- Email: [email protected]
Effect of dielectric anisotropy of quantum wells on reflection
Superlattices and Microstructures, VoL 5, No. 1, 1989
65
E F F E C T O F D I E L E C T R I C ANISOTBLOPY O F QUANTUM WELLS ON REFLECTION C. Zhang, M. Kohl and D. Heitmann
Max-Planck-Institut fftr Festk6rperforschung, T000 Stuttgart 80, FRG (Received 8 August 1988)
An investigation is made of the anlsotropic dielectric response function of quantum wells as a function of well width. We have calculated the Fresnel coefficient of reflection from such a structure taking into account the polar nature of the valence band Bloch states. The polarization dependence of the oscillator strength and resonant frequency is discussed. For the interband optical transitions, we used a two-band model to calculate the exciton wavefunction. The oecfllator strength and resonant frequency is strongly affected by the direction of the incident fields because the heavy and light holes are polarized in different directions. When the electric field is along the growth direction of the quantum wells, the Fresnel coefficient contains no optical transitions of s-like heavy hole due to the optical selection rules. The dielectric anisotropy results in a renormalization of the exciton oscillator strength and for light holes gives an additional resonant energy in the reflection. Numerical results for the differential reflectivity in GaAs-GaA1As quantum wells for some typical well width are presented.
I. I n t r o d u c t i o n The optical absorption in quantum well (QW) structures has become an important area of research due to the potential applications in optical device and also as a probe to study new physics in reduced dimensions[I]. Exciton transitions in QW structure in particular are acquiring special importance because the enhanced exciton binding energy and the presence of the barrier region allows excitonic transition to persist up to high temperature and at high transverse electric fields[2]. This has led to a number of structures with possible applications as optical modulators, optical switches, optical couplers etc[3]. Devices based on light incident both parallel and perpendicular to the layers have been proposed and demonstrated[4]. In the parallel geometry a strong polarization dependence of the absorption and emission have been observed. This polarization dependence has potential applications in optical processing technology and also as a probe to study band structure properties. Recently strongly polarized interband transitions in the perpendicular electric field was observed[5]. Reflectance spectroscopy is commonly used to determine excitonic parameters of both Bulk GaAs[6] and of quantum wells[7]. Specifically, reflectance spectra have revealed many features not appeared in photoluminescence, including higher-lying allowed and forbidden in0749-6036/89/010065 + 05 $02.00/0
terband optical transitions and n = l free exciton splitting attributed to minute layer thickness fluctuations. Although it is possible to obtain these information by other techniques, the comparative ease and accuracy of this method at room temperature have established it as a convenient tool for QW characterization. In this paper, we present a theoretical investigation of the optical reflectance of QW. The Freend coefficient of reflection is calculated taking into account the exciton transitions of q w . The different polar properties of the heavy and light hole tra nRitions lead to oscillator strengths and transition energies which are strongly depend on the polarization of the incident light. Another feature of two-dimensional systems is the depolarization shift in the transition energies observed when the light is polarized along the growth direction. These anisotropic properties of the dielectric function and optical selection rules have been included in our calculation. We use a simple model for the energies of the confined electron and hole states in the weak coupling limit (vanishing wavefunction overlap between adjacent wells). The quantum well exciton problem is treated using a variational approach and the valence bands mixing is neglected. An expression for the differential optical reflectance AR/I~ is derived with the inclusion of multiple internal reflections inside QW structure. It is shown that the reflection coefficient is strongly anisotropic; in the lowest order ap© 1989 Academic Press Limited
Superlattices and Microstructures, Vol. 5, No. 1, 1989
66 proximation, AI:I/R is proportional to th anlsotropy of the dielectric function for both perpendicular (Smode) and parallel (P-mode) polarization. Our result for the differentialreflectance is a function of barrier height, well width and the direction of the incident light. W e find that the dielectricanisotropy not only changes the amplitudes of the reflectance minima at the resonant energies but also produces a m a x i m u m in the reflectance. W e have calculated the reflectivity of GaAs-GaAIAs Q W structures for some typical values of sample parameters and incident angles. II. Dielectric Function From the Rrst order perturbation theory, the dielectric response function including excitonic and band to band transitions can be written as
~(~, k) = e~6,~a+ 41re2~ ~
es/~a~
quantum well of width d and barrier height V, the dispersion energy is given as
tan
d +
m i ( V - E) - m~E
=0 (4)
where r n , and mb are the effective masses inside the well and barrier. The exciton binding energy is still difficult to obtain either analytically or numerically. We shall adopt a commonly used variational approach, The exciton transition energy can be obtained by minimizing the total energy with respect to the variational parameters in the trialwave functions[Ill. It is well known that the resonant energy is not equal to band-to-band transitionenergy and the differenceis called depolarization shift which is denoted by ~wa'~a " in Eq.(1). This shift only appears when the field is along the z-directionbecause the Green's function G,z contains a terms -41r~(z-z'). The depolarization shift for the lowest transition can be calculated as[12]
T --,~,~
(1)
Swo -
81rP~e~ ~ wo f
dzV(z)¢(z)
(5)
where ebis the background dielectric constant and .fna~ is the oscillator strength for (n,m) optical transition which can be calculated in the usual way[8,9],
a# /Z. =
~ k!
