Polarization-dependent quantum beats of four-wave mixing in the Luttinger model for bulk semiconductors

Polarization-dependent quantum beats of four-wave mixing in the Luttinger model

for bulk semiconductors

Wenfeng Wang,1,2Klaas Allaart,1,*and Daan Lenstra3

1_{Faculty of Science, VU University Amsterdam, De Boelelaan 1081, 1081HV Amsterdam, The Netherlands} 2_{Faculty of Physics and Electronics, Hubei University, Wuhan, China}

3_{Faculty of EEMCS, Delft University of Technology, Delft, The Netherlands} 共Received 22 February 2007; published 17 July 2007兲

We examine the description of quantum beats in four-wave mixing with bulk semiconductors within the framework of the Luttinger-Kohn model. An analytic expression for their dependence on the relative linear polarization of pump and probe is derived, taking only the band structure and coherent interaction of the light waves with the semiconductor medium into account. Herewith all features seen in experiments are very well reproduced, e.g., the vanishing of the beats for an angle␪0⬇76° between the polarizations of pump and probe. Therefore, as opposed to general belief based on earlier theoretical work, no ad hoc exciton-exciton Coulomb interaction has to be invoked to describe the observed phase and magnitude of the quantum beats for copolar-ized or for cross-polarcopolar-ized test and pump pulses.

DOI:10.1103/PhysRevB.76.035206 PACS number共s兲: 78.66.Fd, 42.50.Md, 42.65.⫺k

I. INTRODUCTION

Polarization-dependent four-wave mixing 共FWM兲 quan-tum beats1–11_{have been observed after simultaneous}

excita-tions of two optical transiexcita-tions, associated with heavy hole and light hole. The signal magnitude and its beat phase were found to depend on the relative linear polarization of the pump and test pulses. This phenomenon has been analyzed by applying semiconductor Bloch equations共SBE兲 for exci-tations in a six-band model with broad spectrum pump-probe pulses.1,2_{These theoretical studies did not reproduce}

essen-tial details of the observations. For instance, they predicted identical FWM intensities for the two cases: pump and probe having either parallel or perpendicular linear polarizations. Subsequently, a successful explanation was claimed to be given by the biexciton theory,3,5–11 _{invoking a}

phenomeno-logical exciton-exciton coupling attributed to Coulombic in-teractions. However, it has remained obscure why the SBE would fail for the phenomenon in a semiconductor.

In the present work, we shall completely neglect the Cou-lomb interaction between the charge carriers, which means that no exciton-exciton Coulomb interaction is introduced. We just solve the Heisenberg equations of motion for the dipole to third order in the light-matter interaction only. This is done within the framework of the Luttinger model in order to correctly account for the structure of the band wave func-tions. An analytic expression for the polarization dependence of the quantum beats is then derived and shown to represent surprisingly well all features seen in experiments. We shall therefore conclude that FWM quantum beats can be ex-plained as a purely coherent light-matter interaction effect, without invoking a phenomenological exciton-exciton inter-action.

In the Luttinger model for III-V semiconductors, one has, ignoring split-off bands, for the lattice periodic functions of the valence bands the degenerate heavy-hole states 兩h1kជ典,兩h2kជ典 with energy Eh,k and the light-hole states 兩l1kជ典,兩l2kជ典 with energy El,k. These are superpositions, de-pending on the Bloch vector kជ, of the p-like functions12

兩3/2,3/2典 = −

冑

1/2共兩X↑典 + i兩Y↑典兲, 兩3/2,1/2典 = −

冑

1/6共兩X↓典 + i兩Y↓典兲 +

冑

2/3兩Z↑典, 兩3/2,− 1/2典 =

冑

1/6共兩X↑典 − i兩Y↑典兲 +

冑

2/3兩Z↓典,

兩3/2,− 3/2典 =

冑

1/2共兩X↓典 − i兩Y↓典兲, 共1兲 and can be written in a compact matrix multiplication form as13

