REVISTA MEXICANA DE FISICA S53 (7) 61–65 DICIEMBRE 2007

Many-body effects in spectral-hole burning for quantum-well lasers

L.C. Lew Yan VoonDepartment of Physics, Wright State University,

Dayton, Ohio 45435, USA.

M. Willatzen and B. LassenMads Clausen Institute, University of Southern Denmark,

DK-6040 Sønderborg, Denmark.

Recibido el 30 de noviembre de 2006; aceptado el 8 de octubre de 2007

An expression for the spectral-hole burning incorporating many-body effects is derived. Numerical results are presented for III-V singlequantum-well laser structures.

Keywords:Semiconductor laser; III-V; quantum well; many-body; spectral-hole burning.

Una expresion para el quemado de hueco espectral incorporando los efectos de muchos cuerpos es derivada. Resultados numericos paralaseres hechos de ununico III-V pozo cuantico son tambien presentados.

Descriptores:Laser semiconductor; III-V; pozo cuantico; efectos de muchos cuerpos; quemado de hueco espectral

PACS: 73.21.Fg; 73.43.Cd; 78.67.De

1. Introduction

Under high photon excitation, laser gain is nonlinear. Thethree main mechanisms are: spectral hole burning, spatialhole burning, and carrier heating [1, 2]. We are here con-cerned with spectral hole burning (SHB): gain reduction dueto carrier depletion near the laser resonance. Previous cal-culations of SHB were based upon a single-particle pic-ture [1–4]; we extend this to include many-body effects.

2. Theory

We have derived an expression for spectral-hole burning in-corporating many-body effects.

The density-matrix equations (DME’s) including many-body effects (using the Hartree-Fock formalism plus ad-ditional scattering mechanisms responsible for relaxationterms) are [5]:

∂ρnn(t)∂t

= − (ρnn(t)− fnn(t))τ1c

−(ρnn(t)− fnn

)

τs

+ Λnn + i(ωR,nmρ∗nm − ω∗R,nmρnm

), (1)

∂ρmm(t)∂t

= − (ρmm(t)− fmm(t))τ1v

−(ρmm(t)− fmm

)

τs

+ Λmm − i(ωR,nmρ∗nm − ω∗R,nmρnm

), (2)

∂ρnm(t)∂t

=(−iωnm− 1

τ2

)ρnm−iωR,nm (ρnn−ρmm) ,

(3)

where ρnn is the quantum leveln (electron) occupation,ρmm is the quantum levelm (hole) occupation,ρnm isthe electron-hole density-matrix component,fnn (fmm) isthe electron (hole) quasi-equilibrium distribution,fnn is the

thermal-equilibrium distribution,Λnn (Λmm) is the externalpumping into electron (hole) leveln (m), τc (τv) is the elec-tron (hole) relaxation time,τ2 is the dipole relaxation time,~ωnm is the energy difference between levelsn andm, andtis time. The generalized Rabi frequencyωR,nm is given by:

ωR,nm =1~

(dnmE(~r, t) +

∑

n′m′Vn′m′,nmρn′m′(t)

), (4)

wherednm is the dipole matrix element between statesnandm, Vn′m′,nm is the Coulomb interaction potential, andE(~r, t) is the electric field given by:

E(~r, t) =12

(E0(t) exp(iωt) + E∗0 (t) exp(−iωt)) . (5)

Next, we make the following simplifying assumptions: sincethe relaxation timeτs for establishing a thermal equilibriumdistribution is in the order of1 ns while the correspondingquasi-equilibrium relaxation timesτc, τv are approximately50 fsec, we may safely neglect the relaxation term involv-ing τs in the DME’s. Similarly, we assume that pumping iseffective only into higher energy levels than those participat-ing in lasing which allows us to neglect the pumping termsΛnn,Λmm. Finally, writing

ρnm(t) = σnm(t) exp(−iωt), (6)

and considering static spectral-hole burning effects only

(i.e,

∂ρnn(t)∂t

=∂ρmm(t)

∂t=

∂σnm(t)∂t

= 0)

,

62 L.C. LEW YAN VOON, M. WILLATZEN, AND B. LASSEN

we obtain:

0 = − (ρnn(t)− fnn(t)) /τ1c

+ i(ω0

R,nmσ∗nm − ω0∗R,nmσnm

), (7)

0 = − (ρmm(t)− fmm(t)) /τ1v

− i(ω0

R,nmσ∗nm − ω0∗R,nmσnm

), (8)

0 = i(ω − ωnm − 1/τ2)σnm

− iω0R,nm (ρnn − ρmm) , (9)

where we introduced the quantityω0R,nm:

ωR,nm = ω0R,nm exp(−iωt) + ω1

R,nm exp(iωt), (10)

and we have neglected contributions to the DME’s propor-tional to exp(±2iωt). Note thatωR,nm is not in general areal quantity; hence, the relationω1

R,nm 6= ω0∗R,nm does not

either apply in general.Equation (9) immediately yields

σnm =ω0

R,nm (ρnn − ρmm)ω − ωnm + i/τ2

≡ χn′m′k′E∗

0

2.s (11)

In the absence of many-body effects, the following expres-sion is found for the off-diagonal term:

σ0nm =

dnmE∗0 (ρnn − ρmm)

2~ (ω − ωnm + i/τ2)≡ χ

(0)n′m′k′

E∗0

2, (12)

andχ(0)n′m′k′ is the free-carrier susceptibility.

