phys. stat. sol. (c) 4, No. 7, 2752–2755 (2007) / DOI 10.1002/pssc.200674703

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Time-resolved four-wave mixing studies of excitons in GaN

K. Yamaguchi1, Y. Toda∗1,2, T. Ishiguro1, S. Adachi1, K. Hoshino3, and K. Tadatomo3

1 Department of Applied Physics, Hokkaido University, N13W8 Kita-ku, Sapporo, Japan2 CRIS, Hokkaido University, N21W10 Kita-ku, Sapporo, Japan3 Faculty of Engineering, Yamaguchi University, Tokiwadai, Ube, Yamaguchi, Japan

Received 5 September 2006, revised 10 December 2006, accepted 12 December 2006Published online 31 May 2007

PACS 42.50.Md, 71.35.Cc, 71.70.Fk, 78.47.+p

Quantum beats in the time-resolved four-wave mixing (FWM) signals arising from GaN excitons are ob-served. Two different types of beats are investigated: beats between the A-exciton (XA) and B-exciton(XB) states; beats between the XA and A-biexciton ((XXA)) states. The former exhibits oscillations alongall directions in the delay- (τ ) and real-time (t) plane while the latter exhibits no beat in the direction par-allel to the diagonal axis (τ = t). The results are well reproduced by a model calculation, showing goodhomogeneity of the sample.

In the past decade coherently excited quantum beats (QBs) in semiconductor have been studied exten-sively. One of the most powerful experimental tools for investigating QB is the four-wave mixing (FWM)with short laser pulses. What makes excitonic QB unique compared to those of atomic systems is their ef-fective features of many-body interactions via Coulomb forces [1]. To understand the role of interactions inthe beats, various types of semiconductor systems containing semiconductor quantum nanostructures havebeen investigated so far [2–7]. Among the systems, wurtzite GaN provides a successful prototype becauseof its well-defined exciton transitions with a large binding energy. The polarization-dependent phase shiftsof the beats in time-integrated FWM (TIFWM) have been observed, which can be fully understood by aweak-boson model based on the spin-dependent interactions between two excitons [7]. The QB betweentwo-exciton and biexciton states has also been reported [8, 9]. However, little is known about the “realtime” behavior because of the difficulty of time-resolved measurements in the ultra-violet regions: the lackof appropriate nonlinear crystals for upconverting the signal, and the lack of commercially available lasersources for stabilizing the interferometer for correlation measurements.

In this work, we perform a time-resolved FWM (TRFWM) experiment for GaN excitons by utilizing amodified first-order correlation technique, in which the correlation amplitudes are obtained at a time, al-lowing us to evaluate the TRFWM without stabilizing the interferometer. We discuss the results associatedwith different two-types of QBs: one is QBs between two exciton states associated with different valencebands, and another is QB between exciton and biexciton states. The results are consistent with the modelcalculation and show good homogeneity of the sample.

The two-pulse FWM experiments in reflection geometry were carried out on a 3-µm-thick GaN filmwhich was grown on (0001)c-planes of sapphire by using a two-flow metal-organic chemical-vapor-deposition(MOCVD) with 40-50 nm thick buffer layers. The sample was mounted on a closed-cycle helium cryostatkept at 10 K. A frequency doubled, mode-locked Ti:sapphire laser with the spectral width of 13 meV wasused as a tunable excitation source. The total excitation power was kept below 1 mW in order to minimizethe fifth-order contributions. The experimental setup is schematically illustrated in Fig. 1(a). The twopulses with wave vectors k1 and k2 were superimposed onto a sample surface using a lens (f = 200 mm).

