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Two-photon absorption spectra of low-dimensional semiconductors
,.;:
Surface
,.
263 t 1092) 5 12-517
Science
.
A,,
.izj ,,,..: ij,
.::j ,,j”-
y
j::?.‘. :.::::.‘:;j :,,,: :: .: . :.. . . . . ‘.:..,.: .;c
,.
,,.
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surface science ;./:.1,:
North-Holland
“,,... ,>::. ,...,. :,,:: ::: ::.::. .::;:I.
Two-photon Akira
Received
Shimizu
3 June
We present (I=
“, Tetsuo
IYYI;
accepted
a unified
0. I. 2, 3. Widths
excitons
absorption
play crucial
theory
of confining
light
structure.
is in striking
This
Ogawa
beam. This
” and
Hiroyuki
for publication
76 August
of two-photon
absorption
potentials
roles in determining
of the incident
spectra of low-dimensional
anisotropy
contrast
are assumed
the spectra.
to one-photon
spectra
rl =
exciton states
an
undoped quasi-d-dimensional (or insulator), which consists of a d-dimensional well region surrounded by barrier regions. We take the xE (5 = 1,. . . , d) [xc ([ = d f 1.. . ,3)] axes along the unconfined Z
ISY2 - Elsevier
Science
than
semiconductors
the exciton
and 2. the spectra
Bohr
strongly
of the exciton
radius. depend
envelope
of an arbitrary
dimension
and the quasi-c/-dimensional on the polarization
functions
a~/
direction
anisotropic
band
absorption
(Qd D) semiconductor
OO~Y-~O2X/Y2/$OS.o(~
I
ho111 anisotropies
Two-photon absorption (TPA) spectroscopy of low-dimensional semiconductors provides us with a wealth of information complementary to onephoton absorption (OPA) spectra. The TPA spectra of three-dimensional (3D) systems were investigated many years ago [1,2]. On the other hand, the TPA spectra of quasi-two-dimensional (Q2D) systems were studied only recently [3-121, and many features characteristic to the Q2D systems have been discovered. We here extend the successful theory for Q2D systems [6] to semiconductors of arbitrary dimension d (= 0. 1, 2, 3). and thereby discuss TPA spectra near the TPA edge as a function of d.
Consider
of low-dimensional
to he smaller
1. Introduction
2. Quasi-d-dimensional
“K
IYYI
When
comes from
Sakaki
semiconductors
Publishers
B.V.
[confined] directions (denoted by the subscript /I [I]), for which the normalization length is 1. [the well width is L ,I. Accordingly, the position vector r is decomposed as r = tr,,. r ) = ({So}. LYJ). WC assume a semiconductor with a onc-photon-allowed direct band gap at the I‘ point, which consists of a single conduction (c) band and two valence (v) bands. Due to the (3 - d)-dimensional confinement, the band structure is modified: (i) the two-fold degeneracy of the valance bands is lifted, and the modified “heavy-hole” (hh) and “light-hole” (Ih) bands are formed, and (ii) subbands are formed in each of these three bands. Under an interband optical excitation, exciton states will be observed due to an attractive tot-cc between an electron and a hole (c-h). We assume that the exciton is of the Wannicr type. whose Bohr radius is larger than L L to form a QdD exciton. Then, the QdD excitons can bc assigned to each pair of c and v (= hh or Ih) subbands. For a Wannier exciton associated with a c-subband cy and a v-subband ,!?. its wavcfunction for zero center-of-mass-motion wavenumber may be proportional [6,13] to the envelope funcand Yamada
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A. Shimizu et al. / Two-photon absorption spectra of low-dimensional semiconductors
tion U,“p of the d-dimensional e-h relative-motion with quantum number(s) v along the unconfined directions and the subband envelope functions 4, and c$~.The exciton has discrete spectra (bound states) when its energy E,“O < EG(a; PI, and has continuous spectra (unbound states) when E,*” 2 E,(a; p), where E,(cr; p) = E, + E, + lP is a band-gap energy with E, and lP the subband quantization energies and E, the bulk band-gap energy. When penetration of the wavefunction into the barrier regions is small, 4, and 4p take decoupled forms such as 4,(r I) = Ilg+oi(x5), where the 4, ‘s denote the subband envelope functions a fong the xc axis, and (Y= ((~~+i,. . . , a,). Due to the finiteness of the spatial extensions of these subband functions, UFp deviates from those of pure d-dimensional excitons when d = 1, 2 [6,13,14]. We take the deviation into account by using a variational form of U,*a for d = 2 [6], and an analytic solution for d = 1 [14].