冢

兩h1kជ典 兩h2kជ典 兩l1kជ典 兩l2kជ典

冣

=

_冑

1 N

冢

− b R 0 − c* − c 0 R b* R b* c* 0 0 c − b R

冣

冢

兩3/2,3/2典 兩3/2,1/2典 兩3/2,− 1/2典 兩3/2,− 3/2典

冣

, 共2兲 with eigenenergies Ei, i = h , l, R = Hh− Eh= El− Hl, and N = R2+兩c兩2+兩b兩2. Here, we use the conventional notations12

b =

冑

3¯␥ប2共m₀兲−1共kx− iky兲kz, c =

冑

3␥¯ប2_共2m

0兲−1关共kx2− ky2兲 − 2ikxky兴, Hh= −ប2共2m0兲−1关␥1k2−␥2共2kz2− k⬜2兲兴,

Hl= −ប2共2m0兲−1关␥1k2+␥2共2kz2− k⬜2兲,兴, 共3兲

with␥1,␥2, and␥¯ the empirical parameters of the Luttinger

model. The dependence of these wave functions 共2兲 on the

Bloch vector kជmakes the situation in a semiconductor essen-tially different from that in atomic systems. The states of the conduction band兩c1kជ典,兩c2kជ典 are, for small momentum kជ, ap-proximated by the two spin-degenerate s-wave-like functions 兩c1kជ典 = 兩S↑典, 兩c2kជ典 = 兩S↓典. 共4兲 PHYSICAL REVIEW B 76, 035206共2007兲

II. MODEL

The light-matter interaction is described in dipole ap-proximation as −eEជ共t兲·rជ. We consider linearly polarized laser light in the x-y plane. The x component of the dipole opera-tor that couples to x-polarized light is expressed as a super-position of particle-hole operators for the band states with kជ dependent coefficients. Explicitly, omitting for a moment the label kជ on the operators 共also, the coefficients are kជ depen-dent兲 and in units of M =具S兩x兩X典,

x =

兺

kជ

关ac1

†

ah1共− bu兲 + a_c1†ah2共wR − cu兲 + a_c1†al1共uR + wc*兲 + a_c1†al2共− wb兲 + a_c2† ah1共uc*− wR兲 + a_c2†ah2共− ub*兲 + a_c2†al1共− wb*兲 − a_c2†al2共wc + uR兲兴 + H.c., 共5兲 y =

兺

kជ

i关ac1† ah1共− bu兲 − ac1†ah2共wRh+ cu兲 + ac1† al1共uRl− wc*兲 + a_c1†al2共wb兲 − a_c2† ah1共wRh+ uc*兲 + a_c2† ah2共ub*兲

+ a_c2†al1共− wb*_{兲 + ac2}†_{al2共− wc + uRl兲兴 + H.c., 共6兲}

with u = −

冑

1 / 2 and w =

冑

1 / 6.

In four-wave mixing experiments, the incident light fields are a pump fieldEជpand a much weaker probe共testing兲 field Eជt. We assume that the electric field strength is composed of a共strong兲 pulse Eជp共rជ, t兲, during a short time␦ around t = 0, and a test pulseEជt共rជ, t兲, preceding it, during a short time span

␦around t = −␶:

Eជ共rជ,t兲 = Eˆp关E˜p共t兲exp共iqជp· rជ−␻t兲 + c.c.兴

+Eˆt关E˜t共t兲exp共iqជt· rជ−␻t兲 + c.c.兴. 共7兲 The envelop functions E˜p and E˜t are smoothly and slowly varying in time and satisfy

兩E˜p共t兲兩 ⬇ 0 for 兩t兩 ⬎␦, 兩E˜t共t兲兩 ⬇ 0 for 兩t +␶兩 ⬎␦. 共8兲 The pulse lengths must be long enough to justify the distinc-tion between resonant and nonresonant terms, therefore, sev-eral times the oscillation period of the fields:

2␦Ⰷ2␲

␻ ⬅ TF. 共9兲

On the other hand, they should be short enough for the delay time between the two pulses to be well defined on the time scale of the beat time:

2␦Ⰶ 2␲

␻h−␻l

⬅ TB. 共10兲

Otherwise, the whole beat phenomenon will be washed out by the average over a too broad range of delay times. In the experiments,2_{one observed T}

B⬇1 ps, while TFis a few fem-toseconds. So, pulses of roughly 100 fs are quite suitable.