∗ Corresponding author: e-mail: toda@eng.hokudai.ac.jp, Phone: +81 11 706 6627, Fax: +81 11 706 6627

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

3.48 3.49 3.50 3.51 3.52

FW

M in

ten

sity (

arb

. u

nits)

XA

XB

XXA

10KCo-linear

Energy (eV)

3.48 3.49 3.50 3.51 3.52

Energy (eV)

(b)

Delay:t

CCD

2k2 k1

k2

k1

Gate pulse

Delay:τd

Lens

(a) Fig. 1 (a) A sketch of experimen-tal setup for time-resolved FWM ex-periment. (b) Typical TIFWM spec-tra at τ=0 ps at 10 K. The dashedline in each figure represents spec-trum of the incident laser pulse. Thespectra of XA and XB are observedclearly at 3.4949 eV and 3.5025 eV,respectively (upper). A biexcitonicresonance due to XA (XXA) appearsat 3.4902 eV for colinear polariza-tion excitation (lower) while this res-onance is absent when cocircular po-larized excitation is used (inset).

The interplay of polarizations induced by each pulse gives rise to a diffracted signal in a direction 2k2-k1

if the delay time (τ ) is smaller than the decoherence time T2 of the polarization.Figure 1(b) shows spectrally-resolved TIFWM spectra for colinearly polarized excitation at τ=0. In

the upper spectrum, the lower- and higher-energy peaks can be identified as excitonic transitions betweenthe bottom of the conduction band and the two topmost valence bands (hh and lh of zinc-blend semicon-ductor), namely A- (XA) and B-exciton (XB) transitions, respectively. As shown in the figure, energypositions of XA and XB lines are well separated within the excitation laser linewidth. The energy separa-tion (∆ωAB) is estimated to be 7.7 meV from the Lorentzian fitting to the data. On the other hand, whentuning the laser energy to near the center of XA, the peak of biexciton transitions (XXA) is clearly observedas shown in the lower spectrum. In the inset, no XXA resonance is observed for cocircular polarized ex-citation due to the selection rule for the biexciton transition, confirming the identification of XXA. Thebiexciton binding energy (δbx) is estimated to be 4.7 meV, which is comparable to the previously reportedvalues [8, 9].

For measuring the time-evolution of the diffracted signal, we employed a modified first-order correlationtechnique in which a third reference pulse with the delay time t was superimposed on the signal (Fig. 1(a)).In order to avoid the difficulty of precisely tuning t with high stability, we detect interferograms obtainedby slightly deforming the coaxial configuration between the gate pulse and the diffracted signal. In thisconfiguration, the quasi-continuous correlation signals as a function of finely-tuned t are obtained at atime. For coarse delay, the gate pulse was mechanically delayed. The interferograms are recorded using a

Fig. 2 Some examples of inter-ferogram at various t (upper), andtheir corresponding 2DFFT spectra(lower). The fringes are characterizedby the first-order correlation func-tion of the FWM signal field gatedby the reference pulse. In 2DFFT,the frequency distribution of the spec-tral peak somewhat reflects the distri-bution of the polarization frequency,thus enabling the spectrally-resolvedanalysis also.

phys. stat. sol. (c) 4, No. 7 (2007) 2753

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Fig. 3 Contour plots of TRFWM for different polarizations and energies of the incident pulses (upper).Solid (dashed) white lines indicate the axis of t=τ (t=2τ ). These results are well reproduced by the simplenumerical calculations within the framework of optical Bloch equations (lower).

conventional CCD camera equipped with an image acquisition system connected to a PC. Some examplesof the CCD images at different t at τ=0 ps are shown in Fig. 2, together with their corresponding two-dimensional fast Fourier transform (2DFFT) spectra. A number of interferometric fringes more than 50cycles are clearly resolved at t ≥0 ps, and the corresponding correlation peak appears in 2DFFT, wherethe peak intensity corresponds to the correlation amplitude, i.e., TRFWM signal.