3. Polarization absorption
anisotropies
47re2 I A I * hm:,c2(L,
x
c
Bloch function of the valence band when k approaches zero along the confinement directions [6]. Note that the E^dependence of WoQpdAD arises only from the interband matrix element. In other words, OPA spectra reflect no anisotropies in the exciton envelope functions. To calculate the TPA rate WTQpdAD, we employ the same method as in the previous theory for Q2D systems [6], which agreed with experiment [9]. For incident light beams of photon energies hw, and hm2, with a common polarization vector P, and vector-potential amplitudes A, and A,, we have obtained the following general formulas for the TPA rate. When the light beams are polarized along an unconfined direction (6 II.?,), wXD( E^IIii) 64n-e4 I A,A, =
hmic4Ei(
I2
L I + L,)3-d
in the two-photon
Suppose that many identical QdD systems are arranged with a period of L I +L,, where L, is the barrier thickness, to constitute a (3 -d)dimensional array. Before we discuss the TPA, the OPA rate WoQpdAD is briefly reviewed for comparison. For an incident light beam of photon energy ho, a polarization vector E^,and a vectorpotential amplitude A, the OPA rate is given as
woQpD =
x s;q hw, + hw,)
1 )
64re4
I(cli*Plc),l*
=
I A,A,
I2
hmzc4E6( L I + L,)3-d
c=hh,lh
xc
x~l(~,l~,>12Cl~~~(r,,=~)12 v 4
l(ClP~l”)J2 I’
(1)
where m, denotes the free-electron mass, Szp the lineshape function of the apv exciton, and (c I E^*p I u), the interband matrix element, with p the momentum operator. Here ( cjI is the
(4
where pL,.,, is the electron-hole reduced mass for the motion along the unconfined directions, pC the 2, element of the momentum operator. On the other hand when the light beams are polarized along a confined direction (; 11i,) wTQpdAD( E^II “J
+L,)“-d
xl!y(Rw),
513
2 PL,, i
I
reduced mass for where kL. I is the electron-hole the motion along the confined directions, and pi the P, element of the momentum operator. These two formulas are the general expressions of the TPA rate for a QdD excitonic system in an arbitrary dimension. In contrast to the OPA rate [eq. (111, the TPA rates have distinctly different forms depending on the polarization direction. Here we note the case of the absence of excitonic effects. In such a case, the formula for the TPA rate are obtained by making the following substitutions in the above equations: U,:‘“(r,,) + 0 for Et:‘” < &(N; p) and U,nfi(,,,) + eik~~-r~~L,;l”Z for E:,+ 2 &(a; p). It is seen from eqs. (2) and (3) that the TPA rate depends on the light polarization through (i) the interband matrix element and the reduced mass (third line of eqs. (2) and (3)), which become anisotropic due to band mixing caused by the low-dimensional confinement, and (ii) the QdDexciton envelope functions (fourth and fifth lines of eqs. (2) and (3)). This is in striking contrast to the OPA rate. That is, the TPA rate strong!\ depends on the light polarization through the strong anisotropies in the exciton enrlelope functions eLIen if the band structure is isotropic. Hence, one can probe the anisotropies in the QdD exciton envclope functions by measuring the TPA spectra. In the following, we will separately discuss characteristic features of the TPA spectra as a function of d.
4. TPA spectra as a function 3.1. Three-dimensional
of d
case
Mahan discussed thoroughly this case [l]. We shall make comments only on important points in this case. In the case of d = 3, there is no “confinement direction”, so that eq. (2) should be used for any polarization direction. The lowest absorption peak is the 2P exciton, whose oscillator strength is small, so that the discrete spectrum is almost smeared out under the usual experimental conditions. As for continuous states. the overall TPA spectra become a smoothly-increasing function of the photon energies.
