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Spatial dispersion of the renormalized polariton-biexciton spectrum
Journal of Luminescence 34 (1986) 337-342 North-Holland, Amsterdam
SPATIAL DISPERSION H.H. KRANZ
337
OF THE RENORMALIZED
POLARITON-BIEXCITON
SPECTRUM
and H. HAUG
Institut Jiir Theoretische Physik der Uniuersitiit Frankfurt,
Robert-Mayer-Str.
8, D-6000 Frankfurt-Main,
Fed. Rep. Germany
Received 12 November 1985 Revised 7 January 1986
The spatial dispersion of the nonlinearly renormalized exciton-photon-biexciton spectrum is calculated by taking into account the polarization selection rules of the transitions. Some branches of the spectrum are found to be unstable against symmetry-broken solutions. The origin of this mechanism of symmetry breaking is an instability of the induced absorption of the two circularly polarized beams that make up the incident linearly polarized beam. In contrast to previously suggested mechanisms of symmetry breaking these branches become gradually more stable with increasing intensity.
1. Introduction It is well established that the two-photon biexciton resonance introduces a nonlinear change of the exciton-polariton spectrum [l]. In lowest approximation the two-photon resonance causes simply a splitting of the polariton spectrum around half of the biexciton frequency, which resembles the splitting of the photon spectrum near the exciton resonance except for the fact that it is intensity-dependent [2]. In higher approximation, however, the pole in the Green’s functions of the interacting three boson fields becomes quadratic and yields, as long as spatial dispersion for the excitons and biexcitons is not taken into account, strange properties of the renormalized spectrum [3,4]. In order to get rid of these unphysical properties we include two features which have not been considered in our previous analysis: (a) Spatial dispersion of the excitons and biexcitons and (b) the polarization selection rules of the underlying optical transitions. The necessity of including spatial dispersion and polarization selection rules has already been stressed by Cho and Itoh [5]. Our analysis differs from that of ref. [5] and includes the total nonlinear polarization. The resulting spectra are calculated for linearly polarized beams that are de0022-2313/86/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
composed into two circularly polarized beams of opposite polarization. First, we calculate the spectra including spatial dispersion and damping for the symmetric case, i.e., we take both the intensities and the wavevectors of the two circular polarizations to be the same. In the frequency range between the biexciton and exciton resonance we find up to five polariton branches. Outside this range the spectra consist of a propagating mode and two degenerate, damped additional modes. At low intensities the spectra reduce to that of the usual exciton polariton. In this limit the extra modes due to the biexcitons are found to be unstable against solutions for which one of the two circularly polarized beams is strongly damped. Under such a situation the unrenormalized exciton-polariton spectrum reappears. The existence of up to five branches obviously leads to the question which additional boundary conditions (ABCs) have to be taken into account at the surface of a dielectric medium. Physically it is clear that at the surface not only the exciton amplitude but also that of the biexcitons has to vanish. The problem of formulating the nonlinear ABCs in the presence of biexcitons is not considered further in the present paper. Secondly, we show that by symmetry breaking the medium can become optically active and thus rotates the polarization of the transmitted beam. B.V.
H. H. Kranz, H. Haug / Spatial dispersion
338
This effect has already been predicted by Cho and Itoh [5]. However, our analysis shows that only finite damping (i.e., only a finite line width of the exciton and biexciton absorption) causes this feature. If the intensity of one of the two circularly polarized beams, say I_, is initially infinitely smaller than the other, i.e., Z,) then the nonlinear absorption will be larger for the weaker beam and thus the amplitude I_ is dimished further. We find in contrast to refs. [5] and [7] that this new nonlinear polarization instability is most pronounced at low intensities, 2. Model Hamiltonian and nonlinear polarization
of the
renormalized
polariton - brexciton spectrum
=fi,-ir,=fin,.
P, = gx(bh)
+ g,,(bi-,B)
= Pi”+
Pi’).
