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Laser induced superconducting properties in semiconductors
Volume 62A, number 2
PHYSICS LETTERS
25 July 1977
LASER INDUCED SUPERCONDUCTING PROPERTIES IN SEMICONDUCTORS M. NOGA Department of Theoretical Physics, Comenius University, 816 31 Brat&lava, Czechoslovakia Received 24 February 1977 We show that an amorphous semiconductor irradiated by a laser light exhibits a sort of the Meissner effect. A guided light in a thin film of an amorphous semiconductor is stopped or modified in its intensity when the semiconductor is illuminated by another light.
The authors [1], in their first work of this series of papers, have pointed out that the interaction of a free electron with a given monochromatic electromagnetic plane wave inside a transparent medium is formally the same as the interaction of the electron with a perfect crystal-like lattice. Thus the Bragg-like diffraction of relativistic electrons has been predicted [1]. In this letter we report on a next, new and interesting phenomenon, namely the creation of a sort of superconducting properties in amorphous semiconductors when they are irradiated by an intensive laser light. The mechanism and physics of this phenomenon is as follows. By irradiating an amorphous semiconductor by a strong laser light one creates for electrons above the mobility gap a perfect fictitious crystal-like lattice. The electrons will be consequently distributed in allowed bands and will move freely without any resistance. This can be achieved by an enough high intensity of the laser light allowing us to neglect a potential between ions and electrons which is present in the macroscopic medium. The ions will merely provide a uniform background of positive charge to give the electric neutrality on the average. The frequency w of the laser light has also to be so high as to prevent ions to follow the rapid oscillations of the laser field. Then only the electrons partake in the rapid motion and dynamical effects of ions can be ignored. The laser light creating these properties in a semiconductor will be called the action light. The semiconductor, when irradiated by the action light, is referred as to the action system. The action system will resist electromagnetic forces having tendencies to destroy its superconducting properties. Therefore it will exhibit an effect analogous to the Meissner effect in the ordinary superconductivity. 102
For example, an external laser light (with the intensity much less than that of the action light) may be stopped or strongly modified in its intensity in the action system. Thus we expect the light by light stopping effect in amorphous semiconductors. It was pointed out to us [2] that such an effect has already been experimentally observed and reported in [3] as the optical stopping effect. From our point of view it is a sort of the Meissner effect. To describe the aforementioned properties of the action system we start by considering a thin film of an amorphous semiconductor of the thickness d. Its two faces are the infinite planes z = 0 and z = d. Let it be permanently irradiated by the action light with the frequency w in the direction e = k/Ikl = (0, 0, 1), where k is the wave vector. The action light excites electrons from the valence band (with a possibility via quasilocalized states in the mobility gap) and from quasilocalized states into the conduction band. A certain fraction of the excited electrons will come back to the lower states to be excited again and again. Thus the density n(x, t) of the excited electrons will fluctuate around its constant (time and space averaged) value n o with the density fluctuation
p(x, t). So n(x, t) = n o + p(x, t).
(1)
n o may possibly exceed the density of thermally excited electrons, if0 ~ 1016 c m - 3 , even by several orders. Excited electrons are assumed to be free carriers with the electric charge - e and the effective mass m. They move in the infinite one dimensional well under the presence of the electromagnetic field intensities E(x, t) and H(x, t). The intensities E(x, t) and H(x, t).
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The intensities E(x, t) and H(x, t), acting on a single electron described by the wave function ~s(X, t), are due to the laser field and motion of all other electrons. Suppose one solves the Schrodinger equation for all electrons selfconsistently. Then he gets the single electron density
Ps(X, t) -- Ss*(X, t) ffs(X, t),
(2)
and the single electron current density
is(X, t) = ~m (~0s(X, t) V ¢:(x, t) - ~:(x, t) V~s(X, t))
+
e-E-A(x, t)Ps(X, t),
(3)
mc
where A(x, t) is the vector potential. The function Ps(X, t) would then be the known distribution function satisfying the Boltzmann-like kinetic equation with a scattering term [e.g. 4]. By making the sum over occupied electron states s, one gets the total electron density
n(x, t) = n o + p(x, t) = ~ P s ( X , t),
25 July 1977
1 OH VX E + c~--= 0,
(10)
v'n=0.
