ARTICLE IN PRESS

0022-2313/$ - se

doi:10.1016/j.jlu

�CorrespondE-mail addr

(M. Nakayama

Journal of Luminescence 122–123 (2007) 841–843

www.elsevier.com/locate/jlumin

Miniband-width effects on Wannier–Stark localization of the firstand second quantized states in a GaAs/AlAs superlattice

Takayuki Hasegawa, Masaaki Nakayama�

Department of Applied Physics, Graduate School of Engineering, Osaka City University, Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japan

Available online 15 March 2006

Abstract

We performed electroreflectance (ER) measurements in order to reveal the transformation processes from the minibands to the

Wannier–Stark (WS) localization states in a GaAs (6.8 nm)/AlAs (0.9 nm) superlattice. The high sensitivity of ER spectroscopy enabled

us to observe the interband optical transitions associated with the second (n ¼ 2) minibands in addition to those associated with the first

(n ¼ 1) minibands. It is found that the electric field strengths for the formation of the WS-localization states of the n ¼ 2minibands are

considerably higher than those of the n ¼ 1minibands. The experimental results are quantitatively analyzed on the basis of the

theoretical calculation of the electric-field-strength dependence of the miniband states using a transfer-matrix method. It is concluded

that the critical electric field strengths for the formation of the WS-localization states clearly correlate with the miniband widths.

r 2006 Elsevier B.V. All rights reserved.

Keywords: Wannier–Stark localization; Superlattice; GaAs/AlAs; Electroreflectance

Under an applied electric field F, the minibands of asuperlattice (SL) are transformed to localization states withthe energy spacing of eFD, where D is a period of the SL. Thisphenomenon is well known as Wannier–Stark (WS) localiza-tion, which has been considerably investigated from aspects ofphysics and device applications [1–4]. The WS localizationleads to Stark-ladder transitions: E0 �meFD, where E0 is aninterband transition energy within an individual quantum well(QW) and m is a Stark-ladder index indicating an obliquetransition in real space. Although a lot of researches have beenreported, little has been known about the WS-localizationbehavior of the miniband with the quantum number of n

larger than 2. This is mainly due to less sensitivity ofphotocurrent spectroscopy that has been conventionally usedfor the detection of optical transitions. The electronic states ofthe SL generally consist of multiple minibands, so that it ismeaningful to investigate the electric-field-strength depen-dence of both the n ¼ 1 and 2minibands in order to revealcomprehensively the properties of the WS localization. This isthe motivation of the present work. Electroreflectance (ER)

e front matter r 2006 Elsevier B.V. All rights reserved.

min.2006.01.304

ing author. Tel./fax: +81 6 6605 2739.

ess: nakayama@a-phys.eng.osaka-cu.ac.jp

).

spectroscopy has been adopted to observe the opticaltransitions of the SL because of its very high sensitivityespecially for the Stark-ladder transitions [5–8]. The systematicresults obtained by ER spectroscopy demonstrate that thecritical electric field strength for the formation of the WS-localization states of the n ¼ 2miniband is considerablyhigher than that of the n ¼ 1miniband. The experimentalresults are theoretically analyzed.A sample of SL was grown on an n-type (0 0 1)-GaAs

substrate by molecular-beam epitaxy. The SL consists of100 periods of undoped GaAs (6.8 nm)/AlAs (0.9 nm) andis placed in the center of a p–i–n diode structure. The n-type and p-type layers are Si- and Be-doped Al0.4Ga0.6Aslayers with the thickness of 1.0 mm. The sample wasprocessed into a mesa structure with a size of1.0� 1.0mm2, and a gold-film electrode having a lightwindow (0.7� 0.7mm2) was formed on the surface.In ER measurements, the probe light was produced by

combination of a halogen lamp and a monochromator witha spectral resolution of 0.5 nm.The applied bias wasmodulated with amplitude of 100mV and a frequency of450Hz around a given DC bias. The modulated reflectancesignal was detected with a conventional lock-in technique.The electric field strength was characterized by F ¼

ARTICLE IN PRESS

n=2

F=2 kV/cm

10 kV/cm

20 kV/cm

30 kV/cm

n=1

20 kV/cm

5 kV/cm

10 kV/cm

F=2 kV/cm

(a)

(b)

Fig. 1. Calculated results of the squared envelope functions of (a) n ¼ 1

and (b) n ¼ 2 electron states at various electric field strengths in the

modeled GaAs (6.4 nm)/AlAs (0.9 nm) SL, where the depicted envelope

functions belonging to the center QW.