(3)
If we factorize the last term it is straightforward to calculate (b,) and (B) from the coupled equations of motion. The results are (bh)=~[Orn~~-h-~~+I-hlEx,
H = CUU4,4, x
Pl”=
(B)
=&(bh)L+
(4) (b-,)&l,
(5)
where (6)
4=g,2,1EJ2~ det = 9,L?,+L?X_-
I,&?,,-
contributions
I-Q,_.
(7)
to the polarization
are
-&2,$_i-Ih+I-A)EA,
k-)@++A++, pi”
-
(2)
E er are the minimum energies and m,, mT are tee masses of the biexcitons and transverse excitons, respectively. The nonlinear polarization is given by
The resulting
+ &x(
_20
m
2m
m
We treat the linearly polarized light field classically and decompose it into two circularly polarized fields with the amplitudes Ek+ and E,-. The molecule has a r, symmetry in cubic crystals (such as CuCl) and is thus created by an exciton of polarization h and a photon of polarization -A. Taking only resonant terms into account our model Hamiltonian is (tz = 1)
+Qll(k++
(k++k-)2_ir
O,(k++k_)=c,-
&%,
+ h.c.)
~g,,jUW~++~-+ h.c.).
X
(1)
Here, b,, and B, are the annihilation operators of the excitons and biexcitons, respectively. We treat only excitons with the momentum k + of the field and molecules with k, = k, + k_, because we do not include any intraband-scattering processes nor spontaneous decay of excitons. In the following we drop the momentum indices. The Hamiltonian (1) is already in the rotating frame, where the periodic time-dependence of the field is eliminated. The finite lifetimes of the excitons and biexcitons are taken into account through IY, and r,,,, respectively. The shifted exciton and exciton molecule frequencies are ir, - w
i2,(k,)=rT&T
= _ ~(q$:,+I,-I_,) g,’ ]det12
(a,++ S2,~)Z_,E,.
(9)
3. Spectrum for symmetric solutions First, we calculate the modified polariton spectrum for the symmetric solutions I+ = I_ = I. The wave equation describing the propagation of an optical wave through a nonlinear medium is
(10) Taking a plane-wave variation for the electric field E(r, r) and the nonlinear polarization P(r, t) E(r,
t) =e,E,
exp[i(kA.r-
P(r,
t)=e,P,exp[i(kx.r-wt)],
wt)],
(11) (12)
H.H. Kranz, H. Haug / Spatial dispersron of the renormaliredpolariton - biexciton spectrum
we obtain ponents:
the wave equation
for the Fourier
321
com-
339
Ii
_I
(13) Here, the nonlinear polarization depends on intensity I,, and is proportional to the field E,, given in eqs. (8) and (9). Inserting this nonlinear polarization into (13) yields the following eigenvalue equations the modified polariton spectrum: 1 - zk; cd2
+ g,’
f&l&LA -I,
the as eq. for
IMAG [Kl Fig. 2. Dispersion
at a high intensity
I = 10 A5 (eV)’
+ I-,
det =
().
+
(14) We have retained the dependence of the intensity IA and the wavevector k, on the polarization because we make use of it in the discussion of the stability of the symmetric solution k+= k_ for I+= I_ in the next section. In the rest of this section we take k,= k_= k and I+= I_= I. Qualitatively, for the low normalized intensity I = lo-* (eV)2 (that corresponds to an energy flux of about 1.5 kW/cm’) the spectrum in fig. 1 is that of a usual exciton polariton. In the linear-response limit the coupling to the biexciton drops out of the dispersion relation, eq. (14), altogether. The biexciton branches that are drawn in fig. 1 as a dotted line consist of two nearly degenerate modes which are no longer resolved graphically. In
the high-intensity regime, however, the biexcitons strongly modify the spectrum as shown in fig. 2 for 1= lop5 (eVJ2 (A 1.5 MW/cm2). When both damping and dispersion of the excitons and biexcitons are included we find that the regions with negative group velocities of propagating modes disappear that had plagued the spectra in refs. [4] and [6]. Hence, the unphysical features of the spectra in these references are not due to the factorization (bB) = (b)(B) in the calculation of the polarization as has been suggested in ref. [6]. In a consistent treatment of damping and dispersion these unphysical features do not appear. The most obvious modifications of the spectra by biexcitons are: (1) The excitonic branch ends at high intensities near Aw = er. (2) Near Ao = e,/2 the branches are connected differently in the high- and low-intensity regime. It should be possible to measure these features experimentally, e.g., with hyper-raman scattering
181. For the calculation numerical values:
we
used
the
following
e ,_ = 3.2019 eV, er = 3.2022 eV, m, = 5.59 m,, ElTl= 6.3722 eV, E, = 5.59, r, = I’, = 10e4 eV.