Ia~.~Ck,)?eg.(k,)P~.(k,) (2)
where w0 is the transition energy without depolarization shift and @(z) is the exciton envelope function. Comparing to the isotropic case, this shiftcan be much enhanced if e== > > ez~. For thin square quantum well Eq.(5) can be evaluated as,
with
96a~fe2?t ~±
,~,.,.'o= ~
Pnam(kll) = ~_, f dguVPau~' / d z ~ ( z ) ~ ( z ) .
oo
(6)
(3)
/$v
where kll is the momentum component paralld to the layer, u g the I~th spin component of the Bloch function, ~,~(~,,,)the envelope function of n~a(m 0') subband for electrons (holes) and G,~,~(ktt) the Fourier transform of the exciton envelope function. The oscillator strength / is proportional to the value of the exciton envelope function at relative coordinate equal to zero, i.e,the intensity of allowed transitions is proportional to the probability of creating an electron and a hole at the same position. This fact is justifiedin both bulk and two dimensional case[10]. The interband momentum matrix dement < u~'lplu " > can be evaluated by making use of the orthogonality of the spin states and the symmetry properties of the cubic group[ll]. In Eq.(1),7 is the half width of exciton resonance. The broadening of the exciton transition can be understood due the structure inhomogeneity and phonon scattering. [E~(klt ) - E~(ktl)] is band-to-band transition energy of quantum well excitons. It consists of three parts: band gap of the GaAs, free electron and hole dispersion energy and exciton binding energy. For
where a8 = ebtt2/mre (m,. is the reduced mass). Because the depolarization shift is much enhanced for QW compared to bulk case, the resonant energy is strongly polarization dependent. If we choose principal axis such that all the oi~ diagonal dements of the dielectrictenser vanish and use the symmetry properties in x-y plane, we can write e== = %~ = ell = eb + Xll(k,w) = eb+
m~
~
[E~.(k,) - ~ ( h 0 ] 2 - ( ~ + i~) 2
(7)
e,, = ~l = eb + x±(k,w) = eb+
m2o~ ~
[E~(k,) - E~(k,) - *~...]~ - ( ~ + i~) 2 (8)
where .fll,/.L are the oscillator strengths in x-y and z directions. One notes that ell and e± have poles at wit and w T and zeros at w~ and w~ respectively.
Superlattices and Microstructures, VoL 5, No. 1, 1989 Ill. Optical Reflectance Let us consider an electromagnetic wave of the form E(z,k,w)e i(k~-~t), where the direction of the wave vector/~ along the surface is chosen to be the x-axis. Making use of rotational symmetry around the z-axis the Maxwell equations can be decoupled into two parts: (1) the S-polarization part for Ey, Hffi and H, and (2) the P-polarization part for E~,E~ and Hy. For the P-mode the surface admittance is defined as
Yp(k,w) = H,(O) E=(0)
(9)
The Fresnel reflection coefficient of the interface of two contiguous phases is defined as the ratio of the complex amplitudes of the electric field vectors of the reflected and incident waves. Its value is a function of the polarization of the incident beam and the angle of incidence and can be written in terms of Y1, as[13] kl
w
elll
cYI,
67 w h e r e ill is new dielectric function defined as 1 -- 1'lie tii=d
ill = ell 1 + rile likti
(15)
wl ell I i l l ki = [ell-~- - ~--~i.]