(11)
For the sake of convenience we introduce the electron displacement u(x, t) as given by
o(x, t) =- du/dt.
(12)
The system of the partial differential equations (6)(11) is solvable in the linear approximation with respect to p ando. The axial symmetry of the action system requires all physical quantities in eqs. (6)-(11) to be functions only of z and t variables. The linearized pressure tensor Pij has the only non zero elements P33 = P - m n o r
Pll =P22 ~ P '
(av3/az),
(13)
where P is the scalar pressure and r is a transport coefficient related to the viscosity of the electron gass. Our chosen geometry allows to write down the linearised equations in the form
(4)
s
~t(p + n 0 V ' u )
0,
(6a)
and the total electron current density
/(x, t) = n(x, t)o(x, t ) - ~ is(x, t),
(5)
a2u _ at 2
e E - s2 Vp m no
+ ~ a
V(V" u),
(7a)
wherev(x, t) is the electron velocity averaged over the occupied states s. These quantities obey the equations
1 aE 47re"0":au VXH . . . . . . c at c at '
(8a)
(On/at) + V(n. v)= 0,
(6)
V" E = -4nep,
(9a)
(7)
plus the two homogeneons equations (10) and (11). Here s,
doi
-en~+
I c
vX
aPi/,
-):- , 0x, .
where d/dt = (a/at) + (~. v) is the convective derivative, i, ] = 1,2, 3 and Pi/ is the tensor of the pressure. The last equations follow from the Enskog transport equation [e.g. 4]. The electric charge and current densities are -ep and -enu. Thus Maxwell's equations are written down
s2=l
aP m an 0
04)
is the velocity of the compressional waves. The known boundary conditions are (15)
E3(0, t) = E3(d, t) = 0
V X H - 1 aE 41re nu, -- = cot c
(8)
H 3(o, t) = H a(d, t) = 0
V. E = - 41rep,
(9)
(E - e
o) X ~ = ( H - n
o) x e -- 0
(16)
103
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for z = 0 and z = d, and 0u3(0, t)/Ot = au3(d , O/at = 0,
25 July 1977
where (17)
utr(z, t) = u(+)(66t + kz) + u(-)(wt where E 0 and H 0 are intensities outside the film. The boundary conditions for u3(z , t) are incomplete. They should be suplemented by the equation of state relating the pressure P to the density n at z = 0 and z = d. These relations are not known. We only expect that the relations Du3(0, t)/Ozl = 10u3(d, t)/0zl,
(17a)
may be imposed. In deriving the linearised equations one tacitly assumes that both p(z, t) and u(z, t) are smoothly varying functions o f z and t. This is def'mitely not true for all z ~ (0, d) from the following reasons. Physical arguments tells that, e.g., Vp must have &function type discontinuities at least for z = 0 and z = d. General theory of the hyperbolic partial differential equations with a boundary consisting of two curves gives, as a rule, solutions with discontinuous derivatives [e.g. 5]. Thus the linearized equations (6a)-(9a) may describe dynamics of the action system pretty well for all z except for a few isolated points, where their solutions exhibit singularities. There result two types of waves in the action system, longitudinal and transverse. We start first with the longitudinal waves. For this case the eqs. (6a), (9a), (10), and (11)are integrated with the results
O = - n o (Ou3/Oz), E 3 = 4zre noU3,
(18) H 3 = 0.
(19)
The transverse waves Utr(Z , t) must be expressed as a sum of two terms with the arguments (cot - kz) (for waves propagating in z direction) and ( w t + kz) (for reflected waves) respectively. This allows us to carry out the integrations ofeqs. (Sa) and (10) with the resuits 47re n o
662
Err "-Co2 _ c2k2 Utr(Z, t),
- kz).
