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Interband light absorption by a semiconductor in a pulsing electric field
Volume 48, number 3
OPTICS COMMUNICATIONS
1 December 1983
INTERBAND LIGHT ABSORPTION BY A SEMICONDUCTOR IN A PULSING ELECTRIC FIELD V.A. SINYAK Institute of Applied Physics, Academy of Sciences of the MoMavian SSR, Kishinev, 277028, USSR Received 5 August 1983
The light absorption by a semiconductor, placed in a pulsing electric field of arbitrary amplitude and pulse duration is considered. The calculation of the absorption coefficient is carried out on the ground of the effective mass method for direct optical transitions between two simple parabolic bands. The calculated absorption coefficient is valid for photon energies both greater and lesser than the semiconductor forbidden bandwidth.
1. Introduction During the last years more and more attention is given to the investigation o f how a strong and short pulse of an external field act on semiconductors, in connection with the development of laser techniques and of a method to get high intensities of the electric and magnetic fields. The way the processes in a pulsing external field occur is essentially different from the way the same processes occur in a steady-state regime. Besides this, in the regime of short and ultrashort pulses of the external field in semiconductors, new interesting phenomena, such as self-induced transparency, photon echo, optical nutation and bistability appear [1,2]. In this paper the light absorption by the semiconductor, placed in a pulsing electric field of arbitrary amplitude E and pulse duration r is considered. The calculation of the absorption coefficient is carried out on the ground of the effective mass method for direct optical transitions between two simple parabolic bands. The absorption of light with frequency cJ is taken into account in the first order o f the perturbation theory. The electric field o f the pulse given by the vector-potential A = --a th kt, is taken exactly into account, by using the exact wave function of the electron in this field. It is shown that the light absorption coefficient in a pulsing field appreciably differs from the absorption coefficient both in the ab200
sence of the field and in tile case of a constant homogeneous electric field. Actually, the light absorption by a semiconductor in the absence of external fields obeys such a regularity: at the frequencies of the absorbed light w < Cg/h the absorption is small. The light absorption coefficient sharply increases, on increasing the frequency c~ and satisfying the relationship hco ~ Cg. If a constant and homogeneous electric field is applied to the semiconductors the way of light absorption changes. In this case the light absorption with frequencies co lesser than 6g/h is possible. It is connected with the tunnelling o f the electron from the upper edge of the valence band to the forbidden band, where it picks up energy, on absorbing the photon rico and then tunnels farther, up to the conductivity band bottom. In a constant and homogeneous electric field the shift of the absorption threshold to the red side occurs [3]. In our paper the calculated absorption coefficient in a pulsing electric field is valid for photon energies both lesser and greater than the width of the forbidden gap. When the pulse duration r tends to infinity, one gets the known result for the absorption coefficient in a constant and homogeneous electric field [3]. When r ~ 0 we get the absorption coefficient in the absence of external fields as for so short a time the electron will not have time enough to gather energy on the account of the external field and tunnel from 0 030-4018/83/0000--0000/$ 03.00 © 1983 North-HoUand
Volume 48, number 3
OPTICS COMMUNICATIONS
the valence band to the conductivity one. At small intensities of the external pulsing electric field and short pulse durations r, the correction to the absorption coefficient in the absence of the field is proportional to
1 December 1983
and in the valence band is described by the wave function
• egnp(r, t) = ~npUnp(r),
(4)
7-6"
2. The wave functions Let us find the wave functions of the electron and the hole of a semiconductor in a pulse electric field, given by the vector-potential A = - a th kt,
(1)
where 1/k = r is the pulse duration. To the vectorpotential (1) corresponds the electric field 6 = El ch2kt, where E = ak/c and c is the light velocity. Evidently, (1) is an isolated pulse of the electric field, tending to zero when t ~ -+~. One gets another limit o f the field (1) at putting a = Elk and letting k go to zero
A = - ( E / k ) th kt
~ - Et, k-*O
(2)
i.e. the vector-potential, describing a constant and homogeneous electric field. Note that the vectorpotential (1) is widely used in quantum electrodynamics when calculating various quantum processes (e.g. [4,51). Let us suppose the bands to be simple, i.e. the minimum in the conductivity band and the maximum in the valence band are situated in the same point p = 0 of the Brillouin zone. We shall search for such solution of the Schr6dinger equation in the field (1), that they pass into the wave functions of the free electron, when a ~ 0, and coincide with the wave functions in the field (2), when k ~ 0. On taking into account the stressed above, the wave function of the electron in the field (1) can be written in the following way:
¢(r, t) = V-1/2exp(ipr/h)(ch kt) leap~merck
x
i [" e2a 2
exp g[2--m- 2 kthkt- (p0 +
e2"2
P0c = Cg + p2 /2mc,
POo = 6o = - p 2 /2mh
where Cg is the semiconductor forbidden band width. It should be noted, that the wave functions (4) describe the movement of the charge carriers in the pulsing electric field (4) in semi-conductors with a wide forbidden band Cg. At small Cg one should use the two-bands equation of Kane's model, which coincides formally with Dirac's equation in the case of simple bands, at changing the velocity c on the other, and namely: s = (Cg/2m)1/2 [6]. The wave functions of an electron for a narrow-band semiconductor in the field (1) are given in ref. [4].
