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Dependence on dimensionality of excitonic optical nonlinearity in quantum confined structures
~)
Solid State Communications, Vol. 78, No. 4, pp. 279-282, 1991. Printed in Great Britain.
DEPENDENCE
0038-1098/9153.00+.00 Pergamon Press plc
ON DIMENSIONALITY OF EXCITONIC OPTICAL NONLINEARITY IN QUANTUM CONFINED STRUCTURES
T. Takngahara NTT Basic Research Laboratories, Musashino-shi, Tokyo 180, JAPAN (Received 8 February 1991 by T. Tsuzuki) The dependence on dimensionality of the excitonic optical nonlinearity and the figures of merit in quantum confined structures is clarified for the resonant thirdorder optical nonlinearity. An important parameter which determines the dimensionality dependence of the figures of merit is found to be the ratio of the homogeneous linewidth to the exciton binding energy or the ratio of the exciton coherence length t o the exciton Bohr radius. It is predicted that for the III-V compound semiconductors the exciton confinement in low-dimensional structures d = l , 2 d:dimensionality) is favorable for enhancing the figures of merit, whereas r the I-VII compound semiconductors the exciton confinement does not always improve the figures of merit.
Recently the enhanced excitonic optical nonlinearity in low-dimensional materials has attracted much attention from the viewpoint of fundamental physics and from the interest of application to optical devices 1,2. In GaAs-A1GaAs quantum well structures, a large X(s) value of the order of 10 -2 ,,, 10 -1 esu was observed 3 and this absorptive nonlinearity was interpreted in terms of phase space filling of free carriers and excitons 4. A large X(3) value was predicted theoretically in semiconductor microcrystaJlitess,s and a X(3) value of the order of 10 -5 esu was observed for CuC1 microcrystallites embedded in a NaCI matrix 7. On the other hand, a large number of organic materials having a conjugated 7relectron system have been synthesized and their large optical nonlinearity with very short response time has been investigated s. The most important mechanism for enhancing the optical nonlinearity is usually considered to be the quantum confinement effect which leads to a discrete level structure and causes the concentration of oscillator strength to the lowest energy transitions. Since the density of electronic states becomes sharper and sharper by reducing the dimensionality of structures, it seems that a sharp excitonic feature will be realized in low-dimensional materials. In this respect, the zero-dimensional structures, namely quantum dots seem to be most favorable for enhancing the optical nonlinearity. However, the spectral broadening in low-dimensional structures is significant due to the increasingly important role of the interface scattering and due to the enhanced electron-phonon coupling 9,1°. Thus the enhancement of the optical nonlinearity cannot be achieved simply by the exciton confinement in low-dimensional structures. It is essentially important to take into account the homogeneous broadening in discussing the effect of quantum confinement on the optical nonlinearity. There has been so far no systematic study on the dimensionality dependence of the excitonic optical nonlinearity which takes account of the above aspects. In this paper we will focus our attention on the efficiencies of the third-order optical nonlinearity which are usually called the figures of merit
and will clarify their dependence on dimensionality of structures. As a consequence, we will find out for the first time an important parameter which determines the dimensionality dependence of the figures of merit. The third-order nonlinear susceptibility 3((3) is calculated from the third-order perturbation theory with summation over the intermediate states which can be one-exciton states or two-exciton states. A detailed expression of X(3) is given in Ref.ll and consists of two types of contribution; one of which involves only one-exciton states as the intermediate state and the other includes two-exciton states as well. In general, the magnitude of X(s) is determined by the partial cancellation between the two types of contribution and it is rather difficult to obtain an analytically closed expression. However, from various numerical calculations using realistic parameters, we have found that the magnitude of X(3) in the vicinity of the exciton resonance can be safely approximated by the contribution from the one-exciton states only. More explicitly, the maximum value of X(3) at the exciton resonance can be approximated as
1 '~T'( e" ,~2 f~,
where m0 is the free electron mass, V the system volume, /iwi the exciton transition energy, 711i(7-L~) the longitudinal (transverse) relaxation rate, fxi the exciton oscillator strength, the summation is taken over the exciton levels lying within the homogeneous linewidth of the ground exciton state and a factor due to the local field correction is omitted. The oscillator strength of the free exciton state is determined by the exciton coherence volume and this coherence volume is closely related to the homogeneous linewidth of the excitonic transition 12. The homoseneous linewidth refiects the energy fluctuation of the exciton state and 279
280
DIMENSIONALITY OF EXCITONIC OPTICAL NONLINEARITY