(16)
where
I n F~.(15), ei l ~ ( k l ) d and e - l : m ( l l ) i l ' t ~ l t s ,
respectively the phue change and damping on passing through the QW region of thickness d. The explicit expressions for Re(k2) and Im(kl) can be obtained
lt.e(t=)
where
with Kx and K2 given by (10)
ella e±n + ele~, -
cYi, ] 1/2
If1 =
w2 - q~
~A~'~
l
,,~-I
le±l 2
I .R (ii)
and the subscripts 1 and 2 denote the dielectrics of barrier and well region respectively. In Eq.(ll) the dielectric function of the barrier region is assumed to be isotropic and static, denoted by cA. The phase changes on reflection at the interface 1-2 is
[ Sm(rli) ]
' [-.<.+,/.<;+2,..] (17)
+ elll
=
II `±
K2 =
.R.I
"ll <± qisinl 8 le.i 2 -
_
jw-
2
II cl
where ea and e: represent, respectively, the real and imaginary part of the dielectric function and 8 is the az~le of incidence. In order for Eq.(13-18) to describe modes localized inside the QW, K1 must be greater than zero, or the frequency must be lower than the cutoff frequency wc given by
(12) I I FellR elR + elle± q2c2sin20]112
The function Y,(l,o~) can be obtained by solving selfconsistently the field and the current distributions of the system{Ill, The perturbation formula is we~ ~X(k,w) Well Y" = cT2 + ~ 2 = cT2
(13)
where e representsthe dielectricfunction including all excitations. If we substitute Eq.(13) in (10), we have Fresnel coefficient written in a usual way r = (kie~ + k , e ~ ) " For reflection from quantum wells of width d, r can be obtained from rxz and rix by adding up multiple reflections and transmissions (t12 = 1. - rx2) kl elll r = ~
kl if211
elll
elll
(18)
<"°=L
I
]
(19)
One of the effect of the dielectric anisotropy of QW is: if e l < ell, the QW exciton polariton can exist up to a high frequency of incident photon even at small 0. The resonant frequency of quantum well exciton polariton is given by
tanh(k2d + 612 + 621) = ~' 2 -- e . ~ ) l , / 2 2[Clle±(q2 - eAc--:-)(q (ell + e±)q 2 - e±(ell + eA)-'~-
(20)
The reflectivity at interface 1-2 is defined as R12 = ]r1212 and
R
=
Irl ~
=
R12 + R21e 4Im(ki)d + 2 ~ e 2 I m ( k 2 ) d c O S O 1 (14)
1 + R12R21e4I'n(k2) ~ + 2V/-~-~le2Im(kl)dCOS02 (21) where 81 = [612 '- 621 + 2Re(k2)a~ and 82 = [612 +
68
Superlattices and Microstructufes, Vol. 5, No. 1, 1989
1.5
1.5
f.-
1.0 C~
1.0
/
d
0.5
0.5
0'0786
I
I
790
795
0.C
800
804
786
wtnm) Fig.1 Plot d - ~ ~ a function d normalized frequency ~ for well width equal to 50tl, the ia-plane component of wave vector of incident light q® is 0.3106/cm, the two minima corresponding to the light and heavy hole exciton transitions and the maximum at high energy is due to the dielectric anisotropy. Solid curve is for anlsotropic model and broken curve is for isotropic model.
i
i
790
795 ~(nm}
804
800
Fig.2 Same as Fig.1 but q, = 0.4. 108/cm.
1.5
1.0 ¢D
821 - 2Re(k2)d]. Let us denote R,® and Ro respectively the reflectivity with and without excitonic excitation. Absolute measurement of the reflectivity R is very difficultin experimental systems, but the ratio Rc=/g, can be measured accurately with relative ease owing to a cancellation of c o m m o n error. Un£ortunately, the exact theoretical expression for R, Eq.(21), is too complicated to give any direct insight into how the dielectric anisotropy and Q W excitonic excitation affect the reflectivity of a real system. W e have evaluated Eq.(21) numerically the reflectivity for several different well widths and incident angles and the resalts are plotted in Fig.l-4. If the well width is small we can seek a perturbation expansion for AR/R due to the change in e from excitonic transition, we write
0.5
0.0
I
I
750
I
1
760
I
770
w(nm)
Fig.3 Same as Fig.1 but d=100~l.
1.5
10 5
k2=
f
t3
( +X,)-j-Lcb+x±j (22)
0.5 with o
k2 =
ebb- -- q2sin28
(23)
and
0.0
1 [
~2
Xil_X±q28in2{~ ]
(24)
L
750
I
I
i
760
w{nm}
Fig.4 Same as Fig.2 but d=100)l.