3. The transition matrix element and the absorption
coefficient Let a weak and plane electromagnetic wave be described by the vector-potential A l(r, t) = A 0 k exp (i(xr - cot)),
(5)
where A 0 is the amplitude, ). is the unitary polarisation vector, x = conic and co are the wave vector and the frequency correspondingly, A 0 = k - 1 (2kN#ico)l/2, where N is the number of photon in cm ~ , and n is the index of refractions. The matrix element of the perturbation energy
I?I = i(fie/mc)Ao(XV ) exp (-i(cot - xr) }
~t]/
2mec2] J J
where ~np is the function determined from (3), Unp(r) is the periodical part o f Bloch functions for the band n. For the conductivity band n = c and m e = me, and for the valence band n = v and m e = - m h (m h is the effective mass of the hole). In the formula (3) one must change the masses this way and also put
(3) '
where PO = p2/2me, V is the volume and m e is the effective mass of the electron. The state of the electron in the conductivity band
connected with the absorption o f a photon for the transition (v, P l ) -~ (c, P2) on wave functions (4) has the shape
moc = A o(e/mc)(~,p co) 9 ( t ).
(6)
Here Pcv = ih~2olfdrUcoVUuo is the matrix element of the transition for the pulse, I2 0 is the volume of an 201
Volume 48, number 3
OPTICS COMMUNICATIONS
elementary cell, 9 ( 0 is the part of the matrix element, depending on time. On calculating eq. (6) we neglected the wave vector o f the photon. One can easily integrate upon time in eq. (6), after changing the integration variable in the following way th kt = x. Then [7 ] +~
f
1=
+1
dtg(t)=k-lf
dxexp(-i/3x/2)
1 December 1983
When k -+ 0 one gets the case of the constant and homogeneous electric field. In the case of small ~ one can neglect the dependence upon the field in the argument of the Beta-function on the ground of eq. (9). Besides that, considering that 13/7 ~ 10 . 2 when k ~ 1012 s-1 and E "~ 103 V/ cm, as a result of integration with rico ~> 6g from (7) one gets for the absorption coefficient K(oo, k, E) = K(oo, O)
X (1 + x)-l+i("r~)(1 -- X) -l+i('r-a)
X (1 + e4E4/4h2k6#2).
= k - 1 2 2i'/- 1 exp(i/3/2) X B [i(7 + ce), i(7 - a)] 1F1 [i(7 + a); 2i7; -i/3], where BOa, v) = P(ta)F(v)/P(/a + v) is the Beta-function, 1F1 (a, c, x) is the hypergeometric function, ~t-1 = mc 1 + mh 1 a = (1/2rik)( Cg + p2/21a + e2a2/2tac 2 - boa),
3/= eap /2hk pz?,
~ = e2 a 2/hklac 2 .