4.2. Quasi-zero-dimensional
ca,se
In the case of d = 0, an e-h pair is confined in all directions, so that eq. (3) should be used for any polarization direction. Since excitonic effects are negligible as compared with strong effects of subband quantization (recall that we assume small L L), eq. (3) is reduced to w c)Ol> -II’A ZZ
64~~7’1 A,A,I’ tznz;c“E;;(
L
+l,,)ipc’
I(Cl@), c /
P.2>
I’ -
C-1) From the subband integrals in this equation, WC find the subband selection rule; (i) CV;- B, = odd for the polarization direction and (ii) cyLI ~ /3,, = even = 0 for the other directions. Therefore the TPA spectra consist of discrete lines, each corrcsponding to a pair of subbands. 4.3. Quasi-two-dimensional
ca,se
When d = 2. eqs. (2) and (3) reduce to the formulas given in ref. [61, as it should be, and the TPA spectra strongly depend on the polarization direction i. Detailed results of this d = 2 case arc given in ref. [6]. Here we stress again that the large anisotropies of TPA spectra arise from the anisotropies of the cxciton cnvelopc functions. i.c., from the fourth and fifth lines in cq. (2) or eq. (3). To visualize this fact, we plot in fig. 1 this “envelope-function part”, assuming v = lh. Fig. la shows the spectra for i 11i, and fig. lb fin g IIis. A GaAs/Al,,.,, Ga,,,,OAs multiple quantum well structure of 1, ~ = 90 A. and p,,, ,, = O.Ohm,, [I.51 is assumed there. The discrete spectra arc represented by vertical lines, the height of which indicates the integrated TPA intensity divided bq the effective Kydberg energy defined by K,,,,, = pLlh,,eJ/2tl’E’. where t is a dielectric constant. We find the strong anisotropies in the TPA spec-
515
A. Shimizu et al. / Two-photon absorption spectra of low-dimensional semiconductors
(a)
Q2D system cl-lhl transition
z :
e, ‘! 8 ._
0
5 k
15
10
+ flwz - E;%;
P)
1
i II % 3
2
5 6 _c a
1
20
@I :
Q2D system cl -lh2 transition
4
E 2
% -5
r
5 d k?
? // it
5
0
-5
5
0
GUI + fLwz- E&%;
(&I)
20
15
10 B)
(&,,,)
Fig. 1. Envelope-function parts of normalized TPA spectra, the fourth and fifth lines of eqs. (2) and (31, for a GaAs/Al,,,,,Ga,,,,As multiple quantum well structure of L I = 90 A. Contributions from light-hole excitons only (both discrete and continuous states) are plotted (a) when E^is parallel to an unconfined direction, and (b) when i is parallel to the confined direction. Quasi-continuous spectra just below the continuous ones are not shown. The vertical lines represent the discrete spectra, the height of which indicates the integrated TPA intensity divided by the effective Rydberg energy I?,,,,. The horizontal axis is also scaled by this energy. The dashed lines represent the TPA spectra in the absence of exciton effects.
tra as well as the strong enhancements of TPA due to exciton effects for both cases of the polarization directions. 4.4. Quasi-one-dimensional case Excitonic effects become more drastic when the exciton binding energies be-
d = 1 because
QlD system cl -1hl transition
larger. For E^IIf,, the subband selection rule is cy6- /I5 = even = 0, and the derivative of I/@ in eq. (2) was given in ref. [14], which is n&zero only for odd-parity excitons. On the other hand, for f IIf,, the subband selection rule is the same as that in the d = 0 case, and UVapin eq. (3) was given also in ref. [141, which is nonzero only for even-parity excitons. Looking at v = lh only come
(4
QlD system c l -lh2 transition
lb)
n=2 (odd)
-. n=2(even)
l
=___ ---mm
--------mm____.
\
Fig. 2. Envelope-function parts of normalized TPA spectra for a multiple quantum wire structure of z(, = 0.09ehZ/~,h ,,e’, which corresponds to 4R,,,, for the binding energy of the lowest exciton. Contributions from Ih excitons are plotted (a) when E^is parallel to the unconfined direction, and (b) when i is parallel to a confined direction. The vertical scale of these figures is different from that of fig. 1. The quantum number n here is n of ref. 1141 plus one.
and assuming z. = O.O~EA’/~,,, ,,e’ (z,, is a cutoff of a 1D regularized Coulomb potential [14] and is proportional to L I), we have plotted the TPA spectra (the fourth and fifth lines of eqs. (2) and (3)) for f IIi’t in fig. 2a and for i I/ Pz in fig. 2b. In comparison with the case of d = 2, the relative magnitude of the discrete spectra to the continuous spectra becomes larger for d = 1. This is due to increased oscillator strengths in QlD systems [14]. If we look at the continuous spectra, it is interesting that exciton effects enhance the TPA for E^)I P,, whereas they reduce the TPA for E^1)3,. By contrast, the TPA is always enhanced by exciton effects for d 2 2 (see fig. 1 and refs. [ 1,611. The reduction of the continuous TPA spectra by exciton effects in a direct-allowed (two-photonforbidden) gap quantum wire has the same origin as the reduction of OPA of a direct-allowed-gap quantum wire; that is, an anomalously strong concentration of the oscillator strength on the lowest exciton state. The enhancement, on the other hand, has the same origin as that of OPA of a direct-forbidden-gap quantum wire; odd-parity excitons have large slopes at r,, = 0 because the e-h Coulomb attraction becomes more effective for d = 1. In the case of the TPA of the QlD system, one can observe both the reduction (for E^1).i?,) and enhancement (for i IIPi) in the same sample simply by rotating the polarization direction.