770
Superlattices and Microstructures, Vol. 5, No. I, 1989 Similarly we expand the interface Fresnel coefficient as o + ~r12 r12 = r12
(25)
where r12 is given by Eq.(10) but without QW excitation and ~r12 is 02 ~rt~. = ~i2 XIIk~- XII~ c2 +
q2sin20 (26)
Now the differential reflectance is
OR 6k
AR = -~0~R (~r120r12 4- ~
2
(27)
Upon using Eqs.(25-29), our final result for A R / R is
AR R
~_~2im[xiik~ 2
Xllw~ + ~ q Z s i n 2 O ] 2c2
(28) We see immediately that all excitation vanishes as the thickness d approaches zero. From this analytical result Eq.(28), it is clear that the term proportional to [XII - X±I represents the effect of dielectric anisotropy on optical reflection. IV. Result and D i s c u s s i o n In this paper, we have calculated the optical reflectance of quantum well systems with the inclusion of excitonic transition and the dielectric anisotropy associated with these transitions. The general result is given in Eq.(21) which is valid for system of any width and barrier height. The dielectric function is chosen to be a Lorentzian with a polarization dependent oscillator strength (or longitudinal-transverse splitting) and transition energy. However, through our work the nonlocal effect is not taken into account, the result should be accurate if the wavelength of the incident light is much longer compared to the well width and effective exciton Bohr radius. In the figures, we have plotted the differential reflectivity for a GaAs-GaA1As QW system (normalized to its high frequency value) as a function of normalized frequency for some typical values of the structural parameters and for several different in-plane incident momentum. The solid lines are for the azrisotropic model and the broken lines for isotropic model. We found that the resonant energies for both heavy and light hole excitons are almost at the same position for both models and the reflectivity shows a minimum at these resonant energies. However, the size of minima is larger for the anisotropic model compared to the isotropic model and are increasing with the wave vector of the incident light. It can be
69
understood that the dielectric anisotropy affects the optical reflectance only through the wave vector inside the quantum well, k2. For small q, the difference became smaller. From perturbation expansion Eq.(28), one can see that (XJl - X i ) is weighted by a factor q2, therefore the isotropic model is applicable in the long wavelength limit. Another interesting feature is the maximum reflection at the high energy region. The position of this maximum is corresponding to depolarization shift in X±. Since X± enters the reflection as a negative quantity, its pole will be the position of maximum reflection. Again one sees that this maximum decreases as q becomes smaller. In conclusion, we have presented a calculation of reflectivity of a quantum well system. The anisotropic nature of the structure will affect the reflectivity especially when q is not small.
Acknowledgement - We thank M. A. Brummell and R. R. Gerhardts for helpful discussion and critical reading of the manuscript. References 1. G. Bastard, Phys. Rev. B26, 1974 (1982) 2. J. S. Weiner, D. S. Chemla, D. A. B. Miller, T. H. Wood, D. Sireo and A. Y. Cho, Appl. Phys. Lett. 46, 619 (1985) 3. T. H. Wood, C. A. Burrus, D. A. B. Miller, D. S. Chemla, T. C. Damen, A. C. Gossard and W. Wiegman, Appl. Phys. Left. 44, 16 (1984) 4. K. Wakita, Y. Kawamura, Y. Yoshikuni and H. Asaki, IEEE-QE, 22(9), 1831 (1986) 5. M. Kohl, D. Heitmann, P. Grambow and K. Ploog, 6. L. Schultheis and J. Lagois, Phys. Rev. B29, 6784 (1984); L. Schultheis and I. Balslev, Phys. Rev. B28, 2292 (1983); Phys. Rev. B37, 10927 (1988) 7. L. Schultheis and K. Ploog, Phys. rev. B30, 1090 (1984); P. J. Pearah, J. Klein, T. Henderson, Ck K. Peng H. Morkoc, D. C. Reynolds and C. W. Litton, 3. Appl. Phys. 59(11) 3847 (1986) 8. R. J. Elliott, Phys. Rev, 108, 1384 (1966) 9. M. Shinada and S. Sugano, J. Phys. Soc. Jpn. 21, 32 (1966) 10. R. ]. Elliott, Phys. Rev. 105, 1384 (1957) 11. Bangfen Zhu, Phys. Rev. B37, 4689 (1988); M. Matsuura and Y. Shinasu]m, J. Phys. Soc. jpn. 53, 3138 (1984); R. L. Greene, K. K. Bajaj and D. E. Phelps, Phys. Rev. B29, 1807 (1984); C. Weisbuch, R. C. Miller, R. Dingle, A. C. Gossard and W. Wiegmann, Solid State Commun. 37, 219 (1981) 12. T. Ando, A. B. Fowler and F. Stern, Rev. Mod. Phys. 54, 437 (1982) 13. M. Nakayama, J. Phys. Soc. Jpn. 39,265 (1974)
65
E F F E C T O F D I E L E C T R I C ANISOTBLOPY O F QUANTUM WELLS ON REFLECTION C. Zhang, M. Kohl and D. Heitmann
Max-Planck-Institut fftr Festk6rperforschung, T000 Stuttgart 80, FRG (Received 8 August 1988)
An investigation is made of the anlsotropic dielectric response function of quantum wells as a function of well width. We have calculated the Fresnel coefficient of reflection from such a structure taking into account the polar nature of the valence band Bloch states. The polarization dependence of the oscillator strength and resonant frequency is discussed. For the interband optical transitions, we used a two-band model to calculate the exciton wavefunction. The oecfllator strength and resonant frequency is strongly affected by the direction of the incident fields because the heavy and light holes are polarized in different directions. When the electric field is along the growth direction of the quantum wells, the Fresnel coefficient contains no optical transitions of s-like heavy hole due to the optical selection rules. The dielectric anisotropy results in a renormalization of the exciton oscillator strength and for light holes gives an additional resonant energy in the reflection. Numerical results for the differential reflectivity in GaAs-GaA1As quantum wells for some typical well width are presented.