in turn represents the wave-vector fluctuation according to the dispersion relation. This wave-vector fluctuation restricts the exciton coherent motion within a range of the order of inverse the wave-vector fluctuation. In other words, the oscillator strength of the K = 0 exciton is redistributed among all states within the homogeneous linewidth. The redistributed oscillator strength per one exciton state can be written in terms of the exciton coherence length Ic as
f x = f 0 ( l cd D ) d I f d D (0)[ 2 ,
(2)
where f0 is the oscillator strength of the band-to-band transition at the F point and FdD(r~ --rb) denotes the envelope function of the electron-hole relative motion ( d = l , 2 and 3). The coherence length for each dimension is given by
v~
~
'
t° -
ii D =
~
'
rr~z
V2M1DAaD,
(3)
where A d D and M a D are the homogeneous linewidth and the translational mass of the exciton in the ddimensional structure. Then the ratio among maxim u m values of the imaginary part of X (3) for threedimensional(3D), two-dimensional(2D) and one-dimensional(ID) systems is obtained as
Vol. 78, No. 4
favorable for enhancing the magnitude of the optical nonlinearity X(s). However, the situation is more delicate concerning the figures of merit of the optical nonlinearity. The figures of merit are usually defined by the changes in the absorption coefficient a and the refractive index n induced per one electron-hole pair(exciton) in a unit volume, which are denoted by a and ,/, respectively and given as a = as + aN, n = no + rlN, where a0(n0) is the absorption coefficient(refractive index) under no excitation and N is the number density of electron-hole pairs(excitons). These quantities are related to the imaginary and real parts of X(s), respectively as ~ 32rr2 hw2 Imx(s)
~ = - - - - , ~c 2
6XoT
,~=
16r2hw ReX (3) ~C
O~oT
1 2(13D'~3 ~ 2D '~ 1 ~D N~D = -,-¢ , , NiO = 2(lc ) L,,N----~--g= 21c L z L v,
(7)
= 0~DpIF~D(0)P. (I~D)~IF~D(0)P. t~DIF'D(0)P [[
L'a3D
L ~2DA2 Z/l[
~'~2D
"L
z
L ~IDA2 i~Iii
ID
'
(4) where Lz is the quantum well thickness of the 2D system and L~L~ the cross-sectional area of the 1D quantum wire. It is to be noted that for 2D and 1D systems, X(3) is estimated by supposing a filling factor of 100 ~ , namely quantum wells and quantum wires fill all the space without changing the electronic structure. Introducing the exciton Bohr radius a dD associated with FaD(re -- rh) and a dimensionless parameter/3 defined by/3 = mv~--~/(1 + m , ~ ) with the electron-to-hole mass ratio m,h in the bulk material, the ratio (4) can be further reduced to 6~ 3
16~ 2
M3D
~D(XsD)r/2 : 7 r Z , ~ D ( ~ 2 0 ) a ( ~ D ) 4 M2D
LzLy T~ID( ~ I D )5/2 (-dlBD)2V -M-~1D'
(o)
where e is the dielectric constant of the material and r is the exciton lifetime(71~-z). It should be noted that the figures of merit do not contain explicitly the exciton lifetime and structure parameters like the quantum well thickness and the filling factor, reflecting the more fundamental character of these quantities compared to XO) . Now the dimensionality dependence of the figures of merit will be derived on the basis of the findin_~ that the saturation density N° of the absorption coefficient is closely related to the exciton coherence length. In fact we find that
Imx0)(3D)[~.~ : Imx(S)(2D)[~.~ : ImxO)(iD)[~.~ 3DA2
,
(5)
where the linewidth and length are scaled by the effective Rydberg(Ry • ) and the exciton Bohr radius(a 3D B ) in the bulk material, respectively and this scaling is indicated by an overbar. For the two-dimensional exciton, it is well known that ~ D = 1/2 zs, whereas for the one-dimensional exciton, we have ~ D ~ 0.2 for Lz ~-- Ly ~- 0.3 14. Thus we can expect a l a r g e enhancement of the X(s) value in low-dimensional materials so far as the homogeneous linewidth is not very much increased. As seen above, the exciton confinement is generally
where the factor 2 comes from the definition of X(3) employed in Ref.ll. These results can be interpreted that an exciton has a coherent range of the order of lc with respect to the center-of-mass motion and the creation of another exciton within that range is inhibited. In the 2D case, N0 due to the phase space filling mechanism4 is calculated as l/N°2 D = 321r(a 2BD )2 Lz/7. This implies that the kinematical exciton-exciton interaction due to the Pauli exclusion principle restricts the exciton coherence length to a few times the exciton Bohr radius. Actually the dynamical interaction of excitons with various elementary excitations shortens the coherence length furthermore and increases the saturation density. The expression (7) is most general including all the contributions in terms of the exciton coherence length. In order to envisage qualitatively the dimensionality dependence of the figures of merit, we compare 2D and 3D cases. First of all, 2D 3 D ,~v 3D • we have .IV'. / N ~s - Ic / L , , supposing Ic3 D -~, Ic2 D . Here the quantum well thickness Lz can be replaced by a~D because the characteristic length that distinguishes 2D and 3D systems is the exciton Bohr radius and the two-dimensional character becomes remarkable for Lz <_ a~D. Then noticing the relation l/N° = a/as, we can deduce that lc/a 3s D appears as a characteristic parameter which describes the dimensionality dependence of the figures of merit of the optical nonlinearity. Through a more detailed algebra, we find that a(3D) : a(2D) : a(1D) = T/(3D) : r/(2D) : r/(1D) 6~r~ 3 8~ ~ MSD : 7r~ = ( ~ 3 D p / ~ : ( ~ 2 D ) 2 ( ~ D ) 2 M2D ( ~ , D p / ~
, D V M~D