The result (21) explains why we have omitted a term with H in eq. (7a). Next we make use of eq. (7a) to get ~2utr _
Ot2
662 602 p
662 _ c2k 2 Utr'
(22)
where COp,
662 --_47re2no/m, P
(23)
is the frequency of the plasma oscillation [e.g. 6]. The (22) and the boundary conditions (16) require the relation k2 = 6o2
c2
2)_602 n 2 COp
7
(24)
where n r is the refractive index. The last relation is exactly the formula for determining the refractive index in classical theory of optical properties of metals [e.g. 6]. The results (20) and (21) tell us that the action system behaves for the transverse waves as a substance with the dielectric constant e = nr2 , magnetic permeability/a = 1 which has optical properties of metals. On the other hand, the relation (19) tells that the action system aquires properties of a perfect dielectricure, for the component of the electrical displacement D 3 =0. The longitudinal waves u 3 satisfy the differential equation a2u.__~ 3 _s2Au 3 + 2u at 2 COp 3 - K ~- Au 3 = 0,
(25)
with the boundary conditions (15) and (17). The boundary conditions require u3(z , t) to be a superposition of standing waves
(20)
u3(z, t) = = gv(t) sin -~ z,
(26)
Ittr = 66247re_n°c66kc2 k 2 {ug+)(66t + kz ) - U(tr-) (66t - kz ) } X e, (21) 104
where gu(t) are functions of time. Eq. (25) gives the
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PHYSICS LETTERS
exponential extinction of the longitudinal waves with the time constant r ~K
(27)
25 July 1977
6= -aa{/c bt ,
9~ = V × a{
--- 0.
V.~
(33) (34)
The inhomogeneous equation gives
providing that a2u3/~z 2 :# 0. Thus'the dynamical stability requires
1 a2.~
A a f f + V ( V ' s ~ ) + 4*re2--
c 2 ~t 2 ~2u3/Oz2 = 0,
rnc2 [n o + p(z, t)] .~ = 0.
(28)
to be satisfied for each z E (0, d) except for isolated points. The solution to eq. (25) respecting the relation (28) and the boundary conditions (15) and (17) is
where u03 and ct3 are integration constants. These constants may, of course, depend on both characteristics of a semiconductor and the action light. They enter our theory as phenomenological parameters. From eq. (29) we get the persistent oscillating electric current density
-eJ3(z,t)=enoCOpUO3(Iz-d l - d } sin(~Opt + ~3) (30) and the fluctuation density of electrons (31)
rent (30) is shifted by 7r/2 with respect to the phase of E3(z, t). Therefore the time averaged rate of doing work by E 3 is zero. The action system exhibits indeed properties typical for ordinary superconductors. These properties extinct with the relaxation time r given by eq. (27), when the action light is switched off. Suppose that an external weak electromagnetic field is applied on the action system. Then the electron density n(x, t) will be not changed in the first order of perturbation theory. Only the electron current density is changed as given by e
(36)
K = (K1, K2, 0).
(37)
The potential M is represented by t) = e
(38)
where ~ ---½(6opt + a3) is chosen as a new independent variable, s~ satisfies eq. (34) if and only if K . f ( ~ ) = 0.
where the sign + is for z E (0, d/2) and the sign - for z E (cl/2, d). Note that the phase ~Opt + ot3 of the cur-
9 (x, t) =](z, t) + ~ n(z, t) s~ (x, t),
a{. V p = 0.
Consider an electromagnetic plane wave, e.g. a laser light, guided through the action system perpendicularly to z direction. Its wave vector K has the components
U3(Z, t) = U03 Z --~ [--'~ COS(COpt+ 0t3) ,
p(z, t) = +noU03 coS(Wpt + a3) ,
(35) The continuity equation applied for the current 9(x, t) imposes the restriction
(32)
where a{ is the vector potential due to the applied external field. In this case the three homogeneous Maxwell equations are satisfied by the substitutions
Let us first consider a solution to eq. (35) when p = 0. The solution is a monochromatic plane wave (with the frequency ~2) given by
a~(x, t) = a ei(k'x- nt) where K and ~2 are related by i(2 __ (
2/c2) (1 -
_= (s 2/c 2)
and a is a constant vector. Since the refraction index nr(~ ) is imaginary for ~ < ~Op, in this case the amplitude of the guided light will decrease exponentially along its propagation direction. The penetration depth is
X = (c/~2(~o;/~22
1)- 1/2.