The experimental results for different polarizations and energies of the incident pulses are shown inthe upper part of Fig. 3 where the TRFWM signals are plotted as a function of τ (vertical axis) and as afunction of t (horizontal axis). Figure 3 (a) shows TRFWM signals under the condition of simultaneouslyexcited XA and XB (see the upper part of Fig. 1(b)). The periodic oscillations appears in each timesequence and their maxima are located at t = τ + nTAB , t = 2τ , and τ = t + nTAB , where TAB is theoscillation period of the beat and n is an integer. The TAB = 0.51 ps is consistent with ∆ωAB . On theother hand, TRFWM for simultaneously excited XA and XXA (see the lower part of Fig. 1(b)) also showsQB but its temporal behavior is quite different as shown in Fig. 3(b). The beat maxima are located only atthe t = τ + (n + 1/2)Tbx, where Tbx = 0.83 ps is consistent with δbx. In addition, no beat appears alongthe axis parallel to t=τ , indicating that TIFWM signal shows no biexcitonic beats. The results reflect thefact that the sample used in this study yields a homogeneous excitonic system since QB is absent in theTIFWM in homogeneous systems, as discussed in detail below. Finally, Fig. 3(c) shows TRFWM signalswith the same excitation condition as Fig. 3(b) but for cocircular polarization excitation (see the inset ofFig. 1(b)), thus showing absence of the beats. Note that the absence of beats in the FWM signal allows usto evaluate the temporal behavior of single exciton state. The result exhibits a decay time of 0.58 ps (TA

2 =1.2 ps) obtained from a single exponential fitting.

In order to understand the temporal behavior of QB arising from various types of excitonic transitions,we derive the expressions within the framework the optical Bloch equations for the δ-function pulses withparallel polarization. We start with the homogeneously broadened two-level (2L) system of one excitonic

2754 K. Yamaguchi et al.: TRFWM of excitions in GaN

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.pss-c.com

transition. The third-order optical polarization P(3)x (τ ≥ 0) is described by

P (3)x (t, τ ≥ 0) ∝ Θ(t − τ)|µx|4e−iΩx(t−τ)e−iΩ∗

xτ with Ωx = ωx − iγx, (1)

where Ω denotes the complex Rabi frequency consisting of exciton energy (ωx) and its relaxation time(γx). The corresponding TRFWM signal is represented by IFWM(t, τ ≥ 0) ∝ |P (3)

x (t, τ ≥ 0)|2. The P(3)α

(τ ≥ 0) for 3L A and B exciton system and for 4L biexciton system are described, respectively, by [10]

P(3)AB(t, τ ≥ 0) ∝ Θ(t − τ)

[2|µA|4e−iΩA(t−τ)e−iΩ∗

Aτ + 2|µB |4e−iΩB(t−τ)e−iΩ∗Bτ

]

+ Θ(t − τ)|µA|2|µB |2[e−iΩA(t−τ)e−iΩ∗

Aτ + e−iΩB(t−τ)e−iΩ∗Bτ

], (2)

and [11]

P(3)bx (t, τ ≥ 0) ∝ P (3)

x (t, τ)[2 − |Mbx|2ei(δbx+iγbx)(t−τ)

], (3)

where Mbx denotes the simplified exciton-biexciton transition matrix. Note that we neglect Coulombinteractions between two excitons in the 3L system for simplicity. A simplified oscillation part of 3LTRFWM signal is represented by 1 + cos(∆ωABt)1 + cos(∆ωAB(t − τ)), thus showing the beatmaxima along each axis of time-sequence (Fig. 3(e)). On the other hand, the exciton-biexciton TRFWMsignal shows an oscillation represented approximately by the function − cos(δbx(t − τ) (see Fig. 3(e)).As a consequence, the signal shows no oscillation along the axis of t=τ as far as the excitonic systemobserved is regarded as a homogeneous broadening. When the system exhibits inhomogeneous broadening,the TRFWM becomes a echo signal located at t = 2τ , giving rise to QB even along the axis of t=τ [3, 8].

In summary, we have performed a time-resolved four-wave mixing (TRFWM) measurement of GaN byutilizing a spatial correlation technique. The signals allow us to explore the dependence of the temporalbehaviors of quantum beats on excitonic transitions. The results show good agreements with the analyticalmodel calculations that reflect a homogeneous broadening system.

Acknowledgements Y.T. thanks to the grant-in-aid for JSPS and for COE21.

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