5. Concluding
remarks
In summary, we present a unified theory of two-photon absorption (TPA) spectra of low-dimensional semiconductors of arbitrary dimension d (= 0, 1, 2, 3). When the widths of the confining potentials are smaller than the exciton Bohr radius. the excitons become quasi-d-dimensional ones which play an important role in determining the spectra. When d = I and 2, the TPA spectra strongly depend on the polarization direction of the light beam. This strong polarization dependence of the TPA spectra comes from hoti~ anisotropies in the QdD exciton envelope functions and anisotropic band structure, which fact is in remarkable contrast to the OPA case. In the
cast of TPA, one observes different exciton states of different enrlelope functions by simply rotating the polarization direction. Lastly WC again note that the QdD cxcitonic effects on the TPA spectra are clarified assuming a small L , in comparison with the cxciton Bohr radius. In this case, the subband separation energy is much larger than the cxciton binding energy. Otherwise the e-h Coulomb interaction admixes many subband pairs into the excitonic states, and the continuum excitons associated with a lower energy subband pair interfer with discrete cxciton states associated with a higher energy subband pair. This situation is also of interest in the interpretation of experimental results in a dimensional-crossover regime. Extension of our theory to this regime is now progressing and will be reported elsewhere.
Acknowledgements The authors would like to thank Dr. T. Takagahara, Dr. H. Kanbc and Dr. K. Nakamura for discussions. Two of the authors (T.O. and A.!%) also thank Dr. Y. Horikoshi, Dr. T. Kimura, R. Murakami and Y. Takigawa for encouragement.
Note added in proof Low-dimensional confinement of electronic and excitonic states had been believed to he detected in cxperimcnts as optical anisotropics and/or blue shifts of OPA spectra. However. Bauer and Sakaki [Surf. Sci. 367 (1993)] have shown that these cannot be a proof of the confinement. and another probe of the four-dimcnsionality is being looked for. Our results have shown that TPA spectroscopy is a sensitive probe of the low-dimensionality because the TPA spectra arc directly related with the low-dimensional envelope functions.
References [I] G.D.
Mahan.
[7] (‘.C.
Ixe
Phya. Rev.
and H.Y.
170 (I%X)
X2.5.
Fan. Phys. Rev. 13 0 (lU7-i) i.il)l.
A. Shimizu et al. / Two-photon absorption spectra of low-dimensional semiconductors [3] A. Shimizu and K. Fujii, OYO BUTURI 60 (1991) 128 [in Japanese]. [4] H.N. Spector, Phys. Rev. B 35 (1987) 5876. [5] A. Pasquarello and A. Quattropani, Phys. Rev. B 38 (1988) 6206. [6] A. Shim& Phys. Rev. B 40 (1989) 1403. [7] D. FrGhlich, R. Wille, W. Schlapp and G. Weimann, Phys. Rev. Lett. 61 (1988) 1878. [8] W.H. Knox, D.S. Chemla, D.A.B. Miller, J.B. Stark and S. Schmitt-Rink, Phys. Rev. Lett. 62 (1989) 1189. [9] K. Tai, A. Mysyrowicz, R.J. Fisher, R.E. Slusher and A.Y. Cho, Phys. Rev. Lett. 62 (1989) 1784. [lo] M. Nithisoontorn, K. Unterrainer, S. Michaelis, N.
[ll] [12] [13] [14] [15]
511
Sawaki, E. Gornik and H. Kano, Phys. Rev. Lett. 62 (1989) 3078. I.M. Catalano, A. Cingolani, R. Cingolani, M. Lcpore and K. Ploog, Phys. Rev. B 40 (1989) 1312. K. Fujii, A. Shimizu, J. Bergquist and T. Sawada, Phys. Rev. Lett. 65 (1990) 1808. D.A.B. Miller, J.S. Weiner and D.S. Chemla, IEEE J. Quantum Electron. QE-22 (1986) 1816. T. Ogawa and T. Takagahara, Phys. Rev. B 43 (1991) 14325; in press; Surf. Sci. 263 (1992) 507. J.C. Maan, G. Belle, A. Fasolino, M. Altarelli and K. Ploog, Phys. Rev. B 29 (1984) 7085.