Employing next Eq. (11) in Eq. (7) gives:

ρnn−fnn=− 2τ1cτ2

1+[τ2(ω−ωnm)]2|ω0

R,nm|2(ρnn−ρmm) . (13)

Similarly, it is found using Eq. (8) that

ρmm − fmm

=2τ1vτ2

1 + [τ2(ω − ωnm)]2|ω0

R,nm|2 (ρnn − ρmm) . (14)

Combining Eqs. (13)–(14) yields

ρnn − ρmm =(fnn − fmm)

1 + 2τ2(τ1c+τ1v)1+[τ2(ω−ωnm)]2 |ω0

R,nm|2. (15)

We note that the difference in the diagonal density matrixelements is equal to the difference in the quasi-Fermi-Diracdistributions in the absence of spectral-hole burning.

Next, combining Eqs. (4), (6), and (10) allows us to write

ω0R,nm=

E∗0dnm

2~

(1+

1dnm

∑

n′m′Vn′m′,nm

σn′m′

E∗0/2

)

=E∗

0dnm

2~

(1+

1dnm

∑

n′m′Vn′m′,nmχn′m′k′

), (16)

where the definition ofχn′m′k′ from Eq. (11) has been usedin obtaining the last equality. Following the idea of Haug andKoch [5], we use the simplest non-trivial(0, 1)-Pade approx-imation so as to obtain

ω0R,nm =

E∗0dnm

2~1

1− q1nm, (17)

q1nm =1

dnm

∑

n′m′k′Vn′m′,nmχ

(0)n′m′k′ , (18)

whereq1nm the Coulomb enhancement factor, We are now ina position to obtain an expresssion for the susceptibility. ThepolarizationP (t) is determined

P (t) =1V

∑nm

[ρnmd∗nm + c.c.]

=1V

∑nm

[σnmd∗nm exp(−iωt) + c.c.], (19)

and the susceptibilityχ is defined as:

P (t) = ε0χE(~r, t) = ε0χE∗

0

2exp(−iωt) + c.c., (20)

Finally, the many-body gain (MBG) obtained by solvingthe semiconductor Bloch equations within the Hartree–Fockapproximation and with phenomenological scattering termsis given by [5]

g = ω

õ0

ε0n2

1Lw

∑nm

∫d2k

(2π)2|dnm|2 (21)

× (1−Re(q1nm)) ~/τ2 + ~(ω − ωnm)Im(q1nm)(~[ω − ωnm])2 + (~/τ2)

2

× (fnn − fmm)

1 + 2τ2(τ1c+τ1v)(1+[τ2(ω−ωnm)]2) |ω0

R,nm|2,

whereLw is the well width, andωnm are the renormalizedinterband energies.

The SHB coefficientεSHB is defined by the relation:

g = g0 (1− εSHBS) , (22)

S =ε0nng

2~ω|E0|2, (23)

whereS is the photon density,E0 is the electric field ampli-tude, andg0 is the linear gain whenS = 0. Note that

|ω0R,nm|2 =

ω|dnm|22~ε0nng

11− 2Re(q1nm)

S,

with n, ng being the refractive index and group refractive in-dex, respectively, and introducing

αnm =τ2 (τ1c + τ1v)

(1 + [τ2(ω − ωnm)]2)ω|dnm|2~ε0nng

, (24)

Rev. Mex. Fıs. S53 (7) (2007) 61–65

MANY-BODY EFFECTS IN SPECTRAL-HOLE BURNING FOR QUANTUM-WELL LASERS 63

the SHB coefficient may now be expressed as:

εSHB =1g0

ω

õ0

ε0n2

1Lw

∑nm

∫d2k

(2π)2αnm|dnm|2

× (1− Re(q1nm)) ~/τ2 + ~(ω − ωnm)Im(q1nm)(~[ω − ωnm])2 + (~/τ2)

2

× fnn − fmm

1− 2Re(q1nm). (25)

Such an explicit expression does not appear to have been pre-sented before.

The band structure was obtained using a one-band non-parabolic model (i.e., electron, light-holes and heavy-holesstates are uncoupled). Nonparabolicity is given by the fol-lowing equations:

m∗(E)m∗ =

Eg

Eg + E, E =

~2k2

2m∗(E), (26)

E =

[−1 +

√1 +

4Eg

~2k2

2m∗

]Eg

2, (27)

wherem∗ is the effective mass andEg the band gap. Theoptical matrix elements used are wave-vector dependent:

d2cv(ν,k) =

{d2bulk(k)(1− |e · k|2) c− hh

d2bulk(k)( 1

3 + |e · k|2) c− lh, (28)

dmn(k) = dmn(0)Emn(0)Emn(k)

, (29)

wheree is the light polarization andhh (lh) stands for heavy(light) hole.