We consider the case that the pump pulseEជp is linearly polarized in x direction, while the linear polarization of the test pulse makes an angle ␪0 with that of the pump pulse:

Eˆt= xˆ cos␪0+ yˆ sin␪0.

The detected intensity of the FWM signal is2

IFWM⬀

冕

dt兩PជFWM共t兲兩2. 共11兲

Here,PជFWMis the component of the third order polarization of second order inEជp and first order inEជt:

PជFWM⬀ exp关i共2qជp− qជt兲 · rជ兴. 共12兲 The Hamiltonian is the independent particle part,

H0=

兺

i,kជ

共Ecikជa_cikជ† acikជ+ Ehikជahikជ

†

ahikជ+ Elikជalikជ

†

alikជ兲, 共13兲 plus the interaction of the carriers with the light field,

HI= − e关Ex共rជ,t兲 · x + Ey共rជ,t兲 · y兴, 共14兲 with x and y the operators of Eqs.共5兲 and 共6兲. The

polariza-tion is obtained by solving the Heisenberg equapolariza-tions of mo-tion,

d dtPជ =

1

iប关Pជ,H0+ HI兴−, 共15兲 to third order in the light-matter interaction. For convenience of the notation, we introduce

␻hk=共Ecikជ− Ehikជ兲/ប and␻lk=共Ecikជ− Elikជ兲/ប, 共16兲 which are independent of the band indices i due to the two-fold degeneracy of the bands and independent of the direc-tion of kជ due to the assumed intrinsic isotropy of the bulk semiconductor.

III. ANALYSIS

To illustrate some key features of the calculation, we first consider the solution of a particle-hole operator at a time t

⬙

, after the passage of the first 共test兲 pulse Eជt only. The inte-grated equation of motion gives, for band indices r = 1, 2 , q = 1 , 2,

a_crkជ† 共t

⬙

兲ahqkជ共t

⬙

兲 = e iប

冕

−⬁

t⬙

dt

⬘

关E˜t共t

⬘

兲ei共qជt·rជ−␻t⬘兲₊E˜

t

*_共t

_⬘

_兲e−i共qជt·rជ−␻t⬘兲兴ei␻hk共t⬙−t⬘兲

兺

s=1

2

关具hskជ兩Eˆt· rជ兩crkជ典a_hskជ† 共− ⬁兲ahqkជ共− ⬁兲

−具hqkជ兩Eˆt· rជ兩cskជ典a_crkជ† 共− ⬁兲acskជ共− ⬁兲兴. 共17兲

WANG, ALLAART, AND LENSTRA PHYSICAL REVIEW B 76, 035206共2007兲

As the pulse is supposed to be sufficiently long, Eq.共9兲, the

nonresonant term with rapidly oscillating integrand factor exp兵i共␻hk+␻兲t

⬘

其 is discarded. Without initial correlations, the statistical expectation values of the operators are

具ahskជ † 共− ⬁兲ahqkជ共− ⬁兲典 =␦sqnhkជ⬇ 1, 具acrkជ † _{共− ⬁兲acskជ共− ⬁兲典 =␦} rsnckជ⬇ 0. 共18兲 We therefore obtain a_crkជ† 共t

⬙

兲ahqkជ共t

⬙

兲 ⬇ e iបe

−iqជt·rជ具hqkជ兩Eˆt_{· r}ជ兩crkជ典ei␻hkt⬙

⫻

冕

−⬁

t⬙

dt

⬘

E˜t쐓共t

⬘

兲ei共␻−␻hk兲t⬘. 共19兲 Because of conditions共8兲 and 共10兲, we may also apply the

approximation

冕

dt

⬘

E˜t쐓共t

⬘

兲e i共␻−␻_hk兲t⬘_{⬇ e}−i共␻lk−␻hk兲␶

冕

_dt

⬘

E˜ t 쐓_共t

_⬘

_兲ei共␻−␻_lk兲t⬘ .

