (2) and Eq. (2) in Eq. (1) and then integrating to o, we obtain for the total directional flux
FLUX both parts with respect
(12) For conventional semiconductors in the transparency window, the refractive index does not depend strongly on w. For this case and under the condition given by Eq. (7) d(w, Q) is a strongly oscillating function of o. Hence, when integrating, we may replace this function by its o-average, d(R), over the interval (0, zo). It can be shown for simple geometries that this procedure is equivalent to the well known multiple-reflection approximation. We are forced to perform such averaging also for an additional reason. The emitted wave is fully incoherent. Due to this, the absorption c( in Eq. (2) has to be averaged with respect to the random phase of the wave. Since this phase will enter GIadditively with the regular phase, which is proportional to w, we may replace the random-phase average by an o average. In our case, c*(w) does not vanish due to intra-band electron scattering processes. For sufficiently high temperatures, the intra-band scattering is dominated by electron-photon scattering and is almost elastic. Also, the frequencies of interest satisfy the condition
Thermal radiation law for a small semiconductor body
261
where r is the transport relaxation time. Under these conditions, c*(w) can be calculated explicitly by using the transition probability method,s i.e. (13) F(x) = x sinh(x)&(x),
(14)
where or is the plasma frequency and K,(x) is the McDonald function. After substitution of Eqs. (13), (14) into Eq. (12) replacing the integration variable by o = 2kT x x, and using the formula6 m emXx2K2(x)dx = S/S, s0 we arrive at the final result Z(T, LI) =
d(Q) 0.0444 x aT=T=P' c-c
(15)
where 0 is the Stefan-Boltzmann constant and Tp = hop/k. Comparing Eq. (15) with the Stefan-Boltzmann law,’ we find that the small semiconductor body is described by an effective emittance that is strongly dependent on the temperature as follows: e&T,
f2) =
d(R)
0.0444 x -
CT
T=
x2
T2 ’
For a weakly-doped semiconductor, there is a rather broad temperature interval for which the carrier concentration N and hence TiaN are constant.’ Taking into account that l/r a T’.‘.’ we obtain t&T, Q)cc T-o.5. (17) REFERENCES 1. R. Siegel and J. R. Howell, Thermal Radiation Heat Transfer, McGraw-Hill, New York, NY (1981). 2. Y. S. Toulukian ed., Thermophysical Properties of Matter, Vol. 8, Thermal Radiative Properties, IFI/Plenum, New York, NY (1972). 3. H. P. Baltes and F. K. Kneubuehl, Helv. Phys. Acta 45, 481 (1972); H. P. Baltes, Infrared Phys. 16, 1 (1976); B. Steinle, H. P. Baltes, and M. Pabst, ibid. 16, 25 (1976). 4. L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 8, Electrodynamics of Continuous Media, Pergamon Press, New York, NY (1984). 5. K. Seeger, Semiconductor Physics, Springer Series in Solid State Science 40, Springer, Heidelberg (1982). 6. G. N. Watson, A Treatise on the Theory of Bessel Functions, University Press, Cambridge (1966).