(7:o) ` M D. (7:°) M,D = (36~') 1D : 2 ( ~ 0 ) 2 M3D " "~bD M ~o '
(8)
Vol. 78, No. 4
DIMENSIONALITY OF EXCITONIC OPTICAL NONLINEARITY
where the maximum values of a and T/around the exciton resonance are compared. It is confirmed that the dimensionality dependence of the fisures of merit is determined by the parameter ~,dD or [~D -- IdD/a3BD. These two parameters are related to each other as/XdD = O(1)fl2/(TdcD)2 , where O(1) is a number of the order of unity dependent on the dlmensionality. The physical interpretation of these results is as follows. The quantum confinement influences both the electron-hole relative motion and the center-of-mass motion of excitons. In lower-dimensional structures the exciton Bohr radius becomes smaller and the exciton oscillator strength is enhanced. This effect is reflected on the normalized exciton Bohr radius appearing in the denominators of (8). At the same time, the quantum confinement restricts the exciton coherence volume and reduces the optical nonlinearity. This effect is reflected on the different powers of le/aB in (8). Thus the overall dimensionality dependence of the figures of merit is determined by the combination of these counterbalancing effects. According to (8), we can distinguish two typical cases depending on the parameter /~d/~ or ~D. The first ease corresponds to the situation that AdD > 1 or D < 1. This situation holds in materials whose exton binding energy is comparable to or smaller than the homogeneous linewidth or whose exeiton Bohr radius is longer than the exciton coherence length and seems to be realized in the III-V compound semiconductors. In this case we have a(3D) < a(2D) < a(1D) and the exciton confinement is effective for enhancing the figures of merit of the optical nonlinearity. The second case corresponds to the situation that AdD < 1 or i_dD > 1. This situation seems to be realized in the I-VII compound semiconductors. In this case we find a(3D) > a(2D) > a(1D) and the quantum confinement of excitons is not favorable for improving the figures of merit. As an example of the first case, we consider GaAs. For this material, we estimate that
~
r/(3D) : t/(2D): t/(1D) 18.8 97.5
146
(9)
= ( ~ 3 0 ) s / 2 : (~20)2 : (~ID)3/2, where we put a~° = 0.5 and a]~° = 0.2 in (8) and the carrier masses in the bulk are used for all dunensionalities. Since Ado (d=1,2 and 3) is of the order of unity, the figures of merit can be improved in the lowdimensional structures. When we employ a reasonable value of 1/2(~ 2.hmeV) for the linewidth A for both 2D and 3D cases, we have y(2D)/y(3D) ~- 3.6. This enhancement factor is close to the observed value of 3 is confirming the validity of our result (8). As an example of the second case, we examine CuC1. Since the bulk CuCI has a large exciton binding energy Ry* of 213 meV ]e, ~ becomes as small as 10 -2 for A _--__2 meV. Also because the exciton Bohr radius is very small in the bulk CuCl and will be affected little by the quanturn confinement, we can put ~]D ~ ~ D ="~ 1 in (8) and we have 1.09 1.19 1.21 a(3D) : a ( 2 D ) : a(1D) ~ (~3D)5/2 : (~2D)2 : (~1D)3/2.
(10) When we put ~3D ~ /~2D ~ /~ID -- 10 -2, corresponding to A 3 D --~ A2D -~ A I D ~--- 2 meV, a(3D) shows the largest magnitude. Thus for CuCI, the exciton confine-
281
ment is not favorable for enhancing the figures of merit of the optical nonlinearity. The homogeneous linewidth or the exciton coherence length is found to be important in determining the dimensionality dependence of the figures of merit of the optical nonlinearity. Now we discuss the dimensionality dependence of the homogeneous linewidth. In general, the homogeneous linewidth can be written as AdD(T,N) = AOdDJr- %,a dDT + 7 xdD xNdD,
(11)
where T is the temperature, NdD is the exciton density in the d-dimensional structure and A° o is the residual linewidth due to elastic scattering from impurities and interface roughness. These linear dependences on the temperature and on the exciton density have been observed recently in 3D and 2D GaAs samples z7-2°. The temperature coefficient 7~D is determined by the interaction with acoustic phonons and has been found to be decreasing when the dimensionality is reduced. As for the line-broadenlng due to the exciton-exciton collision, given by the third term of (11), the 2D case has a several times larger value than the 3D case because of the much weaker screening of the Coulomb interaction in the 2D case compared to the 3D case 2z. Extrapolating the above arguments, we can expect that the phonon part and the exciton-exciton collision part in (11) have an opposite dependence on the dimensionality. Consequently, the homogeneous linewidth apart from the residual part A 0 D w o u l d show a rather weak dependence on the dimensionality. The residual part A° D is dependent on the interface quality of the lowdimensional structures and is considered to be larger for the lower dimensional systems due to the increasingly important role of the interfaces in determining the quantum confined states. This trend is expected to be more pronounced for materials with larger exciton Bohr radius, e.g., for the III-V and II-VI compound semiconductors. On the other hand, for materials with smaller exciton Bohr radius, e.g., for the I-VII compound semiconductors, A° o would not be sensitive to the dimensionality. Conse-q-uently, for the III-V semiconductors the homogeneous linewidth would be