In the same fashion, the action of the pump pulse to second order inEជpis treated as a twofold integral over times t

⬙

and t

⵮

centered around t = 0. We further introduce a relaxation or dephasing constant ␭ for the polarization, which simulates the effect of Coulomb collisions. Treating all particle-hole operators in Eqs.共5兲 and 共6兲 in this way, we obtain, for the

FWM component of the polarizationPជ= xˆx + yˆy, PជFWM共t兲 ⬇ e 3 iប3e i共2qជ_p−qជt兲·rជ

兺

兩k兩 ei共␻hk−␻兲␶

冕

t₀ t dt

⵮

E˜p共t

⵮

兲 ⫻

冕

t₀ t⵮ dt

⬙

E˜p共t

⬙

兲

冕

−⬁ t⬙ dt

⬘

E˜t쐓共t

⬘

兲e−␭共t+␶兲 ⫻兵e−i␻hkt关Aជ_{+ B}ជ_ei共␻hk−␻lk兲␶兴

+ e−i␻lkt关Bជ_{+ C}ជ_ei共␻hk−␻lk兲␶兴其, 共20兲 with Aជ= 2

冕

d⍀kជ

兺

ijrq 具hi,kជ兩rជ兩cj,kជ典具cj,kជ兩x兩hr,kជ典具hr,kជ兩Eˆt· rជ兩cq,kជ典 ⫻具cq,kជ兩x兩hi,kជ典, Bជ= 2

冕

d⍀kជ

兺

ijrq 具hi,kជ兩rជ兩cj,kជ典具cj,kជ兩x兩lr,kជ典具lr,kជ兩Eˆt· rជ兩cq,kជ典 ⫻具cq,kជ兩x兩hi,kជ典, Cជ= 2

冕

d⍀kជ

兺

ijrq 具li,kជ兩rជ兩cj,kជ典具cj,kជ兩x兩lr,kជ典具lr,kជ兩Eជt· rជ兩cq,kជ典 ⫻具cq,kជ兩x兩li,kជ典.

Here, we split the summation over kជ into a summation over the modulus 兩k兩 and integration over its angles. The latter determines the dependence of the FWM intensity on the

angle between the polarizations of the pulses represented by the factor Eˆt· rជ. The matrix elements of the dipole compo-nents are the coefficients in Eqs. 共5兲 and 共6兲 and the angle

integration can easily be done analytically. The detected in-tensities of the FWM signal then become

I储= Fe−2␭␶ 1 2␭

兺

_兩k兩

再

208 + 192 cos共␻hk−␻lk兲␶ + 4␭ 2 4␭2_+共␻ hk−␻lk兲2 关192 + 208 cos共␻hk−␻lk兲␶兴 + 2␭共␻hk−␻lk兲 4␭2_+共␻ hk−␻lk兲2 40 sin共␻hk−␻lk兲␶

冎

, 共21兲 I_⬜= Fe−2␭␶1 2␭

兺

_兩k兩

再

37 − 12 cos共␻hk−␻lk兲␶ + 4␭ 2 4␭2_+共␻ hk−␻lk兲2 关− 12 + 37 cos共␻hk−␻lk兲␶兴 − 2␭共␻hk−␻lk兲 4␭2_+共␻ hk−␻lk兲2 17.5 sin共␻hk−␻lk兲␶

冎

, 共22兲 in which the factor F contains the modulus squared of the prefactor in Eq. 共20兲 with integrals over the envelop

func-tions of the pulses. The summation共integration兲 extends over the range of兩k兩 values involved in the excitations that are in experiments identified as heavy-hole and light-hole excitons. As we do not describe excitons and their widths explicitly, we adopt a summation over beat frequencies ␻hk−␻lk in a small range of兩k兩 values. The value of the relaxation param-eter ␭ can also be clearly read off from the experimental decay of the FWM signal as a function of the delay time␶. With this, one verifies that the first two terms in expressions 共21兲 and 共22兲 dominate. For an angle␪₀between both polar-izations, we therefore find the approximation