I. I n t r o d u c t i o n The optical absorption in quantum well (QW) structures has become an important area of research due to the potential applications in optical device and also as a probe to study new physics in reduced dimensions[I]. Exciton transitions in QW structure in particular are acquiring special importance because the enhanced exciton binding energy and the presence of the barrier region allows excitonic transition to persist up to high temperature and at high transverse electric fields[2]. This has led to a number of structures with possible applications as optical modulators, optical switches, optical couplers etc[3]. Devices based on light incident both parallel and perpendicular to the layers have been proposed and demonstrated[4]. In the parallel geometry a strong polarization dependence of the absorption and emission have been observed. This polarization dependence has potential applications in optical processing technology and also as a probe to study band structure properties. Recently strongly polarized interband transitions in the perpendicular electric field was observed[5]. Reflectance spectroscopy is commonly used to determine excitonic parameters of both Bulk GaAs[6] and of quantum wells[7]. Specifically, reflectance spectra have revealed many features not appeared in photoluminescence, including higher-lying allowed and forbidden in0749-6036/89/010065 + 05 $02.00/0
terband optical transitions and n = l free exciton splitting attributed to minute layer thickness fluctuations. Although it is possible to obtain these information by other techniques, the comparative ease and accuracy of this method at room temperature have established it as a convenient tool for QW characterization. In this paper, we present a theoretical investigation of the optical reflectance of QW. The Freend coefficient of reflection is calculated taking into account the exciton transitions of q w . The different polar properties of the heavy and light hole tra nRitions lead to oscillator strengths and transition energies which are strongly depend on the polarization of the incident light. Another feature of two-dimensional systems is the depolarization shift in the transition energies observed when the light is polarized along the growth direction. These anisotropic properties of the dielectric function and optical selection rules have been included in our calculation. We use a simple model for the energies of the confined electron and hole states in the weak coupling limit (vanishing wavefunction overlap between adjacent wells). The quantum well exciton problem is treated using a variational approach and the valence bands mixing is neglected. An expression for the differential optical reflectance AR/I~ is derived with the inclusion of multiple internal reflections inside QW structure. It is shown that the reflection coefficient is strongly anisotropic; in the lowest order ap© 1989 Academic Press Limited
Superlattices and Microstructures, Vol. 5, No. 1, 1989
66 proximation, AI:I/R is proportional to th anlsotropy of the dielectric function for both perpendicular (Smode) and parallel (P-mode) polarization. Our result for the differentialreflectance is a function of barrier height, well width and the direction of the incident light. W e find that the dielectricanisotropy not only changes the amplitudes of the reflectance minima at the resonant energies but also produces a m a x i m u m in the reflectance. W e have calculated the reflectivity of GaAs-GaAIAs Q W structures for some typical values of sample parameters and incident angles. II. Dielectric Function From the Rrst order perturbation theory, the dielectric response function including excitonic and band to band transitions can be written as
~(~, k) = e~6,~a+ 41re2~ ~
es/~a~
quantum well of width d and barrier height V, the dispersion energy is given as
tan
d +
m i ( V - E) - m~E
=0 (4)
where r n , and mb are the effective masses inside the well and barrier. The exciton binding energy is still difficult to obtain either analytically or numerically. We shall adopt a commonly used variational approach, The exciton transition energy can be obtained by minimizing the total energy with respect to the variational parameters in the trialwave functions[Ill. It is well known that the resonant energy is not equal to band-to-band transitionenergy and the differenceis called depolarization shift which is denoted by ~wa'~a " in Eq.(1). This shift only appears when the field is along the z-directionbecause the Green's function G,z contains a terms -41r~(z-z'). The depolarization shift for the lowest transition can be calculated as[12]
T --,~,~
(1)
Swo -
81rP~e~ ~ wo f
dzV(z)¢(z)
(5)
where ebis the background dielectric constant and .fna~ is the oscillator strength for (n,m) optical transition which can be calculated in the usual way[8,9],
a# /Z. =
~ k!