The absorption coefficient, determined by the interband transition is equal to
K(~o, k, a) = (e2 /4rrm2 cmok 2 )
× fdplB[i(~ +a), i(7 -
(10)
On calculating (10) we took into account only the first two terms of the array for the degenerated hypergeometric function. One can see from (10), that the correction to the absorption coefficient K(u), 0) in the absence o f the field is proportional with the pulse duration at sixth power (k = l/r). When k -+ ~, eq. (10) also passes into eq. (8), i.e. for such short times the electron does not have time enough to gather energy on account o f the electric field and tunnel from the valence band to the conductivity one. The author wishes to thank V.V. Baltaga for the translation in english.
a)][ 2
References X I1F1 [i(7 + a); 2i7; -i~]l 2 [kpco[2.
(7)
When a -+ 0, eq. (7) passes into the expression for the absorption coefficient in the absence o f the field
K(,o, O) = ( 2e2 /m2 cnoo)lXpcol 2 X (21a/fi2)3/2(fioo- gg)l/2,
(8)
as 1F1 [i(3' + a); 2i7; --i~][a_..,0 = 1, B [i(7 + a), i(7 - a)]la_, 0 = 6 [ ( 1 / 2 r k ) ( 6 g +p2/2/a - ~ w ) ] .
202
(9)
[1 ] L Allen and J.H. Eberly, Optical resonance and two-level atoms (New York, 1975). [2] S.A. Mockalenko, P.I. Khadzhi and A.H. Rotaru, Solitoni i nutacia v excitonnoi oblasti spectra (Shtiinza, Kishinev, 1980). [3] O.V. Keldysh, JETP 34 (1958) 1138. [4] A.I. Nikishov, YF 11 (1970) 1072. [5 ] A.A. Grib, S.G. Mamaev and B.M. Mostepanenko, Kvantovie effekti v intensivnih vneshnih poljh (Atomizdat, Moskva, 1980). [6] A.G. Aronov and G.E. Pikus, JETP 51 (1966) 281. [7 ] I.S. Gradshtein and I.M. Rigik, Tablizi integralov, summ, rjdov i proizvedenii (GIFML, Moskva, 1963). [8] A.I. Anselm, Vvedenie v teoriy poluprovodnikov (GIFML, Moskva, 1978).
OPTICS COMMUNICATIONS
1 December 1983
INTERBAND LIGHT ABSORPTION BY A SEMICONDUCTOR IN A PULSING ELECTRIC FIELD V.A. SINYAK Institute of Applied Physics, Academy of Sciences of the MoMavian SSR, Kishinev, 277028, USSR Received 5 August 1983
The light absorption by a semiconductor, placed in a pulsing electric field of arbitrary amplitude and pulse duration is considered. The calculation of the absorption coefficient is carried out on the ground of the effective mass method for direct optical transitions between two simple parabolic bands. The calculated absorption coefficient is valid for photon energies both greater and lesser than the semiconductor forbidden bandwidth.
1. Introduction During the last years more and more attention is given to the investigation o f how a strong and short pulse of an external field act on semiconductors, in connection with the development of laser techniques and of a method to get high intensities of the electric and magnetic fields. The way the processes in a pulsing external field occur is essentially different from the way the same processes occur in a steady-state regime. Besides this, in the regime of short and ultrashort pulses of the external field in semiconductors, new interesting phenomena, such as self-induced transparency, photon echo, optical nutation and bistability appear [1,2]. In this paper the light absorption by the semiconductor, placed in a pulsing electric field of arbitrary amplitude E and pulse duration r is considered. The calculation of the absorption coefficient is carried out on the ground of the effective mass method for direct optical transitions between two simple parabolic bands. The absorption of light with frequency cJ is taken into account in the first order o f the perturbation theory. The electric field o f the pulse given by the vector-potential A = --a th kt, is taken exactly into account, by using the exact wave function of the electron in this field. It is shown that the light absorption coefficient in a pulsing field appreciably differs from the absorption coefficient both in the ab200