Surface
,.
263 t 1092) 5 12-517
Science
.
A,,
.izj ,,,..: ij,
.::j ,,j”-
y
j::?.‘. :.::::.‘:;j :,,,: :: .: . :.. . . . . ‘.:..,.: .;c
,.
,,.
,.:..:
surface science ;./:.1,:
North-Holland
“,,... ,>::. ,...,. :,,:: ::: ::.::. .::;:I.
Two-photon Akira
Received
Shimizu
3 June
We present (I=
“, Tetsuo
IYYI;
accepted
a unified
0. I. 2, 3. Widths
excitons
absorption
play crucial
theory
of confining
light
structure.
is in striking
This
Ogawa
beam. This
” and
Hiroyuki
for publication
76 August
of two-photon
absorption
potentials
roles in determining
of the incident
spectra of low-dimensional
anisotropy
contrast
are assumed
the spectra.
to one-photon
spectra
rl =
exciton states
an
undoped quasi-d-dimensional (or insulator), which consists of a d-dimensional well region surrounded by barrier regions. We take the xE (5 = 1,. . . , d) [xc ([ = d f 1.. . ,3)] axes along the unconfined Z
ISY2 - Elsevier
Science
than
semiconductors
the exciton
and 2. the spectra
Bohr
strongly
of the exciton
radius. depend
envelope
of an arbitrary
dimension
and the quasi-c/-dimensional on the polarization
functions
a~/
direction
anisotropic
band
absorption
(Qd D) semiconductor
OO~Y-~O2X/Y2/$OS.o(~
I
ho111 anisotropies
Two-photon absorption (TPA) spectroscopy of low-dimensional semiconductors provides us with a wealth of information complementary to onephoton absorption (OPA) spectra. The TPA spectra of three-dimensional (3D) systems were investigated many years ago [1,2]. On the other hand, the TPA spectra of quasi-two-dimensional (Q2D) systems were studied only recently [3-121, and many features characteristic to the Q2D systems have been discovered. We here extend the successful theory for Q2D systems [6] to semiconductors of arbitrary dimension d (= 0. 1, 2, 3). and thereby discuss TPA spectra near the TPA edge as a function of d.
Consider
of low-dimensional
to he smaller
1. Introduction
2. Quasi-d-dimensional
“K
IYYI
When
comes from
Sakaki
semiconductors
Publishers
B.V.
[confined] directions (denoted by the subscript /I [I]), for which the normalization length is 1. [the well width is L ,I. Accordingly, the position vector r is decomposed as r = tr,,. r ) = ({So}. LYJ). WC assume a semiconductor with a onc-photon-allowed direct band gap at the I‘ point, which consists of a single conduction (c) band and two valence (v) bands. Due to the (3 - d)-dimensional confinement, the band structure is modified: (i) the two-fold degeneracy of the valance bands is lifted, and the modified “heavy-hole” (hh) and “light-hole” (Ih) bands are formed, and (ii) subbands are formed in each of these three bands. Under an interband optical excitation, exciton states will be observed due to an attractive tot-cc between an electron and a hole (c-h). We assume that the exciton is of the Wannicr type. whose Bohr radius is larger than L L to form a QdD exciton. Then, the QdD excitons can bc assigned to each pair of c and v (= hh or Ih) subbands. For a Wannier exciton associated with a c-subband cy and a v-subband ,!?. its wavcfunction for zero center-of-mass-motion wavenumber may be proportional [6,13] to the envelope funcand Yamada
Science
Foundntton.
All
rights
rcser~vti
A. Shimizu et al. / Two-photon absorption spectra of low-dimensional semiconductors
tion U,“p of the d-dimensional e-h relative-motion with quantum number(s) v along the unconfined directions and the subband envelope functions 4, and c$~.The exciton has discrete spectra (bound states) when its energy E,“O < EG(a; PI, and has continuous spectra (unbound states) when E,*” 2 E,(a; p), where E,(cr; p) = E, + E, + lP is a band-gap energy with E, and lP the subband quantization energies and E, the bulk band-gap energy. When penetration of the wavefunction into the barrier regions is small, 4, and 4p take decoupled forms such as 4,(r I) = Ilg+oi(x5), where the 4, ‘s denote the subband envelope functions a fong the xc axis, and (Y= ((~~+i,. . . , a,). Due to the finiteness of the spatial extensions of these subband functions, UFp deviates from those of pure d-dimensional excitons when d = 1, 2 [6,13,14]. We take the deviation into account by using a variational form of U,*a for d = 2 [6], and an analytic solution for d = 1 [14].