318 ,’ 18
”
lo’
/,.I. ld2 5 IMAG [Kl
10‘
j, Id3
ld2 REAL [Kl
lo’
10
Fig. 1. Dispersion at a low normalized intensity I = lo-* (eV)‘. hw (Re k) and Aw (Im k), respectively. In all figures the wavevector is given in units of the inverse Bohr radius.
4. Stability of the symmetric mode k += k _ Cho and Itoh [5] found new modes with k+# k_ for CuCl although I+= I_. In their calculation they generalized the dielectric function of ref. [2]
H. H. Kranz, H. Haug
340
to two fields. Instead
(eq. (8) with I+= B,=
-gldI
/ Spatial dispersion
I_=
polariton - biexciton spectrum
I) they used (16)
Whereas we can reproduce their results with P,, these symmetry-broken solutions apparently disappear when we use either P(l) or both terms P(l) and P(*) No lifetime effects were included as in the calculation of Cho and Itoh [5]. However, one would expect symmetry-broken modes provided that a fluctuation in the intensity I+> I_ results in a difference Im k_- Im k+> 0. In this case, the polarization with the smaller intensity I_ will be damped stronger and therefore the original fluctuation will be enhanced. In the reverse case the fluctuation will be damped out and the symmetric mode is stable. Since the present mechanism of symmetry breaking is based on a nonlinear increase of absorption of one of the fields it is necessary to include the finite lifetime of excitons and/or biexcitons. The relevant quantity for the stability of the symmetric modes is
SE
renormalized
of
;m_,. m h
(Im k+-
of the
Im k_)(
I++
I_)
(Im k++ Im k_)(Z+-
Z-)
(17)
’
for I+= I_ and k,= k_. S can be calculated by linearizing the dispersion function (LHS of eq. (14)) in AZ = I+- I_ and Ak = k+- k_. If S 2 0
, I
318 -30
-20
-10
0
s Fig. 4. S (eq. dotted implies broken
Stability analysis of the spectrum in fig. 3. The functions (17)) for the two propagating biexciton modes (solid and lines) and the exciton mode (dashed line). S 5 - 10m2 that the mode is unstable with respect to symmetrysolutions k, + k_.
then the symmetric mode is stable, otherwise it is unstable. For higher intensity, fig. 2, we find all modes to be either stable or only marginally unstable (S 2 -lo-*). For the lower intensity, fig. 3, however, the two propagating biexciton modes are unstable except for a small region. The propagating polariton mode is also unstable near half the biexciton energy c,. The situation becomes more transparent by looking at the function S itself for these three modes in fig. 4. The corresponding curves for the higher intensity show at most a small negative peak near Ao = e,/2 of about S 5 -lo-* and are therefore not shown. Some consequences
318 1 ,,,_ 104
d_Li
103
10* REAL
10’
loo
lKI
Fig. 3. Stable parts (solid lines) and unstable parts (dotted lines) of the spectrum at a low intensity I = 10-s (eV)2.