sence of the field and in tile case of a constant homogeneous electric field. Actually, the light absorption by a semiconductor in the absence of external fields obeys such a regularity: at the frequencies of the absorbed light w < Cg/h the absorption is small. The light absorption coefficient sharply increases, on increasing the frequency c~ and satisfying the relationship hco ~ Cg. If a constant and homogeneous electric field is applied to the semiconductors the way of light absorption changes. In this case the light absorption with frequencies co lesser than 6g/h is possible. It is connected with the tunnelling o f the electron from the upper edge of the valence band to the forbidden band, where it picks up energy, on absorbing the photon rico and then tunnels farther, up to the conductivity band bottom. In a constant and homogeneous electric field the shift of the absorption threshold to the red side occurs [3]. In our paper the calculated absorption coefficient in a pulsing electric field is valid for photon energies both lesser and greater than the width of the forbidden gap. When the pulse duration r tends to infinity, one gets the known result for the absorption coefficient in a constant and homogeneous electric field [3]. When r ~ 0 we get the absorption coefficient in the absence of external fields as for so short a time the electron will not have time enough to gather energy on the account of the external field and tunnel from 0 030-4018/83/0000--0000/$ 03.00 © 1983 North-HoUand
Volume 48, number 3
OPTICS COMMUNICATIONS
the valence band to the conductivity one. At small intensities of the external pulsing electric field and short pulse durations r, the correction to the absorption coefficient in the absence of the field is proportional to
1 December 1983
and in the valence band is described by the wave function
• egnp(r, t) = ~npUnp(r),
(4)
7-6"
2. The wave functions Let us find the wave functions of the electron and the hole of a semiconductor in a pulse electric field, given by the vector-potential A = - a th kt,
(1)
where 1/k = r is the pulse duration. To the vectorpotential (1) corresponds the electric field 6 = El ch2kt, where E = ak/c and c is the light velocity. Evidently, (1) is an isolated pulse of the electric field, tending to zero when t ~ -+~. One gets another limit o f the field (1) at putting a = Elk and letting k go to zero
A = - ( E / k ) th kt
~ - Et, k-*O
(2)
i.e. the vector-potential, describing a constant and homogeneous electric field. Note that the vectorpotential (1) is widely used in quantum electrodynamics when calculating various quantum processes (e.g. [4,51). Let us suppose the bands to be simple, i.e. the minimum in the conductivity band and the maximum in the valence band are situated in the same point p = 0 of the Brillouin zone. We shall search for such solution of the Schr6dinger equation in the field (1), that they pass into the wave functions of the free electron, when a ~ 0, and coincide with the wave functions in the field (2), when k ~ 0. On taking into account the stressed above, the wave function of the electron in the field (1) can be written in the following way:
¢(r, t) = V-1/2exp(ipr/h)(ch kt) leap~merck
x
i [" e2a 2
exp g[2--m- 2 kthkt- (p0 +
e2"2
P0c = Cg + p2 /2mc,
POo = 6o = - p 2 /2mh
where Cg is the semiconductor forbidden band width. It should be noted, that the wave functions (4) describe the movement of the charge carriers in the pulsing electric field (4) in semi-conductors with a wide forbidden band Cg. At small Cg one should use the two-bands equation of Kane's model, which coincides formally with Dirac's equation in the case of simple bands, at changing the velocity c on the other, and namely: s = (Cg/2m)1/2 [6]. The wave functions of an electron for a narrow-band semiconductor in the field (1) are given in ref. [4].
3. The transition matrix element and the absorption
coefficient Let a weak and plane electromagnetic wave be described by the vector-potential A l(r, t) = A 0 k exp (i(xr - cot)),
(5)
where A 0 is the amplitude, ). is the unitary polarisation vector, x = conic and co are the wave vector and the frequency correspondingly, A 0 = k - 1 (2kN#ico)l/2, where N is the number of photon in cm ~ , and n is the index of refractions. The matrix element of the perturbation energy
I?I = i(fie/mc)Ao(XV ) exp (-i(cot - xr) }
~t]/
2mec2] J J
where ~np is the function determined from (3), Unp(r) is the periodical part o f Bloch functions for the band n. For the conductivity band n = c and m e = me, and for the valence band n = v and m e = - m h (m h is the effective mass of the hole). In the formula (3) one must change the masses this way and also put
(3) '
where PO = p2/2me, V is the volume and m e is the effective mass of the electron. The state of the electron in the conductivity band
connected with the absorption o f a photon for the transition (v, P l ) -~ (c, P2) on wave functions (4) has the shape
moc = A o(e/mc)(~,p co) 9 ( t ).