3. Polarization absorption
anisotropies
47re2 I A I * hm:,c2(L,
x
c
Bloch function of the valence band when k approaches zero along the confinement directions [6]. Note that the E^dependence of WoQpdAD arises only from the interband matrix element. In other words, OPA spectra reflect no anisotropies in the exciton envelope functions. To calculate the TPA rate WTQpdAD, we employ the same method as in the previous theory for Q2D systems [6], which agreed with experiment [9]. For incident light beams of photon energies hw, and hm2, with a common polarization vector P, and vector-potential amplitudes A, and A,, we have obtained the following general formulas for the TPA rate. When the light beams are polarized along an unconfined direction (6 II.?,), wXD( E^IIii) 64n-e4 I A,A, =
hmic4Ei(
I2
L I + L,)3-d
in the two-photon
Suppose that many identical QdD systems are arranged with a period of L I +L,, where L, is the barrier thickness, to constitute a (3 -d)dimensional array. Before we discuss the TPA, the OPA rate WoQpdAD is briefly reviewed for comparison. For an incident light beam of photon energy ho, a polarization vector E^,and a vectorpotential amplitude A, the OPA rate is given as
woQpD =
x s;q hw, + hw,)
1 )
64re4
I(cli*Plc),l*
=
I A,A,
I2
hmzc4E6( L I + L,)3-d
c=hh,lh
xc
x~l(~,l~,>12Cl~~~(r,,=~)12 v 4
l(ClP~l”)J2 I’
(1)
where m, denotes the free-electron mass, Szp the lineshape function of the apv exciton, and (c I E^*p I u), the interband matrix element, with p the momentum operator. Here ( cjI is the
(4
where pL,.,, is the electron-hole reduced mass for the motion along the unconfined directions, pC the 2, element of the momentum operator. On the other hand when the light beams are polarized along a confined direction (; 11i,) wTQpdAD( E^II “J
+L,)“-d
xl!y(Rw),
513
2 PL,, i
I
reduced mass for where kL. I is the electron-hole the motion along the confined directions, and pi the P, element of the momentum operator. These two formulas are the general expressions of the TPA rate for a QdD excitonic system in an arbitrary dimension. In contrast to the OPA rate [eq. (111, the TPA rates have distinctly different forms depending on the polarization direction. Here we note the case of the absence of excitonic effects. In such a case, the formula for the TPA rate are obtained by making the following substitutions in the above equations: U,:‘“(r,,) + 0 for Et:‘” < &(N; p) and U,nfi(,,,) + eik~~-r~~L,;l”Z for E:,+ 2 &(a; p). It is seen from eqs. (2) and (3) that the TPA rate depends on the light polarization through (i) the interband matrix element and the reduced mass (third line of eqs. (2) and (3)), which become anisotropic due to band mixing caused by the low-dimensional confinement, and (ii) the QdDexciton envelope functions (fourth and fifth lines of eqs. (2) and (3)). This is in striking contrast to the OPA rate. That is, the TPA rate strong!\ depends on the light polarization through the strong anisotropies in the exciton enrlelope functions eLIen if the band structure is isotropic. Hence, one can probe the anisotropies in the QdD exciton envclope functions by measuring the TPA spectra. In the following, we will separately discuss characteristic features of the TPA spectra as a function of d.
4. TPA spectra as a function 3.1. Three-dimensional
of d
case
Mahan discussed thoroughly this case [l]. We shall make comments only on important points in this case. In the case of d = 3, there is no “confinement direction”, so that eq. (2) should be used for any polarization direction. The lowest absorption peak is the 2P exciton, whose oscillator strength is small, so that the discrete spectrum is almost smeared out under the usual experimental conditions. As for continuous states. the overall TPA spectra become a smoothly-increasing function of the photon energies.