larger in the lower dimensionality, whereas for the I-VII semiconductors the linewidth would be rather insensitive to the dimensionality. It is quite intriguing to extend the above argument to the zero-dimensional materials, namely quantum dots system. In the same scale as the second line of (8), the relative magnitude of a(OD) appears as f(OD)/~OD, where f(OD) is the enhancement factor of the oscillator strength and its expression is given in Ref.5. Here we are considering the mesoscopic enhancement regime where the inter-subband energy spacings are larger than the homogeneous linewidth and the oscillator strength is concentrated to the lowest exciton transition. In this regime, the enhancement factor f(OD) increases almost proportionally to the volume of the quantum dot. However, we cannot have an indefinitely large value of the figures of merit by increasing the quantum dot size. This is because the exciton spectrum changes from the discrete level scheme to a quasi-continuous level scheme as the quantum dot size is increased and the concentration of oscillator strength to a sharp excitonic transition does not occur. The crossover between the two schemes occurs around the quantum dot radius R for which the inter-subband energy is comparable to tile homogeneous linewidth, namely ti2[(4.49) 2 - ( 3 . 1 4 ) 2 ] / ( 2 M R 2) ~ A, where 4.49 and 3.14 are the first zeros of the spherical Bessel func-
282
DIMENSIONALITY OF EXCITONIC OPTICAL NONLINEARITY
tions jl and j0, respectively. The critical radius Rc~ estimated in this way gives a measure of the coherence length of the zero-dimensional exciton. For CuCI microcrystallitesI the criticalradius Rcr is estimated to be about 75 A and 92 .~ for A = 3 m e V and 2meV, respectively. These values happen to be in accordance with the radius at which the plot of saturation density versus radius deviates from the R -2's dependence 7. A more comprehensive theoretic.alstudy on the size dependence of X (3) and the figures of merit for semiconductor microcrystalliteswill be reported elsewhere 2~. In this paper the interest has been focused predominantly on the resonant third-order excitonic optical nonlinearity. The nonlinear susceptibilityat the exciton resonance was approximated by a sum_~,~tion of the contribution from the one-exciton states in which the contribution from the two-exciton states was effectively incorporated in terms of the exciton coherence length. Although we believe that the qualitative features of the dimensionality dependence of the figures
of merit are grasped within this appr~imation, it is truly desirable to examine the problem on the basis of an analytical expression which incorporates both contributions from one- and two-exciton states exactly. This is left for the future study. At the same time, the non-resonant optical nonlinearity is also very important due to its potentiality for realizing the ultrafast optical switching. In the extremely off-resonant case some interesting scaling laws concerning the odd
and even order o~)tical nonlinearities were discussed neglecting the Coulomb interaction 23. However, since the Coulomb interaction modifies the exciton density of states through the so-called Sommerfeld factor 13, a more detailed study including the Coulomb interaction and the many-body correlation effect is necessary to clarify the dimensionality dependence of the nonresonant optical nonlinearity. In conclusion, the figures of merit for the resonant excitonic optical nonlinearity are derived and their dependence on the dimensionality is clarified. An important parameter which determines the dimensionality dependence of the figures of merit is found to be the ratio of the homogeneous linewidth to the exciton binding energy or the ratio of the exciton coherence length to the exciton Bohr radius. It is predicted that for the III-V compound semiconductors the exciton confinement in low-dimensional structures (d=l,2 d:dimeusionality) is favorable for enhancing the figures of merit, whereas for the I-VII compound semiconductors the exciton confinement does not always improve the figures of merit. For the II-VI coml~ound semiconductors the situation would be intermediate between these two regimes. Acknowledgement - The author would like to thank Drs. N. Uesugi and J. Yumoto for stimulating discussions.
References 1. Semiconductors and Semimetals, Vol.24, ed. R. 13. Dingle(Academic, New York, 1987). 2. Optical Nonlinearities and Instabilities in Semi14. conductors, ed. H. Haug(Academic, New York,
1988). 3. D. S. Chemla, D. A. B. Miller, P. W. Smith, A. C. Gossard and W. Wiegrnann, IEEE J. Quantum Electron. QE-20, 265(1984). 4. S. Schraitt-Rink, D. S. Chemla and D. A. B. Miller, Phys. Rev. B32, 6601(1985). 5. T. Takngahara, Phys. Rev. B36, 9293(1987). 6. E. Hanamura, Phys. Rev. B37, 1273(1988). 7. Y. Masumoto, M. Yarnazaki and H. Sugawara, Appl. Phys. Lett. 53, 1527(1988). 8. Nonlinear Optical and Electroactive Polymers, eds. P. N. Prasad and D. 1~ Ulrich (Plenum, New York,
1988).
9. P. Roussignol, D. Ricard, C. Flytzanis and N. Neuroth, Phys. Rev. Left. 62, 312(1989). 10. T. Takagahara, in Ultrafast Phenomena VI, eds.
T. Yajima, K. Yoshihara, C. B. Harris and S. Shionoya (Springer, Berlin, 1988), p.337. 11. T. Takagahara, Phys. Rev. B39, 10206 (1989). 12. J. Feldmann, G. Peter, E. O. G6bel, P. Dawson, K. Moore, C. Foxon and R. J. FAliott, Phys. Rev. Lett. 59, 2337(1987).