(1) From eq. (14) follows that the biexciton features should disappear from the spectra in the limit of vanishing intensities. According to our stability analysis this happens because the extra biexciton modes become unstable due to some fluctuation in the intensity, say I+> I_. In the limit ZZ-+ 0, we see from eq. (14) that the surviving polarization indeed has the dispersion of a polariton without any renormalization by biexcitons. (2) At an intermediate thickness of the sample this mechanism gives rise to a Faraday rotation of
H.H. Kranz, H. Haug / Spatial dispersion
of the renormalizedpolariton
- biexciton spectrum
341
the incident linearly polarized beam. This effect disappears for intensities higher than a critical value somewhere between I = lop8 (eV)2 and 10e5 (eV)2. Above this range all modes are stable with respect to symmetry-breaking and the Faraday rotation is suppressed.
Appendix IL12
An important feature of the spectra is the occurrence of almost degenerate modes with relative distances in the wavenumber down to about lo-‘. These modes rearrange or end by merging depending on energy and intensity. Standard iterative methods to determine the modes that make use of derivatives, like Newton’s procedure, failed. Since the changes of sign are rather abrupt, the procedure often was trapped in some local minimum or jumped to another branch. An iteration method that gave good results in a limited energy range made use of the fact that the dispersion function (LHS of eq. (14)) can be written either as a third degree polynomial of the k2 with (k2)* as a parameter or as a second degree polynomial of (k’)* with k2 as a parameter. The major part of the spectra has been calculated with a generalized two-dimensional regula falsi: we constructed a circle of search points around a trial point that was extrapolated from the already calculated part of the spectrum. With these search point we looked for the intersections of the circle with lines where the real or imaginary part of the dispersion function vanishes. By linear interpolation an improved trial point was determined and the procedure was repeated. A relative accuracy of about low9 was used. It was, of course, important to decide whether some of the modes end by merging near ~,/2 and cr or, only get so close that they can no longer be separated numerically. The behaviour of the modes near er could readily by analyzed be expanding the dispersion function around s2,s2, - I = 0 up to second order in the wavevector. This expansion, however, is very poor near c,/2. So we plotted the lines where the real and imaginary part of the dispersion function vanishes for fixed energies. The intersections of these two lines give the real
l.LIL
1.418
1.416
l.L?Ll 1.L22
IMAG IKI
r
P
l.L2L .lo“
L,g52i
9.86
9.87
988
IMAG [Kl
9.89 Xl?
Fig. 5. Lines along whic)l the real part (solid line) and imaginary part (dotted line) of the dispersion function (LHS eq. (14)) vanish for fixed energies E. The intersections give real and imaginary parts of the wavevector (indicated arrows). (a) E = 3.18622 eV; (b) E = 3.18615 eV.
the of the by
and imaginary part of the wavevector (denoted by arrows in fig. 5a). In fig. 5b these two lines no longer intersect, i.e., the two modes indeed have merged and ended for low intensities around o = eJ2 as can be seen in fig. 1.
Acknowledgements This work has been supported by the Deutsche Forschungsgemeinschaft through the Sonderforschungsbereich Frankfurt/Darmstadt, and by the Commission of the European Communities (under the experimental phase of the European Stimulation Action).
References [l] See, e.g., H. Haug and S. Schmitt-Rink, 9 (1984) 3.
Prog. Quant.
Electr.
342
H.H. Kranz, H. Haug / Spatial dispersion
[2] R. MPrz, S. Schmitt-Rink and H. Haug, Z. Phys. B 40 (1980) 9. [3] I. Abram and A. Maruani, Phys. Rev. B 26 (1982) 4759. [4] H. Haug, J. Lumin. 30 (1985) 171. [5] K. Cho and T. Itoh, Solid State Commun. 52 (1984) 287.
ofthe renormalizedpolariton
- biexciion spectrum
[6] CC. Sung and C.M. Bowden, Phys. Rev. A 29 (1984) 1957. [7] M. Inoue, preprint. [S] Such experiments are discussed, e.g., in the review article by B. HGnerlage, R. Levy, J.B. Grun, C. Klingshirn and K. Bohnert. Phys. Rep. 124 (1985) 111.