T. Hasegawa, M. Nakayama / Journal of Luminescence 122–123 (2007) 841–843842

(Vb�Va)/Li, where Vb is the built-in voltage that is 0.9V inthe present case, Va is the applied bias voltage, and Li is thetotal length of the undoped layers. The value of Vb wasestimated from the applied voltage at which the energies ofthe WS-localization states converge to the miniband center[5]. All the measurements were performed at 10K.

At first, in order to understand intuitively the transfor-mation process from the miniband to the WS-localizationstate, we describe the theoretical calculation of theenvelope functions in the SL at various electric fieldstrengths with use of a transfer-matrix (TM) method. In aTM method, Airy functions have been usually used [7];however, the Airy function cannot be applied to calculatethe eigenstates under almost flat-band conditions becauseof the divergence characteristics. In the present work, themain subject is the transformation process from theminiband to the WS-localization state, so that we adopteda different method that has been proposed in our previousreport [8]. In this method, the SL is approximated by asystem of QWs with a step-like potential including anelectrostatic potential having an average value of eachconstituent layer. The quantum-confined Stark effect [9],which cannot be included in this model, is negligiblebecause the relevant electric field is not so high: Fo60 kV/cm. In the TM calculation, we used the band parametersincluding band nonparabolicity according to Refs. [8,10].The modeled SL consists of 15 sets of GaAs (6.4 nm)/AlAs(0.9 nm) with edge layers of AlAs. The GaAs-layerthickness was slightly changed from the designed value(6.8 nm) in the sample growth for adjusting the calculatedtransition energies to the observed results.

Fig. 1 shows the calculated results of the squared envelopefunctions of (a) n ¼ 1 and (b) n ¼ 2 electron states belongingto the center QW at various values of F, where the envelopefunctions in the QWs at the edges are removed because thefiniteness of the SL modifies their shapes. At F ¼ 2 kV/cm,the envelope functions of the n ¼ 1 and 2 states extendentirely over the SL, which reflects the formation of theminiband. It is evident that the envelope function of then ¼ 1 state tends to be localized around the center QW atF ¼ 5kV/cm. On the other hand, the localization of theenvelope function of the n ¼ 2 state becomes remarkable at20kV/cm that is much higher than F for the localization ofthe n ¼ 1 state. On the basis of the calculated results of theenvelope functions and the transition energies shown inFig. 3, we discuss the experimental results for the formationof the WS-localization states below.

Fig. 2 shows the normalized ER spectra of the GaAs(6.8 nm)/AlAs (0.9 nm) SL at various values of F. Thearrows indicate the calculated energies of the opticaltransitions using the well-established Kronig–Penney(KP) model in the framework of the effective massapproximation, where Hnenh (Lnenh) represents the opticaltransition between the neth electron miniband and the nhthHH (LH) miniband. The symbols G and p indicate thetransitions at the G and p points, respectively, where pmeans the mini-Brillouin edge of p/D. In ER spectra, fine

structures of the optical transitions related to n ¼ 2 statesin addition to n ¼ 1 states are observed. With increasing F,the profiles of the ER spectra change from the minibandfeatures and become complicated. These changes of the ERspectra correspond to the transformation processes fromthe minibands to the WS-localization states.In order to analyze the transformation processes, the

image maps of the ER spectra as a function of F, which areshown in Fig. 3(a) for the n ¼ 1minibands and Fig. 3(b) forthe n ¼ 2minibands, are meaningful. The ER intensity isrepresented by the gray scale shown on the right side, andthe calculated energies of Hnenh (Lnenh) transitionsutilizing the TM method are indicated by solid (dashed)curves. The intensity of the ER-image map in Fig. 3(b) ismagnified by a factor of 20 in order to make it clear. Sincewe focus on the transformation process from the mini-bands to the WS-localization states, the precise estimationof the transition energies is not important. It is evident thatthe transformation processes of the n ¼ 1 and

ARTICLE IN PRESSE

R I

nten

sity

(ar

b. u

nits

)

2.12.01.91.81.71.6

Photon Energy (eV)

× 20H11

L11

H22

L22

( Γ )