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15.
16. 17. 18. 19. 20. 21.
22. 23.
M. Shinada and S. Sugano, J. Phys. Soc. Jpn. 21, 1936(1966). L. Banyal, I. Galbralth, C. Ell and H. Haug, Phys. Rev. B36, 6099(1987). S. It. Park, J. F. Morhange, A. D. Jeffery, R. A. Morgan, A. Chavez-Pirson, H. M. Gibbs, S. W. Koch, N. Peyghambarian, M. Derstine, A. C. Gossard, J. H. English and W. Wiegmann, Appl. Phys. Lett. 52, 1201(1988). S. Nikitine, Prog. Semicond. 6, 235(1962). L. Schultheis, J. Kuhl, A. Honold and C. W. Tu, Phys. Rev. Lett. 57, 1797 (1986). L. Schultheis, A. Honold, J. Kuhl, K. K6hler and C. W. Tu, Phys. Rev. B. 34, 9027 (1986). A. Honold, L. Schultheis, J. Kuhl and C. W. Tu, Appl. Phys. Lett. 52, 2105 (1988). A. Honold, L. Schultheis, J. Kuhl and C. W. Tu, Phys. Rev. B40, 6442 (1989). T. Ando, A. B. Fowler and F. Stern, Rev. Mod. Phys. 54, 437 (1982). T. Talmgahara, unpublished. C. Flytzanis, in Nonlinear Optical Effects in Oranic Polymers, eds. J. Messier, F. Kajzar, P. rasad and D. Ulrich (Kluwer Academic, Dordrecht, 1989), p.1.
Solid State Communications, Vol. 78, No. 4, pp. 279-282, 1991. Printed in Great Britain.
DEPENDENCE
0038-1098/9153.00+.00 Pergamon Press plc
ON DIMENSIONALITY OF EXCITONIC OPTICAL NONLINEARITY IN QUANTUM CONFINED STRUCTURES
T. Takngahara NTT Basic Research Laboratories, Musashino-shi, Tokyo 180, JAPAN (Received 8 February 1991 by T. Tsuzuki) The dependence on dimensionality of the excitonic optical nonlinearity and the figures of merit in quantum confined structures is clarified for the resonant thirdorder optical nonlinearity. An important parameter which determines the dimensionality dependence of the figures of merit is found to be the ratio of the homogeneous linewidth to the exciton binding energy or the ratio of the exciton coherence length t o the exciton Bohr radius. It is predicted that for the III-V compound semiconductors the exciton confinement in low-dimensional structures d = l , 2 d:dimensionality) is favorable for enhancing the figures of merit, whereas r the I-VII compound semiconductors the exciton confinement does not always improve the figures of merit.
Recently the enhanced excitonic optical nonlinearity in low-dimensional materials has attracted much attention from the viewpoint of fundamental physics and from the interest of application to optical devices 1,2. In GaAs-A1GaAs quantum well structures, a large X(s) value of the order of 10 -2 ,,, 10 -1 esu was observed 3 and this absorptive nonlinearity was interpreted in terms of phase space filling of free carriers and excitons 4. A large X(3) value was predicted theoretically in semiconductor microcrystaJlitess,s and a X(3) value of the order of 10 -5 esu was observed for CuC1 microcrystallites embedded in a NaCI matrix 7. On the other hand, a large number of organic materials having a conjugated 7relectron system have been synthesized and their large optical nonlinearity with very short response time has been investigated s. The most important mechanism for enhancing the optical nonlinearity is usually considered to be the quantum confinement effect which leads to a discrete level structure and causes the concentration of oscillator strength to the lowest energy transitions. Since the density of electronic states becomes sharper and sharper by reducing the dimensionality of structures, it seems that a sharp excitonic feature will be realized in low-dimensional materials. In this respect, the zero-dimensional structures, namely quantum dots seem to be most favorable for enhancing the optical nonlinearity. However, the spectral broadening in low-dimensional structures is significant due to the increasingly important role of the interface scattering and due to the enhanced electron-phonon coupling 9,1°. Thus the enhancement of the optical nonlinearity cannot be achieved simply by the exciton confinement in low-dimensional structures. It is essentially important to take into account the homogeneous broadening in discussing the effect of quantum confinement on the optical nonlinearity. There has been so far no systematic study on the dimensionality dependence of the excitonic optical nonlinearity which takes account of the above aspects. In this paper we will focus our attention on the efficiencies of the third-order optical nonlinearity which are usually called the figures of merit
and will clarify their dependence on dimensionality of structures. As a consequence, we will find out for the first time an important parameter which determines the dimensionality dependence of the figures of merit. The third-order nonlinear susceptibility 3((3) is calculated from the third-order perturbation theory with summation over the intermediate states which can be one-exciton states or two-exciton states. A detailed expression of X(3) is given in Ref.ll and consists of two types of contribution; one of which involves only one-exciton states as the intermediate state and the other includes two-exciton states as well. In general, the magnitude of X(s) is determined by the partial cancellation between the two types of contribution and it is rather difficult to obtain an analytically closed expression. However, from various numerical calculations using realistic parameters, we have found that the magnitude of X(3) in the vicinity of the exciton resonance can be safely approximated by the contribution from the one-exciton states only. More explicitly, the maximum value of X(3) at the exciton resonance can be approximated as