SPATIAL DISPERSION H.H. KRANZ
337
OF THE RENORMALIZED
POLARITON-BIEXCITON
SPECTRUM
and H. HAUG
Institut Jiir Theoretische Physik der Uniuersitiit Frankfurt,
Robert-Mayer-Str.
8, D-6000 Frankfurt-Main,
Fed. Rep. Germany
Received 12 November 1985 Revised 7 January 1986
The spatial dispersion of the nonlinearly renormalized exciton-photon-biexciton spectrum is calculated by taking into account the polarization selection rules of the transitions. Some branches of the spectrum are found to be unstable against symmetry-broken solutions. The origin of this mechanism of symmetry breaking is an instability of the induced absorption of the two circularly polarized beams that make up the incident linearly polarized beam. In contrast to previously suggested mechanisms of symmetry breaking these branches become gradually more stable with increasing intensity.
1. Introduction It is well established that the two-photon biexciton resonance introduces a nonlinear change of the exciton-polariton spectrum [l]. In lowest approximation the two-photon resonance causes simply a splitting of the polariton spectrum around half of the biexciton frequency, which resembles the splitting of the photon spectrum near the exciton resonance except for the fact that it is intensity-dependent [2]. In higher approximation, however, the pole in the Green’s functions of the interacting three boson fields becomes quadratic and yields, as long as spatial dispersion for the excitons and biexcitons is not taken into account, strange properties of the renormalized spectrum [3,4]. In order to get rid of these unphysical properties we include two features which have not been considered in our previous analysis: (a) Spatial dispersion of the excitons and biexcitons and (b) the polarization selection rules of the underlying optical transitions. The necessity of including spatial dispersion and polarization selection rules has already been stressed by Cho and Itoh [5]. Our analysis differs from that of ref. [5] and includes the total nonlinear polarization. The resulting spectra are calculated for linearly polarized beams that are de0022-2313/86/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
composed into two circularly polarized beams of opposite polarization. First, we calculate the spectra including spatial dispersion and damping for the symmetric case, i.e., we take both the intensities and the wavevectors of the two circular polarizations to be the same. In the frequency range between the biexciton and exciton resonance we find up to five polariton branches. Outside this range the spectra consist of a propagating mode and two degenerate, damped additional modes. At low intensities the spectra reduce to that of the usual exciton polariton. In this limit the extra modes due to the biexcitons are found to be unstable against solutions for which one of the two circularly polarized beams is strongly damped. Under such a situation the unrenormalized exciton-polariton spectrum reappears. The existence of up to five branches obviously leads to the question which additional boundary conditions (ABCs) have to be taken into account at the surface of a dielectric medium. Physically it is clear that at the surface not only the exciton amplitude but also that of the biexcitons has to vanish. The problem of formulating the nonlinear ABCs in the presence of biexcitons is not considered further in the present paper. Secondly, we show that by symmetry breaking the medium can become optically active and thus rotates the polarization of the transmitted beam. B.V.
H. H. Kranz, H. Haug / Spatial dispersion
338
This effect has already been predicted by Cho and Itoh [5]. However, our analysis shows that only finite damping (i.e., only a finite line width of the exciton and biexciton absorption) causes this feature. If the intensity of one of the two circularly polarized beams, say I_, is initially infinitely smaller than the other, i.e., Z,) then the nonlinear absorption will be larger for the weaker beam and thus the amplitude I_ is dimished further. We find in contrast to refs. [5] and [7] that this new nonlinear polarization instability is most pronounced at low intensities, 2. Model Hamiltonian and nonlinear polarization
of the
renormalized
polariton - brexciton spectrum
=fi,-ir,=fin,.
P, = gx(bh)
+ g,,(bi-,B)
= Pi”+
Pi’).