Ia~.~Ck,)?eg.(k,)P~.(k,) (2)
where w0 is the transition energy without depolarization shift and @(z) is the exciton envelope function. Comparing to the isotropic case, this shiftcan be much enhanced if e== > > ez~. For thin square quantum well Eq.(5) can be evaluated as,
with
96a~fe2?t ~±
,~,.,.'o= ~
Pnam(kll) = ~_, f dguVPau~' / d z ~ ( z ) ~ ( z ) .
oo
(6)
(3)
/$v
where kll is the momentum component paralld to the layer, u g the I~th spin component of the Bloch function, ~,~(~,,,)the envelope function of n~a(m 0') subband for electrons (holes) and G,~,~(ktt) the Fourier transform of the exciton envelope function. The oscillator strength / is proportional to the value of the exciton envelope function at relative coordinate equal to zero, i.e,the intensity of allowed transitions is proportional to the probability of creating an electron and a hole at the same position. This fact is justifiedin both bulk and two dimensional case[10]. The interband momentum matrix dement < u~'lplu " > can be evaluated by making use of the orthogonality of the spin states and the symmetry properties of the cubic group[ll]. In Eq.(1),7 is the half width of exciton resonance. The broadening of the exciton transition can be understood due the structure inhomogeneity and phonon scattering. [E~(klt ) - E~(ktl)] is band-to-band transition energy of quantum well excitons. It consists of three parts: band gap of the GaAs, free electron and hole dispersion energy and exciton binding energy. For
where a8 = ebtt2/mre (m,. is the reduced mass). Because the depolarization shift is much enhanced for QW compared to bulk case, the resonant energy is strongly polarization dependent. If we choose principal axis such that all the oi~ diagonal dements of the dielectrictenser vanish and use the symmetry properties in x-y plane, we can write e== = %~ = ell = eb + Xll(k,w) = eb+
m~
~
[E~.(k,) - ~ ( h 0 ] 2 - ( ~ + i~) 2
(7)
e,, = ~l = eb + x±(k,w) = eb+
m2o~ ~
[E~(k,) - E~(k,) - *~...]~ - ( ~ + i~) 2 (8)
where .fll,/.L are the oscillator strengths in x-y and z directions. One notes that ell and e± have poles at wit and w T and zeros at w~ and w~ respectively.
Superlattices and Microstructures, VoL 5, No. 1, 1989 Ill. Optical Reflectance Let us consider an electromagnetic wave of the form E(z,k,w)e i(k~-~t), where the direction of the wave vector/~ along the surface is chosen to be the x-axis. Making use of rotational symmetry around the z-axis the Maxwell equations can be decoupled into two parts: (1) the S-polarization part for Ey, Hffi and H, and (2) the P-polarization part for E~,E~ and Hy. For the P-mode the surface admittance is defined as
Yp(k,w) = H,(O) E=(0)
(9)
The Fresnel reflection coefficient of the interface of two contiguous phases is defined as the ratio of the complex amplitudes of the electric field vectors of the reflected and incident waves. Its value is a function of the polarization of the incident beam and the angle of incidence and can be written in terms of Y1, as[13] kl
w
elll
cYI,
67 w h e r e ill is new dielectric function defined as 1 -- 1'lie tii=d
ill = ell 1 + rile likti
(15)
wl ell I i l l ki = [ell-~- - ~--~i.]