1 '~T'( e" ,~2 f~,
where m0 is the free electron mass, V the system volume, /iwi the exciton transition energy, 711i(7-L~) the longitudinal (transverse) relaxation rate, fxi the exciton oscillator strength, the summation is taken over the exciton levels lying within the homogeneous linewidth of the ground exciton state and a factor due to the local field correction is omitted. The oscillator strength of the free exciton state is determined by the exciton coherence volume and this coherence volume is closely related to the homogeneous linewidth of the excitonic transition 12. The homoseneous linewidth refiects the energy fluctuation of the exciton state and 279
280
DIMENSIONALITY OF EXCITONIC OPTICAL NONLINEARITY
in turn represents the wave-vector fluctuation according to the dispersion relation. This wave-vector fluctuation restricts the exciton coherent motion within a range of the order of inverse the wave-vector fluctuation. In other words, the oscillator strength of the K = 0 exciton is redistributed among all states within the homogeneous linewidth. The redistributed oscillator strength per one exciton state can be written in terms of the exciton coherence length Ic as
f x = f 0 ( l cd D ) d I f d D (0)[ 2 ,
(2)
where f0 is the oscillator strength of the band-to-band transition at the F point and FdD(r~ --rb) denotes the envelope function of the electron-hole relative motion ( d = l , 2 and 3). The coherence length for each dimension is given by
v~
~
'
t° -
ii D =
~
'
rr~z
V2M1DAaD,
(3)
where A d D and M a D are the homogeneous linewidth and the translational mass of the exciton in the ddimensional structure. Then the ratio among maxim u m values of the imaginary part of X (3) for threedimensional(3D), two-dimensional(2D) and one-dimensional(ID) systems is obtained as
Vol. 78, No. 4
favorable for enhancing the magnitude of the optical nonlinearity X(s). However, the situation is more delicate concerning the figures of merit of the optical nonlinearity. The figures of merit are usually defined by the changes in the absorption coefficient a and the refractive index n induced per one electron-hole pair(exciton) in a unit volume, which are denoted by a and ,/, respectively and given as a = as + aN, n = no + rlN, where a0(n0) is the absorption coefficient(refractive index) under no excitation and N is the number density of electron-hole pairs(excitons). These quantities are related to the imaginary and real parts of X(s), respectively as ~ 32rr2 hw2 Imx(s)
~ = - - - - , ~c 2
6XoT
,~=
16r2hw ReX (3) ~C
O~oT
1 2(13D'~3 ~ 2D '~ 1 ~D N~D = -,-¢ , , NiO = 2(lc ) L,,N----~--g= 21c L z L v,
(7)
= 0~DpIF~D(0)P. (I~D)~IF~D(0)P. t~DIF'D(0)P [[
L'a3D
L ~2DA2 Z/l[
~'~2D
"L
z
L ~IDA2 i~Iii
ID
'
(4) where Lz is the quantum well thickness of the 2D system and L~L~ the cross-sectional area of the 1D quantum wire. It is to be noted that for 2D and 1D systems, X(3) is estimated by supposing a filling factor of 100 ~ , namely quantum wells and quantum wires fill all the space without changing the electronic structure. Introducing the exciton Bohr radius a dD associated with FaD(re -- rh) and a dimensionless parameter/3 defined by/3 = mv~--~/(1 + m , ~ ) with the electron-to-hole mass ratio m,h in the bulk material, the ratio (4) can be further reduced to 6~ 3
16~ 2
M3D
~D(XsD)r/2 : 7 r Z , ~ D ( ~ 2 0 ) a ( ~ D ) 4 M2D
LzLy T~ID( ~ I D )5/2 (-dlBD)2V -M-~1D'
(o)
where e is the dielectric constant of the material and r is the exciton lifetime(71~-z). It should be noted that the figures of merit do not contain explicitly the exciton lifetime and structure parameters like the quantum well thickness and the filling factor, reflecting the more fundamental character of these quantities compared to XO) . Now the dimensionality dependence of the figures of merit will be derived on the basis of the findin_~ that the saturation density N° of the absorption coefficient is closely related to the exciton coherence length. In fact we find that
Imx0)(3D)[~.~ : Imx(S)(2D)[~.~ : ImxO)(iD)[~.~ 3DA2
,
(5)
where the linewidth and length are scaled by the effective Rydberg(Ry • ) and the exciton Bohr radius(a 3D B ) in the bulk material, respectively and this scaling is indicated by an overbar. For the two-dimensional exciton, it is well known that ~ D = 1/2 zs, whereas for the one-dimensional exciton, we have ~ D ~ 0.2 for Lz ~-- Ly ~- 0.3 14. Thus we can expect a l a r g e enhancement of the X(s) value in low-dimensional materials so far as the homogeneous linewidth is not very much increased. As seen above, the exciton confinement is generally