3. Results

Prior to numerical results, the analytic form of Eq. (25) al-lows a number of results:

FIGURE 1. Coulomb enhancement factor for various quantum-wellwidths andn = 3× 1018m−3.

FIGURE 2. Gain for a 100A GaAs/AlGaAs QW laser withn = 3× 1018m−3 andT = 314 K.

• With only one subband and withoutk-space disper-sion,

εSHB =αnm

1− 2Re(q1nm). (30)

This predictsεSHB > 0 if Re(q1nm) < 0.5.

• It also shows that for increasingRe(q1nm) < 0.5, εSHB

increases.

• εSHB is well-behaved at the transparency point despitebeing divided by the linear gain as is seen in Eq. (25).

• The single-particle expression forεSHB is then identi-cal toαnm.

In order to obtain a quantitative measure of the influ-ence of the Coulomb enhancement factor, we have car-ried out sample calculations of the net modal gain in sin-gle quantum-well laser structures. We present results for a100A GaAs/AlGaAs QW laser withn = 3 × 1018m−3. Wesetτ1c = τ1v = τ2 = 50 fs.

We first examine the Coulomb enhancement factor itself.The dependence of the real part ofq1nm on the well width isshown in Fig. 1; we note that the imaginary part ofq1nm isalmost negligible and is not given. In the absence of many-body effects,Re(q1nm) = 0.

The Fermi wave number is less than0.05 A−1

for thechosen carrier density, hence explaining the rapid decay inq1nm. For small wave vector, it is seen that the Coulomb en-hancement factor is strongly dependent upon the well width.We also note that, sinceRe(q1nm) < 0.5, then, according tothe simple model, the SHB coefficient would be positive andlarger than using a single-particle model. We will find thelatter statement not to be valid in our numerical calculations(see below).

Many-body gain (MBG) and free-carrier gain (FCG) areshown in Fig. 2. For the FCG calculation, we have assumedthat only one of the subbands is occupied, while for the MBGcalculation, we have also included the more realistic caseof distributing the carriers over the available subbands usingquasi-Fermi-Dirac distributions.

Rev. Mex. Fıs. S53 (7) (2007) 61–65

64 L.C. LEW YAN VOON, M. WILLATZEN, AND B. LASSEN

FIGURE 3. Spectral-hole burning coefficient for a 100AGaAs/AlGaAs QW laser withn = 3× 1018m−3 andT = 314 K.The bottom figure is using the simple model ofεSHB using Eq. (30).

We recover the well-known result that the MBG is en-hanced over the FCG due to the electron–hole interaction andthe redshift in the spectrum is due to band-gap renormaliza-tion [6]. This is illustrated in Fig. 2 for the one-subband cal-culation but is generally true. However, a multisubband cal-culation gives a smaller gain than assuming only one subbandpopulated at the same total carrier population. The reason isbecause a multisubband population spreads the gain distribu-tion and, consequently, reduces the peak gain. Even then, weget a peak gain of the order of200 cm−1.

Spectral-hole burning coefficients with (MBG) and with-out (FCG) many-body effects are shown in Figs. 3–4 fortwo different temperatures. Spectral-hole burning changesgain by 1 − 10 cm−1 maximum for a photon density ofS = 1021 m−3. Here, a multisubband calculation was notfound to necessarily reduce the SHB. Also, Coulomb en-hancement (i.e., q1nm) leads to a slight decrease in the value

FIGURE 4. Same as for Fig. 3 but withT = 4.2K.

of εSHB (Fig. 3) above the band gap, the downshift in energybeing due to the band-gap renormalization. Furthermore, thesimple model of Eq. (30) is seen to be adequate for describingthe dispersion of the SHB coefficient though it overestimatesthe effect. The latter consequence is consistent with the ne-glect of wave-vector dependence.

4. Summary

We have obtained an expression for the spectral-hole burningcoefficient incorporating many-body effects. The discrete-level model is found to be approximately valid. Explicit cal-culations show that Coulomb enhancement increases the gainand decreases the spectral-hole burning coefficient.

Acknowledgments

This work was supported by a Research Challenge grantthrough Wright State University.

Rev. Mex. Fıs. S53 (7) (2007) 61–65

MANY-BODY EFFECTS IN SPECTRAL-HOLE BURNING FOR QUANTUM-WELL LASERS 65

1. T. Takahashi and Y. Arakawa,IEEE J. Quantum Electron.27(1991) 1824.

2. M. Willatzen, A. Uskov, J. Mørk, H. Olesen, B. Tromborg, andA.-P. Jauho,IEEE Trans. Photon. Lett.3 (1991) 606 .

3. S. Schuster and H. Haug,J. Opt. Soc. Am. B13 (1996) 1605.

4. S. Balle,Optics Lett.27 (2002) 1923.

5. H. Haug and S. Koch,Quantum Theory of the Optical and Elec-tronic Properties of Semiconductors(World Scientific, Singa-pore, 2003).

6. C. Ell and H. Haug,Phys. Stat. Solidi B159(1990) 117.

Rev. Mex. Fıs. S53 (7) (2007) 61–65