(39)
The intensity of the guided light may be modified by this mechanism in addition to others two investigated below. Next we consider a solution to eq. (35) with p 4:0 and given by eq. (31). The relation (36) requires 105
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a{3(x, t) = 0.
(40)
So C 3(x, t) = 0, andi~(x, t) has to be parallel to z. The action system exhibits a sort of a stiffness to remain a perfect dielectricum and to preserve its persistent current ]3(z, t)unchanged. This phenomenon reminds distantly the Meissner effect in ordinary superconductivity. The unpolarized guided light is thus truncated of its s~ 3 component. Its intensity has to drop at least to the value I = 1012, where I 0 is the intensity of the guided light in a semiconductor in absence of the action light. The degree of stopping introduced in [3] as ln(Io//) is ln(I0//) = In 2. This is the second mechanism for the optical stopping effect. Setting (38) into eq. (35) we get Mathieu's equation [7] d2f(~)/d~2,+ (7 +- 3' cos 2~)f(~) = 0,
(41)
where r/ =_4(c2K2]w 2
+ 1),
7 -= 4u03.
(42)
Eq. (41) has, according to theorem of Floquet [e.g. 7], the general solution f+(~) = al+ eu~ B(~) +a2+ e-U~B(-~), for z E (0, all2)
(43a)
and
(43b) for z E (d/2, d). Here ai±, i = 1,2, are constant vectors, B(~) is a periodic function with period rr and # is a characteristic root determined by the parameters 77and 3'. The characteristic root/a can be either pure imaginary or complex. In the pure imaginary case the plane wave e ik'xf(~) propagates through the action system without absorption. In the complex case, Re/a :/= 0, the amplitude of the guided light decreases exponentially with time. This gives the third mechanism for the optical stopping effect. By consulting the stability (r/, 3') diagram of Mathieu's equation [7] one finds that this phenomenon happens when r/satisfies the relation 77~ N 2, 106
(44)
25 July 1977
where N is any positive integer. Thus the degree of stopping as a function of K 2 will exhibit a resonance behaviour for those values o f K 2 which satisfy eq. (44). Our conclusions concerning the optical stopping effect are: (i) Three different mechanisms may contribute for the optical stopping effect. The effect is enlarged when two or all of them are mixed together. The degree of stopping, In(/0//), as a function of the wave length of the unpolarized guided light will have rather smooth parts with ln(I0//) = ln2 separated by large peaks. (ii) When the intensity of the action light is lowered so that the guided light may distort electron wave functions in the action system considerably, then the optical stopping effect becomes less visible. (iii) When the action light is switched off, then the fluctuation density of electrons (the presence of which is responsible for the optical stopping effect) extincts exponentially with time. Thus the intensity of the guided light recovers exponentially with time to its value I 0 determined by the optical properties of a semiconductor in the state of its thermal equilibrium. The listed properties of the optical stopping effect are in the perfect agreement with experimental observations [3]. However the found mechanisms responsible for this effect differ completely from those proposed in [3]. Our theory has treated a semiconductor illuminated by the action light macroscopically. That is why we cannot make quantitative predictions on properties Of the optical stopping effect which depend on the microscopic structure of the semiconductor. Theauthor acknowledges valuable discussions with Dr. F. Culik. References
[1] C. Cronstr6m and M. Noga, Phys. Lett. A60 (1977) 137. [2] E. Majernikov~,private communication. [3] M. Kikuchi and A. Matsuda, in: Proc. of the VIth Inter. Conf. on Amorphous and liquid semiconductors, Leningrad (USSR), 1975, p.35; A. Matsuda et al., Appl. Phys. Lett. 24 (1974) 314. [4] R.L. Liboff, Introduction to the theory of kinetic equations (John Wiley, New York, 1969). [5] G.A. Korn and T.M. Korn, Mathematical Handbook (MaeGraw-Hill, New York, 1968).
[6] J.D. Jackson, Classical eleetrodynamies (John Wiley, New York, 1967). [7] L. Brillouin, Wave propagation in periodic structures (MacGraw-Hill,New York, 1946).