where the factor 2 comes from the definition of X(3) employed in Ref.ll. These results can be interpreted that an exciton has a coherent range of the order of lc with respect to the center-of-mass motion and the creation of another exciton within that range is inhibited. In the 2D case, N0 due to the phase space filling mechanism4 is calculated as l/N°2 D = 321r(a 2BD )2 Lz/7. This implies that the kinematical exciton-exciton interaction due to the Pauli exclusion principle restricts the exciton coherence length to a few times the exciton Bohr radius. Actually the dynamical interaction of excitons with various elementary excitations shortens the coherence length furthermore and increases the saturation density. The expression (7) is most general including all the contributions in terms of the exciton coherence length. In order to envisage qualitatively the dimensionality dependence of the figures of merit, we compare 2D and 3D cases. First of all, 2D 3 D ,~v 3D • we have .IV'. / N ~s - Ic / L , , supposing Ic3 D -~, Ic2 D . Here the quantum well thickness Lz can be replaced by a~D because the characteristic length that distinguishes 2D and 3D systems is the exciton Bohr radius and the two-dimensional character becomes remarkable for Lz <_ a~D. Then noticing the relation l/N° = a/as, we can deduce that lc/a 3s D appears as a characteristic parameter which describes the dimensionality dependence of the figures of merit of the optical nonlinearity. Through a more detailed algebra, we find that a(3D) : a(2D) : a(1D) = T/(3D) : r/(2D) : r/(1D) 6~r~ 3 8~ ~ MSD : 7r~ = ( ~ 3 D p / ~ : ( ~ 2 D ) 2 ( ~ D ) 2 M2D ( ~ , D p / ~
, D V M~D
(7:o) ` M D. (7:°) M,D = (36~') 1D : 2 ( ~ 0 ) 2 M3D " "~bD M ~o '
(8)
Vol. 78, No. 4
DIMENSIONALITY OF EXCITONIC OPTICAL NONLINEARITY
where the maximum values of a and T/around the exciton resonance are compared. It is confirmed that the dimensionality dependence of the fisures of merit is determined by the parameter ~,dD or [~D -- IdD/a3BD. These two parameters are related to each other as/XdD = O(1)fl2/(TdcD)2 , where O(1) is a number of the order of unity dependent on the dlmensionality. The physical interpretation of these results is as follows. The quantum confinement influences both the electron-hole relative motion and the center-of-mass motion of excitons. In lower-dimensional structures the exciton Bohr radius becomes smaller and the exciton oscillator strength is enhanced. This effect is reflected on the normalized exciton Bohr radius appearing in the denominators of (8). At the same time, the quantum confinement restricts the exciton coherence volume and reduces the optical nonlinearity. This effect is reflected on the different powers of le/aB in (8). Thus the overall dimensionality dependence of the figures of merit is determined by the combination of these counterbalancing effects. According to (8), we can distinguish two typical cases depending on the parameter /~d/~ or ~D. The first ease corresponds to the situation that AdD > 1 or D < 1. This situation holds in materials whose exton binding energy is comparable to or smaller than the homogeneous linewidth or whose exeiton Bohr radius is longer than the exciton coherence length and seems to be realized in the III-V compound semiconductors. In this case we have a(3D) < a(2D) < a(1D) and the exciton confinement is effective for enhancing the figures of merit of the optical nonlinearity. The second case corresponds to the situation that AdD < 1 or i_dD > 1. This situation seems to be realized in the I-VII compound semiconductors. In this case we find a(3D) > a(2D) > a(1D) and the quantum confinement of excitons is not favorable for improving the figures of merit. As an example of the first case, we consider GaAs. For this material, we estimate that
~
r/(3D) : t/(2D): t/(1D) 18.8 97.5
146
(9)
= ( ~ 3 0 ) s / 2 : (~20)2 : (~ID)3/2, where we put a~° = 0.5 and a]~° = 0.2 in (8) and the carrier masses in the bulk are used for all dunensionalities. Since Ado (d=1,2 and 3) is of the order of unity, the figures of merit can be improved in the lowdimensional structures. When we employ a reasonable value of 1/2(~ 2.hmeV) for the linewidth A for both 2D and 3D cases, we have y(2D)/y(3D) ~- 3.6. This enhancement factor is close to the observed value of 3 is confirming the validity of our result (8). As an example of the second case, we examine CuC1. Since the bulk CuCI has a large exciton binding energy Ry* of 213 meV ]e, ~ becomes as small as 10 -2 for A _--__2 meV. Also because the exciton Bohr radius is very small in the bulk CuCl and will be affected little by the quanturn confinement, we can put ~]D ~ ~ D ="~ 1 in (8) and we have 1.09 1.19 1.21 a(3D) : a ( 2 D ) : a(1D) ~ (~3D)5/2 : (~2D)2 : (~1D)3/2.