(16)
where
I n F~.(15), ei l ~ ( k l ) d and e - l : m ( l l ) i l ' t ~ l t s ,
respectively the phue change and damping on passing through the QW region of thickness d. The explicit expressions for Re(k2) and Im(kl) can be obtained
lt.e(t=)
where
with Kx and K2 given by (10)
ella e±n + ele~, -
cYi, ] 1/2
If1 =
w2 - q~
~A~'~
l
,,~-I
le±l 2
I .R (ii)
and the subscripts 1 and 2 denote the dielectrics of barrier and well region respectively. In Eq.(ll) the dielectric function of the barrier region is assumed to be isotropic and static, denoted by cA. The phase changes on reflection at the interface 1-2 is
[ Sm(rli) ]
' [-.<.+,/.<;+2,..] (17)
+ elll
=
II `±
K2 =
.R.I
"ll <± qisinl 8 le.i 2 -
_
jw-
2
II cl
where ea and e: represent, respectively, the real and imaginary part of the dielectric function and 8 is the az~le of incidence. In order for Eq.(13-18) to describe modes localized inside the QW, K1 must be greater than zero, or the frequency must be lower than the cutoff frequency wc given by
(12) I I FellR elR + elle± q2c2sin20]112
The function Y,(l,o~) can be obtained by solving selfconsistently the field and the current distributions of the system{Ill, The perturbation formula is we~ ~X(k,w) Well Y" = cT2 + ~ 2 = cT2
(13)
where e representsthe dielectricfunction including all excitations. If we substitute Eq.(13) in (10), we have Fresnel coefficient written in a usual way r = (kie~ + k , e ~ ) " For reflection from quantum wells of width d, r can be obtained from rxz and rix by adding up multiple reflections and transmissions (t12 = 1. - rx2) kl elll r = ~
kl if211
elll
elll
(18)
<"°=L
I
]
(19)
One of the effect of the dielectric anisotropy of QW is: if e l < ell, the QW exciton polariton can exist up to a high frequency of incident photon even at small 0. The resonant frequency of quantum well exciton polariton is given by
tanh(k2d + 612 + 621) = ~' 2 -- e . ~ ) l , / 2 2[Clle±(q2 - eAc--:-)(q (ell + e±)q 2 - e±(ell + eA)-'~-
(20)
The reflectivity at interface 1-2 is defined as R12 = ]r1212 and
R
=
Irl ~
=
R12 + R21e 4Im(ki)d + 2 ~ e 2 I m ( k 2 ) d c O S O 1 (14)
1 + R12R21e4I'n(k2) ~ + 2V/-~-~le2Im(kl)dCOS02 (21) where 81 = [612 '- 621 + 2Re(k2)a~ and 82 = [612 +
68
Superlattices and Microstructufes, Vol. 5, No. 1, 1989
1.5
1.5
f.-
1.0 C~
1.0
/
d
0.5
0.5
0'0786
I
I
790
795
0.C
800
804
786
wtnm) Fig.1 Plot d - ~ ~ a function d normalized frequency ~ for well width equal to 50tl, the ia-plane component of wave vector of incident light q® is 0.3106/cm, the two minima corresponding to the light and heavy hole exciton transitions and the maximum at high energy is due to the dielectric anisotropy. Solid curve is for anlsotropic model and broken curve is for isotropic model.
i
i
790
795 ~(nm}
804
800
Fig.2 Same as Fig.1 but q, = 0.4. 108/cm.
1.5
1.0 ¢D
821 - 2Re(k2)d]. Let us denote R,® and Ro respectively the reflectivity with and without excitonic excitation. Absolute measurement of the reflectivity R is very difficultin experimental systems, but the ratio Rc=/g, can be measured accurately with relative ease owing to a cancellation of c o m m o n error. Un£ortunately, the exact theoretical expression for R, Eq.(21), is too complicated to give any direct insight into how the dielectric anisotropy and Q W excitonic excitation affect the reflectivity of a real system. W e have evaluated Eq.(21) numerically the reflectivity for several different well widths and incident angles and the resalts are plotted in Fig.l-4. If the well width is small we can seek a perturbation expansion for AR/R due to the change in e from excitonic transition, we write
0.5
0.0
I
I
750
I
1
760
I
770
w(nm)
Fig.3 Same as Fig.1 but d=100~l.
1.5
10 5
k2=
f
t3
( +X,)-j-Lcb+x±j (22)
0.5 with o
k2 =
ebb- -- q2sin28
(23)
and
0.0
1 [
~2
Xil_X±q28in2{~ ]
(24)
L
750
I
I
i
760
w{nm}
Fig.4 Same as Fig.2 but d=100)l.