(3)
If we factorize the last term it is straightforward to calculate (b,) and (B) from the coupled equations of motion. The results are (bh)=~[Orn~~-h-~~+I-hlEx,
H = CUU4,4, x
Pl”=
(B)
=&(bh)L+
(4) (b-,)&l,
(5)
where (6)
4=g,2,1EJ2~ det = 9,L?,+L?X_-
I,&?,,-
contributions
I-Q,_.
(7)
to the polarization
are
-&2,$_i-Ih+I-A)EA,
k-)@++A++, pi”
-
(2)
E er are the minimum energies and m,, mT are tee masses of the biexcitons and transverse excitons, respectively. The nonlinear polarization is given by
The resulting
+ &x(
_20
m
2m
m
We treat the linearly polarized light field classically and decompose it into two circularly polarized fields with the amplitudes Ek+ and E,-. The molecule has a r, symmetry in cubic crystals (such as CuCl) and is thus created by an exciton of polarization h and a photon of polarization -A. Taking only resonant terms into account our model Hamiltonian is (tz = 1)
+Qll(k++
(k++k-)2_ir
O,(k++k_)=c,-
&%,
+ h.c.)
~g,,jUW~++~-+ h.c.).
X
(1)
Here, b,, and B, are the annihilation operators of the excitons and biexcitons, respectively. We treat only excitons with the momentum k + of the field and molecules with k, = k, + k_, because we do not include any intraband-scattering processes nor spontaneous decay of excitons. In the following we drop the momentum indices. The Hamiltonian (1) is already in the rotating frame, where the periodic time-dependence of the field is eliminated. The finite lifetimes of the excitons and biexcitons are taken into account through IY, and r,,,, respectively. The shifted exciton and exciton molecule frequencies are ir, - w
i2,(k,)=rT&T
= _ ~(q$:,+I,-I_,) g,’ ]det12
(a,++ S2,~)Z_,E,.
(9)
3. Spectrum for symmetric solutions First, we calculate the modified polariton spectrum for the symmetric solutions I+ = I_ = I. The wave equation describing the propagation of an optical wave through a nonlinear medium is
(10) Taking a plane-wave variation for the electric field E(r, r) and the nonlinear polarization P(r, t) E(r,
t) =e,E,
exp[i(kA.r-
P(r,
t)=e,P,exp[i(kx.r-wt)],
wt)],
(11) (12)
H.H. Kranz, H. Haug / Spatial dispersron of the renormaliredpolariton - biexciton spectrum
we obtain ponents:
the wave equation
for the Fourier
321
com-
339
Ii
_I
(13) Here, the nonlinear polarization depends on intensity I,, and is proportional to the field E,, given in eqs. (8) and (9). Inserting this nonlinear polarization into (13) yields the following eigenvalue equations the modified polariton spectrum: 1 - zk; cd2
+ g,’
f&l&LA -I,
the as eq. for
IMAG [Kl Fig. 2. Dispersion
at a high intensity
I = 10 A5 (eV)’
+ I-,
det =
().
+
(14) We have retained the dependence of the intensity IA and the wavevector k, on the polarization because we make use of it in the discussion of the stability of the symmetric solution k+= k_ for I+= I_ in the next section. In the rest of this section we take k,= k_= k and I+= I_= I. Qualitatively, for the low normalized intensity I = lo-* (eV)2 (that corresponds to an energy flux of about 1.5 kW/cm’) the spectrum in fig. 1 is that of a usual exciton polariton. In the linear-response limit the coupling to the biexciton drops out of the dispersion relation, eq. (14), altogether. The biexciton branches that are drawn in fig. 1 as a dotted line consist of two nearly degenerate modes which are no longer resolved graphically. In
the high-intensity regime, however, the biexcitons strongly modify the spectrum as shown in fig. 2 for 1= lop5 (eVJ2 (A 1.5 MW/cm2). When both damping and dispersion of the excitons and biexcitons are included we find that the regions with negative group velocities of propagating modes disappear that had plagued the spectra in refs. [4] and [6]. Hence, the unphysical features of the spectra in these references are not due to the factorization (bB) = (b)(B) in the calculation of the polarization as has been suggested in ref. [6]. In a consistent treatment of damping and dispersion these unphysical features do not appear. The most obvious modifications of the spectra by biexcitons are: (1) The excitonic branch ends at high intensities near Aw = er. (2) Near Ao = e,/2 the branches are connected differently in the high- and low-intensity regime. It should be possible to measure these features experimentally, e.g., with hyper-raman scattering
181. For the calculation numerical values:
we
used
the
following
e ,_ = 3.2019 eV, er = 3.2022 eV, m, = 5.59 m,, ElTl= 6.3722 eV, E, = 5.59, r, = I’, = 10e4 eV.