(10) When we put ~3D ~ /~2D ~ /~ID -- 10 -2, corresponding to A 3 D --~ A2D -~ A I D ~--- 2 meV, a(3D) shows the largest magnitude. Thus for CuCI, the exciton confine-
281
ment is not favorable for enhancing the figures of merit of the optical nonlinearity. The homogeneous linewidth or the exciton coherence length is found to be important in determining the dimensionality dependence of the figures of merit of the optical nonlinearity. Now we discuss the dimensionality dependence of the homogeneous linewidth. In general, the homogeneous linewidth can be written as AdD(T,N) = AOdDJr- %,a dDT + 7 xdD xNdD,
(11)
where T is the temperature, NdD is the exciton density in the d-dimensional structure and A° o is the residual linewidth due to elastic scattering from impurities and interface roughness. These linear dependences on the temperature and on the exciton density have been observed recently in 3D and 2D GaAs samples z7-2°. The temperature coefficient 7~D is determined by the interaction with acoustic phonons and has been found to be decreasing when the dimensionality is reduced. As for the line-broadenlng due to the exciton-exciton collision, given by the third term of (11), the 2D case has a several times larger value than the 3D case because of the much weaker screening of the Coulomb interaction in the 2D case compared to the 3D case 2z. Extrapolating the above arguments, we can expect that the phonon part and the exciton-exciton collision part in (11) have an opposite dependence on the dimensionality. Consequently, the homogeneous linewidth apart from the residual part A 0 D w o u l d show a rather weak dependence on the dimensionality. The residual part A° D is dependent on the interface quality of the lowdimensional structures and is considered to be larger for the lower dimensional systems due to the increasingly important role of the interfaces in determining the quantum confined states. This trend is expected to be more pronounced for materials with larger exciton Bohr radius, e.g., for the III-V and II-VI compound semiconductors. On the other hand, for materials with smaller exciton Bohr radius, e.g., for the I-VII compound semiconductors, A° o would not be sensitive to the dimensionality. Conse-q-uently, for the III-V semiconductors the homogeneous linewidth would be larger in the lower dimensionality, whereas for the I-VII semiconductors the linewidth would be rather insensitive to the dimensionality. It is quite intriguing to extend the above argument to the zero-dimensional materials, namely quantum dots system. In the same scale as the second line of (8), the relative magnitude of a(OD) appears as f(OD)/~OD, where f(OD) is the enhancement factor of the oscillator strength and its expression is given in Ref.5. Here we are considering the mesoscopic enhancement regime where the inter-subband energy spacings are larger than the homogeneous linewidth and the oscillator strength is concentrated to the lowest exciton transition. In this regime, the enhancement factor f(OD) increases almost proportionally to the volume of the quantum dot. However, we cannot have an indefinitely large value of the figures of merit by increasing the quantum dot size. This is because the exciton spectrum changes from the discrete level scheme to a quasi-continuous level scheme as the quantum dot size is increased and the concentration of oscillator strength to a sharp excitonic transition does not occur. The crossover between the two schemes occurs around the quantum dot radius R for which the inter-subband energy is comparable to tile homogeneous linewidth, namely ti2[(4.49) 2 - ( 3 . 1 4 ) 2 ] / ( 2 M R 2) ~ A, where 4.49 and 3.14 are the first zeros of the spherical Bessel func-
282
DIMENSIONALITY OF EXCITONIC OPTICAL NONLINEARITY
tions jl and j0, respectively. The critical radius Rc~ estimated in this way gives a measure of the coherence length of the zero-dimensional exciton. For CuCI microcrystallitesI the criticalradius Rcr is estimated to be about 75 A and 92 .~ for A = 3 m e V and 2meV, respectively. These values happen to be in accordance with the radius at which the plot of saturation density versus radius deviates from the R -2's dependence 7. A more comprehensive theoretic.alstudy on the size dependence of X (3) and the figures of merit for semiconductor microcrystalliteswill be reported elsewhere 2~. In this paper the interest has been focused predominantly on the resonant third-order excitonic optical nonlinearity. The nonlinear susceptibilityat the exciton resonance was approximated by a sum_~,~tion of the contribution from the one-exciton states in which the contribution from the two-exciton states was effectively incorporated in terms of the exciton coherence length. Although we believe that the qualitative features of the dimensionality dependence of the figures
of merit are grasped within this appr~imation, it is truly desirable to examine the problem on the basis of an analytical expression which incorporates both contributions from one- and two-exciton states exactly. This is left for the future study. At the same time, the non-resonant optical nonlinearity is also very important due to its potentiality for realizing the ultrafast optical switching. In the extremely off-resonant case some interesting scaling laws concerning the odd
and even order o~)tical nonlinearities were discussed neglecting the Coulomb interaction 23. However, since the Coulomb interaction modifies the exciton density of states through the so-called Sommerfeld factor 13, a more detailed study including the Coulomb interaction and the many-body correlation effect is necessary to clarify the dimensionality dependence of the nonresonant optical nonlinearity. In conclusion, the figures of merit for the resonant excitonic optical nonlinearity are derived and their dependence on the dimensionality is clarified. An important parameter which determines the dimensionality dependence of the figures of merit is found to be the ratio of the homogeneous linewidth to the exciton binding energy or the ratio of the exciton coherence length to the exciton Bohr radius. It is predicted that for the III-V compound semiconductors the exciton confinement in low-dimensional structures (d=l,2 d:dimeusionality) is favorable for enhancing the figures of merit, whereas for the I-VII compound semiconductors the exciton confinement does not always improve the figures of merit. For the II-VI coml~ound semiconductors the situation would be intermediate between these two regimes. Acknowledgement - The author would like to thank Drs. N. Uesugi and J. Yumoto for stimulating discussions.
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