I共␪0兲 ⬀ e−2␭␶

兺

兩k兩

兵关208 + 192 cos共␻hk−␻lk兲␶兴cos2␪0

+关37 − 12 cos共␻hk−␻lk兲␶兴sin2␪0其. 共23兲

This is our main result. Due to the summation of兩k兩 in a small region, the beat frequency will not be completely sharp, leading to some smoothing as a function of the delay time␶, resulting in Fig.1. The figure and Eq.共23兲 show three

features that are also observed in experiment.2 _{Firstly, the}

FWM signal is stronger in the case of parallel polarization than in case of cross polarization by roughly a factor of 5, in good agreement with the experiment of Bennhardt et al.2and in contrast to earlier theoretical expressions,1 _which

pre-dicted equal strength in both cases. Secondly, we do indeed find a beat behavior as a function of the delay time which has a maximum at zero delay time for the case of parallel polar-ization and an opposite oscillating behavior for orthogonal polarization. Thirdly, we find that the beats are more pro-nounced for parallel polarization than for orthogonal polar-ization. The beats vanish for tan2_共␪

0兲=16, that is, for ␪0

⬇76°. In view of the various approximations made, this is in remarkable agreement with the observations in Ref. 2 All POLARIZATION-DEPENDENT QUANTUM BEATS OF FOUR-… PHYSICAL REVIEW B 76, 035206共2007兲

these features seen in experiments are reflected in Eq.共23兲

without invoking other mechanisms than just the coupling of the carrier共polarization兲 dynamics with the light fields. This is in contrast to, for instance, Ref. 2, where the data were interpreted by introducing a coupling parameter ascribed to disorder.

One may remark that the analytic derivation given here involves the Luttinger-Kohn model wave functions 共2兲 for

isotropic semiconductor, whereas experiments were done with quantum well material. In the latter, the wave functions may be modified by strain13_{and by a confining potential. The}

similarity between our analytic result关Eq. 共23兲兴 and the

ex-perimental findings indicates, however, that the essentials of the kជ· pជ Luttinger-Kohn model structure of heavy-hole and light-hole wave functions still play an important role in quantum wells with a thickness of a few tens of nanometers.

IV. CONCLUSION

We have examined the description of quantum beats in four-wave mixing共FWM兲 with bulk semiconductors within the framework of the Luttinger-Kohn model. An analytic ex-pression for their dependence on the relative linear polariza-tion of pump and probe is derived, taking only the band structure and coherent interaction of the light waves with the semiconductor medium into account. Herewith, all features seen in experiments are very well reproduced. Therefore, as opposed to general belief based on earlier theoretical work, no ad hoc exciton-exciton Coulomb interaction has to be invoked to describe the observed phase and magnitude of the FWM quantum beats.

ACKNOWLEDGMENT

The authors acknowledge financial support by the Free-band Communication Impulse of the technology programme of the Netherlands’ Ministry of Economic Affairs.

*allaart@nat.vu.nl

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FIG. 1. Dependence of quantum beats in four wave-mixing 共FWM兲 on the relative 共linear兲 polarization angle␪0between pump and probe pulses, within the Luttinger-Kohn model. Plotted is the integrated FWM signal as a function of delay time␶ between probe and pump, assuming a beat period of 1 ps. The共upper兲 dash-dotted line for is for parallel polarization,␪0= 0, and the共lowest兲 dashed line is for orthogonal polarization,␪₀=␲/2. The beats vanish for ␪0= 76° 共dotted line兲; the full lines are for intermediate angles 0.32␲ 共upper兲 and 0.45␲ 共lower兲.

WANG, ALLAART, AND LENSTRA PHYSICAL REVIEW B 76, 035206共2007兲