4.2. Quasi-zero-dimensional
ca,se
In the case of d = 0, an e-h pair is confined in all directions, so that eq. (3) should be used for any polarization direction. Since excitonic effects are negligible as compared with strong effects of subband quantization (recall that we assume small L L), eq. (3) is reduced to w c)Ol> -II’A ZZ
64~~7’1 A,A,I’ tznz;c“E;;(
L
+l,,)ipc’
I(Cl@), c /
P.2>
I’ -
C-1) From the subband integrals in this equation, WC find the subband selection rule; (i) CV;- B, = odd for the polarization direction and (ii) cyLI ~ /3,, = even = 0 for the other directions. Therefore the TPA spectra consist of discrete lines, each corrcsponding to a pair of subbands. 4.3. Quasi-two-dimensional
ca,se
When d = 2. eqs. (2) and (3) reduce to the formulas given in ref. [61, as it should be, and the TPA spectra strongly depend on the polarization direction i. Detailed results of this d = 2 case arc given in ref. [6]. Here we stress again that the large anisotropies of TPA spectra arise from the anisotropies of the cxciton cnvelopc functions. i.c., from the fourth and fifth lines in cq. (2) or eq. (3). To visualize this fact, we plot in fig. 1 this “envelope-function part”, assuming v = lh. Fig. la shows the spectra for i 11i, and fig. lb fin g IIis. A GaAs/Al,,.,, Ga,,,,OAs multiple quantum well structure of 1, ~ = 90 A. and p,,, ,, = O.Ohm,, [I.51 is assumed there. The discrete spectra arc represented by vertical lines, the height of which indicates the integrated TPA intensity divided bq the effective Kydberg energy defined by K,,,,, = pLlh,,eJ/2tl’E’. where t is a dielectric constant. We find the strong anisotropies in the TPA spec-
515
A. Shimizu et al. / Two-photon absorption spectra of low-dimensional semiconductors
(a)
Q2D system cl-lhl transition
z :
e, ‘! 8 ._
0
5 k
15
10
+ flwz - E;%;
P)
1
i II % 3
2
5 6 _c a
1
20
@I :
Q2D system cl -lh2 transition
4
E 2
% -5
r
5 d k?
? // it
5
0
-5
5
0
GUI + fLwz- E&%;
(&I)
20
15
10 B)
(&,,,)
Fig. 1. Envelope-function parts of normalized TPA spectra, the fourth and fifth lines of eqs. (2) and (31, for a GaAs/Al,,,,,Ga,,,,As multiple quantum well structure of L I = 90 A. Contributions from light-hole excitons only (both discrete and continuous states) are plotted (a) when E^is parallel to an unconfined direction, and (b) when i is parallel to the confined direction. Quasi-continuous spectra just below the continuous ones are not shown. The vertical lines represent the discrete spectra, the height of which indicates the integrated TPA intensity divided by the effective Rydberg energy I?,,,,. The horizontal axis is also scaled by this energy. The dashed lines represent the TPA spectra in the absence of exciton effects.
tra as well as the strong enhancements of TPA due to exciton effects for both cases of the polarization directions. 4.4. Quasi-one-dimensional case Excitonic effects become more drastic when the exciton binding energies be-
d = 1 because
QlD system cl -1hl transition
larger. For E^IIf,, the subband selection rule is cy6- /I5 = even = 0, and the derivative of I/@ in eq. (2) was given in ref. [14], which is n&zero only for odd-parity excitons. On the other hand, for f IIf,, the subband selection rule is the same as that in the d = 0 case, and UVapin eq. (3) was given also in ref. [141, which is nonzero only for even-parity excitons. Looking at v = lh only come
(4
QlD system c l -lh2 transition
lb)
n=2 (odd)
-. n=2(even)
l
=___ ---mm
--------mm____.
\
Fig. 2. Envelope-function parts of normalized TPA spectra for a multiple quantum wire structure of z(, = 0.09ehZ/~,h ,,e’, which corresponds to 4R,,,, for the binding energy of the lowest exciton. Contributions from Ih excitons are plotted (a) when E^is parallel to the unconfined direction, and (b) when i is parallel to a confined direction. The vertical scale of these figures is different from that of fig. 1. The quantum number n here is n of ref. 1141 plus one.
and assuming z. = O.O~EA’/~,,, ,,e’ (z,, is a cutoff of a 1D regularized Coulomb potential [14] and is proportional to L I), we have plotted the TPA spectra (the fourth and fifth lines of eqs. (2) and (3)) for f IIi’t in fig. 2a and for i I/ Pz in fig. 2b. In comparison with the case of d = 2, the relative magnitude of the discrete spectra to the continuous spectra becomes larger for d = 1. This is due to increased oscillator strengths in QlD systems [14]. If we look at the continuous spectra, it is interesting that exciton effects enhance the TPA for E^)I P,, whereas they reduce the TPA for E^1)3,. By contrast, the TPA is always enhanced by exciton effects for d 2 2 (see fig. 1 and refs. [ 1,611. The reduction of the continuous TPA spectra by exciton effects in a direct-allowed (two-photonforbidden) gap quantum wire has the same origin as the reduction of OPA of a direct-allowed-gap quantum wire; that is, an anomalously strong concentration of the oscillator strength on the lowest exciton state. The enhancement, on the other hand, has the same origin as that of OPA of a direct-forbidden-gap quantum wire; odd-parity excitons have large slopes at r,, = 0 because the e-h Coulomb attraction becomes more effective for d = 1. In the case of the TPA of the QlD system, one can observe both the reduction (for E^1).i?,) and enhancement (for i IIPi) in the same sample simply by rotating the polarization direction.