318 ,’ 18
”
lo’
/,.I. ld2 5 IMAG [Kl
10‘
j, Id3
ld2 REAL [Kl
lo’
10
Fig. 1. Dispersion at a low normalized intensity I = lo-* (eV)‘. hw (Re k) and Aw (Im k), respectively. In all figures the wavevector is given in units of the inverse Bohr radius.
4. Stability of the symmetric mode k += k _ Cho and Itoh [5] found new modes with k+# k_ for CuCl although I+= I_. In their calculation they generalized the dielectric function of ref. [2]
H. H. Kranz, H. Haug
340
to two fields. Instead
(eq. (8) with I+= B,=
-gldI
/ Spatial dispersion
I_=
polariton - biexciton spectrum
I) they used (16)
Whereas we can reproduce their results with P,, these symmetry-broken solutions apparently disappear when we use either P(l) or both terms P(l) and P(*) No lifetime effects were included as in the calculation of Cho and Itoh [5]. However, one would expect symmetry-broken modes provided that a fluctuation in the intensity I+> I_ results in a difference Im k_- Im k+> 0. In this case, the polarization with the smaller intensity I_ will be damped stronger and therefore the original fluctuation will be enhanced. In the reverse case the fluctuation will be damped out and the symmetric mode is stable. Since the present mechanism of symmetry breaking is based on a nonlinear increase of absorption of one of the fields it is necessary to include the finite lifetime of excitons and/or biexcitons. The relevant quantity for the stability of the symmetric modes is
SE
renormalized
of
;m_,. m h
(Im k+-
of the
Im k_)(
I++
I_)
(Im k++ Im k_)(Z+-
Z-)
(17)
’
for I+= I_ and k,= k_. S can be calculated by linearizing the dispersion function (LHS of eq. (14)) in AZ = I+- I_ and Ak = k+- k_. If S 2 0
, I
318 -30
-20
-10
0
s Fig. 4. S (eq. dotted implies broken
Stability analysis of the spectrum in fig. 3. The functions (17)) for the two propagating biexciton modes (solid and lines) and the exciton mode (dashed line). S 5 - 10m2 that the mode is unstable with respect to symmetrysolutions k, + k_.
then the symmetric mode is stable, otherwise it is unstable. For higher intensity, fig. 2, we find all modes to be either stable or only marginally unstable (S 2 -lo-*). For the lower intensity, fig. 3, however, the two propagating biexciton modes are unstable except for a small region. The propagating polariton mode is also unstable near half the biexciton energy c,. The situation becomes more transparent by looking at the function S itself for these three modes in fig. 4. The corresponding curves for the higher intensity show at most a small negative peak near Ao = e,/2 of about S 5 -lo-* and are therefore not shown. Some consequences
318 1 ,,,_ 104
d_Li
103
10* REAL
10’
loo
lKI
Fig. 3. Stable parts (solid lines) and unstable parts (dotted lines) of the spectrum at a low intensity I = 10-s (eV)2.