PHYSICS LETTERS
25 July 1977
LASER INDUCED SUPERCONDUCTING PROPERTIES IN SEMICONDUCTORS M. NOGA Department of Theoretical Physics, Comenius University, 816 31 Brat&lava, Czechoslovakia Received 24 February 1977 We show that an amorphous semiconductor irradiated by a laser light exhibits a sort of the Meissner effect. A guided light in a thin film of an amorphous semiconductor is stopped or modified in its intensity when the semiconductor is illuminated by another light.
The authors [1], in their first work of this series of papers, have pointed out that the interaction of a free electron with a given monochromatic electromagnetic plane wave inside a transparent medium is formally the same as the interaction of the electron with a perfect crystal-like lattice. Thus the Bragg-like diffraction of relativistic electrons has been predicted [1]. In this letter we report on a next, new and interesting phenomenon, namely the creation of a sort of superconducting properties in amorphous semiconductors when they are irradiated by an intensive laser light. The mechanism and physics of this phenomenon is as follows. By irradiating an amorphous semiconductor by a strong laser light one creates for electrons above the mobility gap a perfect fictitious crystal-like lattice. The electrons will be consequently distributed in allowed bands and will move freely without any resistance. This can be achieved by an enough high intensity of the laser light allowing us to neglect a potential between ions and electrons which is present in the macroscopic medium. The ions will merely provide a uniform background of positive charge to give the electric neutrality on the average. The frequency w of the laser light has also to be so high as to prevent ions to follow the rapid oscillations of the laser field. Then only the electrons partake in the rapid motion and dynamical effects of ions can be ignored. The laser light creating these properties in a semiconductor will be called the action light. The semiconductor, when irradiated by the action light, is referred as to the action system. The action system will resist electromagnetic forces having tendencies to destroy its superconducting properties. Therefore it will exhibit an effect analogous to the Meissner effect in the ordinary superconductivity. 102
For example, an external laser light (with the intensity much less than that of the action light) may be stopped or strongly modified in its intensity in the action system. Thus we expect the light by light stopping effect in amorphous semiconductors. It was pointed out to us [2] that such an effect has already been experimentally observed and reported in [3] as the optical stopping effect. From our point of view it is a sort of the Meissner effect. To describe the aforementioned properties of the action system we start by considering a thin film of an amorphous semiconductor of the thickness d. Its two faces are the infinite planes z = 0 and z = d. Let it be permanently irradiated by the action light with the frequency w in the direction e = k/Ikl = (0, 0, 1), where k is the wave vector. The action light excites electrons from the valence band (with a possibility via quasilocalized states in the mobility gap) and from quasilocalized states into the conduction band. A certain fraction of the excited electrons will come back to the lower states to be excited again and again. Thus the density n(x, t) of the excited electrons will fluctuate around its constant (time and space averaged) value n o with the density fluctuation
p(x, t). So n(x, t) = n o + p(x, t).
(1)
n o may possibly exceed the density of thermally excited electrons, if0 ~ 1016 c m - 3 , even by several orders. Excited electrons are assumed to be free carriers with the electric charge - e and the effective mass m. They move in the infinite one dimensional well under the presence of the electromagnetic field intensities E(x, t) and H(x, t). The intensities E(x, t) and H(x, t).
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The intensities E(x, t) and H(x, t), acting on a single electron described by the wave function ~s(X, t), are due to the laser field and motion of all other electrons. Suppose one solves the Schrodinger equation for all electrons selfconsistently. Then he gets the single electron density
Ps(X, t) -- Ss*(X, t) ffs(X, t),
(2)
and the single electron current density
is(X, t) = ~m (~0s(X, t) V ¢:(x, t) - ~:(x, t) V~s(X, t))
+
e-E-A(x, t)Ps(X, t),
(3)
mc
where A(x, t) is the vector potential. The function Ps(X, t) would then be the known distribution function satisfying the Boltzmann-like kinetic equation with a scattering term [e.g. 4]. By making the sum over occupied electron states s, one gets the total electron density
n(x, t) = n o + p(x, t) = ~ P s ( X , t),
25 July 1977
1 OH VX E + c~--= 0,
(10)
v'n=0.