( Γ )( Γ )

( π ) ( π )

( Γ ) ( π )

( π )

× 20

F = 0 (kV/cm)

15

30

45

60

× 10

× 5

× 2

Fig. 2. ER spectra of the GaAs (6.8 nm)/AlAs (0.9 nm) SL at various

electric field strengths. The arrows indicate the calculated energies of the

H11, L11, H22 and L22 transitions at zero electric field strength by using

the KP model.6050403020100

Electric Field (kV/cm)

1.68

1.66

1.64

1.62

1.60

1.58

Ene

rgy

(eV

)H11

L11

n=1

2.10

2.05

2.00

1.95

1.90

1.85

1.80

1.75

Ene

rgy

(eV

)

H22

L22

n =2

Peak

Dip

Peak

Dip

(a)

(b)

Fig. 3. Image maps of ER spectra of the optical transitions associated

with (a) n ¼ 1 and (b) 2 as a function of electric field strength, where the

ER intensity is represented by the gray scale. The solid and dashed curves

indicate the energies of the Hnenh and Lnenh transitions calculated by the

TM method, respectively.

T. Hasegawa, M. Nakayama / Journal of Luminescence 122–123 (2007) 841–843 843

n ¼ 2miniband are visualized in Fig. 3, and the calcu-lated results are almost consistent with the experimentalresults.

We discuss the electric field strength at which the Stark-ladder transitions, of which energies exhibit the lineardependence on F, appears: a so-called critical electric fieldstrength (Fc) for the formation of the WS-localizationstates. The calculated and the experimental results shownin Fig. 3 indicate the existence of Fc. It is estimated fromthe experimental results that the values of Fc are about 5, 7,15, and 20 kV/cm for the H11, L11, H22, and L22transitions, respectively. The total miniband widths (sumwidths of the electron and hole minibands) of H11, L11,H22, and L22 transitions estimated from the KP calcula-tion are 32, 55, 90, and 138meV, respectively, where theratios of the miniband widths relative to the value of theH11 transition are 1:1.7:2.8:4.3. For the Fc describedabove, the ratios relative to the value of the H11 transitionare 1:1.4:3.0:4.0 for the H11, L11, H22, and L22transitions, respectively. Thus, we find the clear correlationbetween the total miniband widths and values of Fc. Thisdemonstrates that the critical electric field strengths for theformation of the WS-localization states are generallydominated by the total miniband widths. The minibandwidth is determined by the coupling strength betweenenvelope functions in constituent QWs; therefore, the widerminiband leads to the higher Fc for the WS localization.

References

[1] E.E. Mendez, F. Agullo-Rueda, J.M. Hong, Phys. Rev. Lett. 60

(1988) 2426.

[2] P. Voisin, J. Bleuse, C. Bouche, S. Gaillard, C. Alibert, A. Regreny,

Phys. Rev. Lett. 61 (1988) 1639.

[3] For a review, M. Nakayama, in: T. Ogawa, Y. Kanemitsu, (Eds.),

Optical Properties of Low-Dimensional Materials, World Scientific,

Singapore, 1995, p.197, and references for the WS localization

therein.

[4] H. Schneider, K. Fujiwara, H.T. Grahn, K.V. Klitzing, K. Ploog,

Appl. Phys. Lett. 56 (1990) 605.

[5] M. Nakayama, I. Tanaka, T. Doguchi, H. Nishimura, K. Kawa-

shima, K. Fujiwara, Solid State Commun 77 (1991) 303.

[6] M. Nakayama, I. Tanaka, H. Nishimura, K. Kawashima,

K. Fujiwara, Phys. Rev. B 44 (1991) R5935.

[7] I. Tanaka, M. Nakayama, H. Nishimura, K. Kawashima,

K. Fujiwara, Phys. Rev. B 46 (1992) 7656.

[8] I. Tanaka, M. Nakayama, H. Nishimura, K. Kawashima,

K. Fujiwara, Phys. Rev. B 48 (1993) 2787.

[9] D.A.B. Miller, D.S. Chemla, T.C. Damen, A.C. Gossard, W.

Wiegmann, T.H. Wood, C.A. Burrus, Phys. Rev. B 32 (1985) 1043.

[10] D.F. Nelson, R.C. Miller, C.W. Tu, S.K. Sputz, Phys. Rev. B 36

(1987) 8063.