(1) From eq. (14) follows that the biexciton features should disappear from the spectra in the limit of vanishing intensities. According to our stability analysis this happens because the extra biexciton modes become unstable due to some fluctuation in the intensity, say I+> I_. In the limit ZZ-+ 0, we see from eq. (14) that the surviving polarization indeed has the dispersion of a polariton without any renormalization by biexcitons. (2) At an intermediate thickness of the sample this mechanism gives rise to a Faraday rotation of
H.H. Kranz, H. Haug / Spatial dispersion
of the renormalizedpolariton
- biexciton spectrum
341
the incident linearly polarized beam. This effect disappears for intensities higher than a critical value somewhere between I = lop8 (eV)2 and 10e5 (eV)2. Above this range all modes are stable with respect to symmetry-breaking and the Faraday rotation is suppressed.
Appendix IL12
An important feature of the spectra is the occurrence of almost degenerate modes with relative distances in the wavenumber down to about lo-‘. These modes rearrange or end by merging depending on energy and intensity. Standard iterative methods to determine the modes that make use of derivatives, like Newton’s procedure, failed. Since the changes of sign are rather abrupt, the procedure often was trapped in some local minimum or jumped to another branch. An iteration method that gave good results in a limited energy range made use of the fact that the dispersion function (LHS of eq. (14)) can be written either as a third degree polynomial of the k2 with (k2)* as a parameter or as a second degree polynomial of (k’)* with k2 as a parameter. The major part of the spectra has been calculated with a generalized two-dimensional regula falsi: we constructed a circle of search points around a trial point that was extrapolated from the already calculated part of the spectrum. With these search point we looked for the intersections of the circle with lines where the real or imaginary part of the dispersion function vanishes. By linear interpolation an improved trial point was determined and the procedure was repeated. A relative accuracy of about low9 was used. It was, of course, important to decide whether some of the modes end by merging near ~,/2 and cr or, only get so close that they can no longer be separated numerically. The behaviour of the modes near er could readily by analyzed be expanding the dispersion function around s2,s2, - I = 0 up to second order in the wavevector. This expansion, however, is very poor near c,/2. So we plotted the lines where the real and imaginary part of the dispersion function vanishes for fixed energies. The intersections of these two lines give the real
l.LIL
1.418
1.416
l.L?Ll 1.L22
IMAG IKI
r
P
l.L2L .lo“
L,g52i
9.86
9.87
988
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9.89 Xl?
Fig. 5. Lines along whic)l the real part (solid line) and imaginary part (dotted line) of the dispersion function (LHS eq. (14)) vanish for fixed energies E. The intersections give real and imaginary parts of the wavevector (indicated arrows). (a) E = 3.18622 eV; (b) E = 3.18615 eV.
the of the by
and imaginary part of the wavevector (denoted by arrows in fig. 5a). In fig. 5b these two lines no longer intersect, i.e., the two modes indeed have merged and ended for low intensities around o = eJ2 as can be seen in fig. 1.
Acknowledgements This work has been supported by the Deutsche Forschungsgemeinschaft through the Sonderforschungsbereich Frankfurt/Darmstadt, and by the Commission of the European Communities (under the experimental phase of the European Stimulation Action).
References [l] See, e.g., H. Haug and S. Schmitt-Rink, 9 (1984) 3.
Prog. Quant.
Electr.
342
H.H. Kranz, H. Haug / Spatial dispersion
[2] R. MPrz, S. Schmitt-Rink and H. Haug, Z. Phys. B 40 (1980) 9. [3] I. Abram and A. Maruani, Phys. Rev. B 26 (1982) 4759. [4] H. Haug, J. Lumin. 30 (1985) 171. [5] K. Cho and T. Itoh, Solid State Commun. 52 (1984) 287.
ofthe renormalizedpolariton
- biexciion spectrum
[6] CC. Sung and C.M. Bowden, Phys. Rev. A 29 (1984) 1957. [7] M. Inoue, preprint. [S] Such experiments are discussed, e.g., in the review article by B. HGnerlage, R. Levy, J.B. Grun, C. Klingshirn and K. Bohnert. Phys. Rep. 124 (1985) 111.