(11)
For the sake of convenience we introduce the electron displacement u(x, t) as given by
o(x, t) =- du/dt.
(12)
The system of the partial differential equations (6)(11) is solvable in the linear approximation with respect to p ando. The axial symmetry of the action system requires all physical quantities in eqs. (6)-(11) to be functions only of z and t variables. The linearized pressure tensor Pij has the only non zero elements P33 = P - m n o r
Pll =P22 ~ P '
(av3/az),
(13)
where P is the scalar pressure and r is a transport coefficient related to the viscosity of the electron gass. Our chosen geometry allows to write down the linearised equations in the form
(4)
s
~t(p + n 0 V ' u )
0,
(6a)
and the total electron current density
/(x, t) = n(x, t)o(x, t ) - ~ is(x, t),
(5)
a2u _ at 2
e E - s2 Vp m no
+ ~ a
V(V" u),
(7a)
wherev(x, t) is the electron velocity averaged over the occupied states s. These quantities obey the equations
1 aE 47re"0":au VXH . . . . . . c at c at '
(8a)
(On/at) + V(n. v)= 0,
(6)
V" E = -4nep,
(9a)
(7)
plus the two homogeneons equations (10) and (11). Here s,
doi
-en~+
I c
vX
aPi/,
-):- , 0x, .
where d/dt = (a/at) + (~. v) is the convective derivative, i, ] = 1,2, 3 and Pi/ is the tensor of the pressure. The last equations follow from the Enskog transport equation [e.g. 4]. The electric charge and current densities are -ep and -enu. Thus Maxwell's equations are written down
s2=l
aP m an 0
04)
is the velocity of the compressional waves. The known boundary conditions are (15)
E3(0, t) = E3(d, t) = 0
V X H - 1 aE 41re nu, -- = cot c
(8)
H 3(o, t) = H a(d, t) = 0
V. E = - 41rep,
(9)
(E - e
o) X ~ = ( H - n
o) x e -- 0
(16)
103
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for z = 0 and z = d, and 0u3(0, t)/Ot = au3(d , O/at = 0,
25 July 1977
where (17)
utr(z, t) = u(+)(66t + kz) + u(-)(wt where E 0 and H 0 are intensities outside the film. The boundary conditions for u3(z , t) are incomplete. They should be suplemented by the equation of state relating the pressure P to the density n at z = 0 and z = d. These relations are not known. We only expect that the relations Du3(0, t)/Ozl = 10u3(d, t)/0zl,
(17a)
may be imposed. In deriving the linearised equations one tacitly assumes that both p(z, t) and u(z, t) are smoothly varying functions o f z and t. This is def'mitely not true for all z ~ (0, d) from the following reasons. Physical arguments tells that, e.g., Vp must have &function type discontinuities at least for z = 0 and z = d. General theory of the hyperbolic partial differential equations with a boundary consisting of two curves gives, as a rule, solutions with discontinuous derivatives [e.g. 5]. Thus the linearized equations (6a)-(9a) may describe dynamics of the action system pretty well for all z except for a few isolated points, where their solutions exhibit singularities. There result two types of waves in the action system, longitudinal and transverse. We start first with the longitudinal waves. For this case the eqs. (6a), (9a), (10), and (11)are integrated with the results
O = - n o (Ou3/Oz), E 3 = 4zre noU3,
(18) H 3 = 0.
(19)
The transverse waves Utr(Z , t) must be expressed as a sum of two terms with the arguments (cot - kz) (for waves propagating in z direction) and ( w t + kz) (for reflected waves) respectively. This allows us to carry out the integrations ofeqs. (Sa) and (10) with the resuits 47re n o
662
Err "-Co2 _ c2k2 Utr(Z, t),
- kz).