(6)
Here Pcv = ih~2olfdrUcoVUuo is the matrix element of the transition for the pulse, I2 0 is the volume of an 201
Volume 48, number 3
OPTICS COMMUNICATIONS
elementary cell, 9 ( 0 is the part of the matrix element, depending on time. On calculating eq. (6) we neglected the wave vector o f the photon. One can easily integrate upon time in eq. (6), after changing the integration variable in the following way th kt = x. Then [7 ] +~
f
1=
+1
dtg(t)=k-lf
dxexp(-i/3x/2)
1 December 1983
When k -+ 0 one gets the case of the constant and homogeneous electric field. In the case of small ~ one can neglect the dependence upon the field in the argument of the Beta-function on the ground of eq. (9). Besides that, considering that 13/7 ~ 10 . 2 when k ~ 1012 s-1 and E "~ 103 V/ cm, as a result of integration with rico ~> 6g from (7) one gets for the absorption coefficient K(oo, k, E) = K(oo, O)
X (1 + x)-l+i("r~)(1 -- X) -l+i('r-a)
X (1 + e4E4/4h2k6#2).
= k - 1 2 2i'/- 1 exp(i/3/2) X B [i(7 + ce), i(7 - a)] 1F1 [i(7 + a); 2i7; -i/3], where BOa, v) = P(ta)F(v)/P(/a + v) is the Beta-function, 1F1 (a, c, x) is the hypergeometric function, ~t-1 = mc 1 + mh 1 a = (1/2rik)( Cg + p2/21a + e2a2/2tac 2 - boa),
3/= eap /2hk pz?,
~ = e2 a 2/hklac 2 .
The absorption coefficient, determined by the interband transition is equal to
K(~o, k, a) = (e2 /4rrm2 cmok 2 )
× fdplB[i(~ +a), i(7 -
(10)
On calculating (10) we took into account only the first two terms of the array for the degenerated hypergeometric function. One can see from (10), that the correction to the absorption coefficient K(u), 0) in the absence o f the field is proportional with the pulse duration at sixth power (k = l/r). When k -+ ~, eq. (10) also passes into eq. (8), i.e. for such short times the electron does not have time enough to gather energy on account o f the electric field and tunnel from the valence band to the conductivity one. The author wishes to thank V.V. Baltaga for the translation in english.
a)][ 2
References X I1F1 [i(7 + a); 2i7; -i~]l 2 [kpco[2.
(7)
When a -+ 0, eq. (7) passes into the expression for the absorption coefficient in the absence o f the field
K(,o, O) = ( 2e2 /m2 cnoo)lXpcol 2 X (21a/fi2)3/2(fioo- gg)l/2,
(8)
as 1F1 [i(3' + a); 2i7; --i~][a_..,0 = 1, B [i(7 + a), i(7 - a)]la_, 0 = 6 [ ( 1 / 2 r k ) ( 6 g +p2/2/a - ~ w ) ] .
202
(9)
[1 ] L Allen and J.H. Eberly, Optical resonance and two-level atoms (New York, 1975). [2] S.A. Mockalenko, P.I. Khadzhi and A.H. Rotaru, Solitoni i nutacia v excitonnoi oblasti spectra (Shtiinza, Kishinev, 1980). [3] O.V. Keldysh, JETP 34 (1958) 1138. [4] A.I. Nikishov, YF 11 (1970) 1072. [5 ] A.A. Grib, S.G. Mamaev and B.M. Mostepanenko, Kvantovie effekti v intensivnih vneshnih poljh (Atomizdat, Moskva, 1980). [6] A.G. Aronov and G.E. Pikus, JETP 51 (1966) 281. [7 ] I.S. Gradshtein and I.M. Rigik, Tablizi integralov, summ, rjdov i proizvedenii (GIFML, Moskva, 1963). [8] A.I. Anselm, Vvedenie v teoriy poluprovodnikov (GIFML, Moskva, 1978).