5. Concluding
remarks
In summary, we present a unified theory of two-photon absorption (TPA) spectra of low-dimensional semiconductors of arbitrary dimension d (= 0, 1, 2, 3). When the widths of the confining potentials are smaller than the exciton Bohr radius. the excitons become quasi-d-dimensional ones which play an important role in determining the spectra. When d = I and 2, the TPA spectra strongly depend on the polarization direction of the light beam. This strong polarization dependence of the TPA spectra comes from hoti~ anisotropies in the QdD exciton envelope functions and anisotropic band structure, which fact is in remarkable contrast to the OPA case. In the
cast of TPA, one observes different exciton states of different enrlelope functions by simply rotating the polarization direction. Lastly WC again note that the QdD cxcitonic effects on the TPA spectra are clarified assuming a small L , in comparison with the cxciton Bohr radius. In this case, the subband separation energy is much larger than the cxciton binding energy. Otherwise the e-h Coulomb interaction admixes many subband pairs into the excitonic states, and the continuum excitons associated with a lower energy subband pair interfer with discrete cxciton states associated with a higher energy subband pair. This situation is also of interest in the interpretation of experimental results in a dimensional-crossover regime. Extension of our theory to this regime is now progressing and will be reported elsewhere.
Acknowledgements The authors would like to thank Dr. T. Takagahara, Dr. H. Kanbc and Dr. K. Nakamura for discussions. Two of the authors (T.O. and A.!%) also thank Dr. Y. Horikoshi, Dr. T. Kimura, R. Murakami and Y. Takigawa for encouragement.
Note added in proof Low-dimensional confinement of electronic and excitonic states had been believed to he detected in cxperimcnts as optical anisotropics and/or blue shifts of OPA spectra. However. Bauer and Sakaki [Surf. Sci. 367 (1993)] have shown that these cannot be a proof of the confinement. and another probe of the four-dimcnsionality is being looked for. Our results have shown that TPA spectroscopy is a sensitive probe of the low-dimensionality because the TPA spectra arc directly related with the low-dimensional envelope functions.
References [I] G.D.
Mahan.
[7] (‘.C.
Ixe
Phya. Rev.
and H.Y.
170 (I%X)
X2.5.
Fan. Phys. Rev. 13 0 (lU7-i) i.il)l.
A. Shimizu et al. / Two-photon absorption spectra of low-dimensional semiconductors [3] A. Shimizu and K. Fujii, OYO BUTURI 60 (1991) 128 [in Japanese]. [4] H.N. Spector, Phys. Rev. B 35 (1987) 5876. [5] A. Pasquarello and A. Quattropani, Phys. Rev. B 38 (1988) 6206. [6] A. Shim& Phys. Rev. B 40 (1989) 1403. [7] D. FrGhlich, R. Wille, W. Schlapp and G. Weimann, Phys. Rev. Lett. 61 (1988) 1878. [8] W.H. Knox, D.S. Chemla, D.A.B. Miller, J.B. Stark and S. Schmitt-Rink, Phys. Rev. Lett. 62 (1989) 1189. [9] K. Tai, A. Mysyrowicz, R.J. Fisher, R.E. Slusher and A.Y. Cho, Phys. Rev. Lett. 62 (1989) 1784. [lo] M. Nithisoontorn, K. Unterrainer, S. Michaelis, N.
[ll] [12] [13] [14] [15]
511
Sawaki, E. Gornik and H. Kano, Phys. Rev. Lett. 62 (1989) 3078. I.M. Catalano, A. Cingolani, R. Cingolani, M. Lcpore and K. Ploog, Phys. Rev. B 40 (1989) 1312. K. Fujii, A. Shimizu, J. Bergquist and T. Sawada, Phys. Rev. Lett. 65 (1990) 1808. D.A.B. Miller, J.S. Weiner and D.S. Chemla, IEEE J. Quantum Electron. QE-22 (1986) 1816. T. Ogawa and T. Takagahara, Phys. Rev. B 43 (1991) 14325; in press; Surf. Sci. 263 (1992) 507. J.C. Maan, G. Belle, A. Fasolino, M. Altarelli and K. Ploog, Phys. Rev. B 29 (1984) 7085.