The result (21) explains why we have omitted a term with H in eq. (7a). Next we make use of eq. (7a) to get ~2utr _
Ot2
662 602 p
662 _ c2k 2 Utr'
(22)
where COp,
662 --_47re2no/m, P
(23)
is the frequency of the plasma oscillation [e.g. 6]. The (22) and the boundary conditions (16) require the relation k2 = 6o2
c2
2)_602 n 2 COp
7
(24)
where n r is the refractive index. The last relation is exactly the formula for determining the refractive index in classical theory of optical properties of metals [e.g. 6]. The results (20) and (21) tell us that the action system behaves for the transverse waves as a substance with the dielectric constant e = nr2 , magnetic permeability/a = 1 which has optical properties of metals. On the other hand, the relation (19) tells that the action system aquires properties of a perfect dielectricure, for the component of the electrical displacement D 3 =0. The longitudinal waves u 3 satisfy the differential equation a2u.__~ 3 _s2Au 3 + 2u at 2 COp 3 - K ~- Au 3 = 0,
(25)
with the boundary conditions (15) and (17). The boundary conditions require u3(z , t) to be a superposition of standing waves
(20)
u3(z, t) = = gv(t) sin -~ z,
(26)
Ittr = 66247re_n°c66kc2 k 2 {ug+)(66t + kz ) - U(tr-) (66t - kz ) } X e, (21) 104
where gu(t) are functions of time. Eq. (25) gives the
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exponential extinction of the longitudinal waves with the time constant r ~K
(27)
25 July 1977
6= -aa{/c bt ,
9~ = V × a{
--- 0.
V.~
(33) (34)
The inhomogeneous equation gives
providing that a2u3/~z 2 :# 0. Thus'the dynamical stability requires
1 a2.~
A a f f + V ( V ' s ~ ) + 4*re2--
c 2 ~t 2 ~2u3/Oz2 = 0,
rnc2 [n o + p(z, t)] .~ = 0.
(28)
to be satisfied for each z E (0, d) except for isolated points. The solution to eq. (25) respecting the relation (28) and the boundary conditions (15) and (17) is
where u03 and ct3 are integration constants. These constants may, of course, depend on both characteristics of a semiconductor and the action light. They enter our theory as phenomenological parameters. From eq. (29) we get the persistent oscillating electric current density
-eJ3(z,t)=enoCOpUO3(Iz-d l - d } sin(~Opt + ~3) (30) and the fluctuation density of electrons (31)
rent (30) is shifted by 7r/2 with respect to the phase of E3(z, t). Therefore the time averaged rate of doing work by E 3 is zero. The action system exhibits indeed properties typical for ordinary superconductors. These properties extinct with the relaxation time r given by eq. (27), when the action light is switched off. Suppose that an external weak electromagnetic field is applied on the action system. Then the electron density n(x, t) will be not changed in the first order of perturbation theory. Only the electron current density is changed as given by e
(36)
K = (K1, K2, 0).
(37)
The potential M is represented by t) = e
(38)
where ~ ---½(6opt + a3) is chosen as a new independent variable, s~ satisfies eq. (34) if and only if K . f ( ~ ) = 0.
where the sign + is for z E (0, d/2) and the sign - for z E (cl/2, d). Note that the phase ~Opt + ot3 of the cur-
9 (x, t) =](z, t) + ~ n(z, t) s~ (x, t),
a{. V p = 0.
Consider an electromagnetic plane wave, e.g. a laser light, guided through the action system perpendicularly to z direction. Its wave vector K has the components
U3(Z, t) = U03 Z --~ [--'~ COS(COpt+ 0t3) ,
p(z, t) = +noU03 coS(Wpt + a3) ,
(35) The continuity equation applied for the current 9(x, t) imposes the restriction
(32)
where a{ is the vector potential due to the applied external field. In this case the three homogeneous Maxwell equations are satisfied by the substitutions
Let us first consider a solution to eq. (35) when p = 0. The solution is a monochromatic plane wave (with the frequency ~2) given by
a~(x, t) = a ei(k'x- nt) where K and ~2 are related by i(2 __ (
2/c2) (1 -
_= (s 2/c 2)
and a is a constant vector. Since the refraction index nr(~ ) is imaginary for ~ < ~Op, in this case the amplitude of the guided light will decrease exponentially along its propagation direction. The penetration depth is
X = (c/~2(~o;/~22
1)- 1/2.