770
Superlattices and Microstructures, Vol. 5, No. I, 1989 Similarly we expand the interface Fresnel coefficient as o + ~r12 r12 = r12
(25)
where r12 is given by Eq.(10) but without QW excitation and ~r12 is 02 ~rt~. = ~i2 XIIk~- XII~ c2 +
q2sin20 (26)
Now the differential reflectance is
OR 6k
AR = -~0~R (~r120r12 4- ~
2
(27)
Upon using Eqs.(25-29), our final result for A R / R is
AR R
~_~2im[xiik~ 2
Xllw~ + ~ q Z s i n 2 O ] 2c2
(28) We see immediately that all excitation vanishes as the thickness d approaches zero. From this analytical result Eq.(28), it is clear that the term proportional to [XII - X±I represents the effect of dielectric anisotropy on optical reflection. IV. Result and D i s c u s s i o n In this paper, we have calculated the optical reflectance of quantum well systems with the inclusion of excitonic transition and the dielectric anisotropy associated with these transitions. The general result is given in Eq.(21) which is valid for system of any width and barrier height. The dielectric function is chosen to be a Lorentzian with a polarization dependent oscillator strength (or longitudinal-transverse splitting) and transition energy. However, through our work the nonlocal effect is not taken into account, the result should be accurate if the wavelength of the incident light is much longer compared to the well width and effective exciton Bohr radius. In the figures, we have plotted the differential reflectivity for a GaAs-GaA1As QW system (normalized to its high frequency value) as a function of normalized frequency for some typical values of the structural parameters and for several different in-plane incident momentum. The solid lines are for the azrisotropic model and the broken lines for isotropic model. We found that the resonant energies for both heavy and light hole excitons are almost at the same position for both models and the reflectivity shows a minimum at these resonant energies. However, the size of minima is larger for the anisotropic model compared to the isotropic model and are increasing with the wave vector of the incident light. It can be
69
understood that the dielectric anisotropy affects the optical reflectance only through the wave vector inside the quantum well, k2. For small q, the difference became smaller. From perturbation expansion Eq.(28), one can see that (XJl - X i ) is weighted by a factor q2, therefore the isotropic model is applicable in the long wavelength limit. Another interesting feature is the maximum reflection at the high energy region. The position of this maximum is corresponding to depolarization shift in X±. Since X± enters the reflection as a negative quantity, its pole will be the position of maximum reflection. Again one sees that this maximum decreases as q becomes smaller. In conclusion, we have presented a calculation of reflectivity of a quantum well system. The anisotropic nature of the structure will affect the reflectivity especially when q is not small.
Acknowledgement - We thank M. A. Brummell and R. R. Gerhardts for helpful discussion and critical reading of the manuscript. References 1. G. Bastard, Phys. Rev. B26, 1974 (1982) 2. J. S. Weiner, D. S. Chemla, D. A. B. Miller, T. H. Wood, D. Sireo and A. Y. Cho, Appl. Phys. Lett. 46, 619 (1985) 3. T. H. Wood, C. A. Burrus, D. A. B. Miller, D. S. Chemla, T. C. Damen, A. C. Gossard and W. Wiegman, Appl. Phys. Left. 44, 16 (1984) 4. K. Wakita, Y. Kawamura, Y. Yoshikuni and H. Asaki, IEEE-QE, 22(9), 1831 (1986) 5. M. Kohl, D. Heitmann, P. Grambow and K. Ploog, 6. L. Schultheis and J. Lagois, Phys. Rev. B29, 6784 (1984); L. Schultheis and I. Balslev, Phys. Rev. B28, 2292 (1983); Phys. Rev. B37, 10927 (1988) 7. L. Schultheis and K. Ploog, Phys. rev. B30, 1090 (1984); P. J. Pearah, J. Klein, T. Henderson, Ck K. Peng H. Morkoc, D. C. Reynolds and C. W. Litton, 3. Appl. Phys. 59(11) 3847 (1986) 8. R. J. Elliott, Phys. Rev, 108, 1384 (1966) 9. M. Shinada and S. Sugano, J. Phys. Soc. Jpn. 21, 32 (1966) 10. R. ]. Elliott, Phys. Rev. 105, 1384 (1957) 11. Bangfen Zhu, Phys. Rev. B37, 4689 (1988); M. Matsuura and Y. Shinasu]m, J. Phys. Soc. jpn. 53, 3138 (1984); R. L. Greene, K. K. Bajaj and D. E. Phelps, Phys. Rev. B29, 1807 (1984); C. Weisbuch, R. C. Miller, R. Dingle, A. C. Gossard and W. Wiegmann, Solid State Commun. 37, 219 (1981) 12. T. Ando, A. B. Fowler and F. Stern, Rev. Mod. Phys. 54, 437 (1982) 13. M. Nakayama, J. Phys. Soc. Jpn. 39,265 (1974)