(39)
The intensity of the guided light may be modified by this mechanism in addition to others two investigated below. Next we consider a solution to eq. (35) with p 4:0 and given by eq. (31). The relation (36) requires 105
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PHYSICS LETTERS
a{3(x, t) = 0.
(40)
So C 3(x, t) = 0, andi~(x, t) has to be parallel to z. The action system exhibits a sort of a stiffness to remain a perfect dielectricum and to preserve its persistent current ]3(z, t)unchanged. This phenomenon reminds distantly the Meissner effect in ordinary superconductivity. The unpolarized guided light is thus truncated of its s~ 3 component. Its intensity has to drop at least to the value I = 1012, where I 0 is the intensity of the guided light in a semiconductor in absence of the action light. The degree of stopping introduced in [3] as ln(Io//) is ln(I0//) = In 2. This is the second mechanism for the optical stopping effect. Setting (38) into eq. (35) we get Mathieu's equation [7] d2f(~)/d~2,+ (7 +- 3' cos 2~)f(~) = 0,
(41)
where r/ =_4(c2K2]w 2
+ 1),
7 -= 4u03.
(42)
Eq. (41) has, according to theorem of Floquet [e.g. 7], the general solution f+(~) = al+ eu~ B(~) +a2+ e-U~B(-~), for z E (0, all2)
(43a)
and
(43b) for z E (d/2, d). Here ai±, i = 1,2, are constant vectors, B(~) is a periodic function with period rr and # is a characteristic root determined by the parameters 77and 3'. The characteristic root/a can be either pure imaginary or complex. In the pure imaginary case the plane wave e ik'xf(~) propagates through the action system without absorption. In the complex case, Re/a :/= 0, the amplitude of the guided light decreases exponentially with time. This gives the third mechanism for the optical stopping effect. By consulting the stability (r/, 3') diagram of Mathieu's equation [7] one finds that this phenomenon happens when r/satisfies the relation 77~ N 2, 106
(44)
25 July 1977
where N is any positive integer. Thus the degree of stopping as a function of K 2 will exhibit a resonance behaviour for those values o f K 2 which satisfy eq. (44). Our conclusions concerning the optical stopping effect are: (i) Three different mechanisms may contribute for the optical stopping effect. The effect is enlarged when two or all of them are mixed together. The degree of stopping, In(/0//), as a function of the wave length of the unpolarized guided light will have rather smooth parts with ln(I0//) = ln2 separated by large peaks. (ii) When the intensity of the action light is lowered so that the guided light may distort electron wave functions in the action system considerably, then the optical stopping effect becomes less visible. (iii) When the action light is switched off, then the fluctuation density of electrons (the presence of which is responsible for the optical stopping effect) extincts exponentially with time. Thus the intensity of the guided light recovers exponentially with time to its value I 0 determined by the optical properties of a semiconductor in the state of its thermal equilibrium. The listed properties of the optical stopping effect are in the perfect agreement with experimental observations [3]. However the found mechanisms responsible for this effect differ completely from those proposed in [3]. Our theory has treated a semiconductor illuminated by the action light macroscopically. That is why we cannot make quantitative predictions on properties Of the optical stopping effect which depend on the microscopic structure of the semiconductor. Theauthor acknowledges valuable discussions with Dr. F. Culik. References
[1] C. Cronstr6m and M. Noga, Phys. Lett. A60 (1977) 137. [2] E. Majernikov~,private communication. [3] M. Kikuchi and A. Matsuda, in: Proc. of the VIth Inter. Conf. on Amorphous and liquid semiconductors, Leningrad (USSR), 1975, p.35; A. Matsuda et al., Appl. Phys. Lett. 24 (1974) 314. [4] R.L. Liboff, Introduction to the theory of kinetic equations (John Wiley, New York, 1969). [5] G.A. Korn and T.M. Korn, Mathematical Handbook (MaeGraw-Hill, New York, 1968).
[6] J.D. Jackson, Classical eleetrodynamies (John Wiley, New York, 1967). [7] L. Brillouin, Wave propagation in periodic structures (MacGraw-Hill,New York, 1946).