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263

FARADAY ROTATION

IN THE LOWEST EXCITON RESONANCE OF GALLIUMARSENIDE Th. Hornung and

R.G. Ulbrlch

Instltut fQr Physlk der Unlverslt~t Dortmund 4600 Dortmund 50 Fed. Rep. Germany Thls paper reports Faraday rotatlon (FR) spectra of the exc~ton polarzton resonance in hlgh purity GaAs. The drastlc variation of the group velocity around the lowest exclton n=1 leads to a giant resonance enhancement in the FR spectrum. The experlmental data are explalned quantltatlvely in a simple model wlthln the polarlton framework. It is posslble to measure the product geff'ELT of the Is exciton polarlton resonance with hxgh preclslon. In the low fleld reglon it is an excellent method to measure geff if ELT is kown. I. INTRODUCTION Magneto-optlcal studies over the past 25 years gave detalled inslght into the electronlc structure of semiconductors. Various experimental technlques, all based on cyclotron resonance and lnterband Faraday rotatlon (FR),allowed a falrly dlrect determlnatlon of band parameters characterizlng free carrlers In the conduction and valence band [i]. Detailed studles of excitons in magnetlc flelds by reflectance spectroscopy gave addltzonal lnformatlon on exchange interactlon energles, spln coupllng schemes, compllcated Zeeman patterns and dlamagnetlc shlfts of bound electron-hole palrs [2]. In the exlstlng studles of near band edge interband FR in direct-gap semlconductors the excltonlc contributlons were treated only phenomenologically in the framework of damped Lorentz osclllators [3]. We report here an experlmental magnetoabsorption study of the exclton polariton resonance in hlgh purlty GaAs at low temperatures. We focus on the enormous enhancement of FR in the lowest exclton resonance. We present a quantltatlve descrxptlon of the observed glant FR angle spectra and obtaln from it the parameters ELT , E T (energy splittlng between the longltud~nal and transverse exclton energles, and transverse exclton energy) and the effectlve g-factor geff of the ls exclton in the range of magnetlc flelds B=O.2 ...2.6T. II. EXPERIMENTAL We used vapor-phase-epltaxlally grown hlgh purity (ND=2'IOI~cm-3) GaAs layers on a substrate of degenerately doped materlal. The thlckness was determlned by (i) dlrect measurements and (li) from comparlson wzth Hills published absorptlon coefflclent ( ~ = 8 x 1 0 3 c m -I In the contlnuum dlrectly above the gap energy Eq=1.5192 eV [4]. Both methods lead to a thlckn~ss of 5.7±0.5 um. The sample was mounted strainfree and immersed in liquid He II In a cryos~at wlth a superconductlve spllt coll magnet. The temperature was 1.5K and the magnetlc fleld could be varled up to 6.4 T. The cw absorptlon measurements were performed uslng a tungsten halogen lamp with polarlzer and ~/4 plate in elther linearly polarized, left hand (LCP), or right hand circularly polarxzed (RCP) conflguratlon. The trans-

0 3784363/83/0000-0000/$03 00 © 1983 North-Holland

mission spectra were a n a l y s e d w i t h a lm grating spectrometer, a GaAs photocathode PM and a photon countlng system. The spectral resolutlon was about 0.04 meV in the band gap region. The FR angles were measured wlth a rotating analyzer using a lock-ln amplifler and a phase-lock loop. The measurement accuracy was ±0.5 degrees. In all experlments the wave-vector of the incident llght beam was antlparallel to the magnetlc fleld and parallel to the

~S

.,2 n=3

n=l

cCx) A

~"

15%

I~IIS

T=15K

16

PHOTON ENERSY1~ [WI Fig. 1

Absorptlon spectra of the 5.7 um thick GaAs sample in the reglon of the lowest exclton resonances n=l,2... E T deDotes the lowest transverse exclton energy, Eg the band-gap energy - onset of the cont£nuum states - and (DO,x),(DO,X) ~ the donor bound exciton and its electronic exclted state, respectlvely.

Figure i shows the absorption spectrum of the 5.7 um thlck GaAs sample in the reglon of the lowest excxton resonances and the onset of the contlnuum at E g . The intrlnslc excltonxc Rydberg serles n=I,2,3 is well resolved; the extrinsic structure at 1.51402 eV is the (D°,X) donor bound exciton together with its rotatlonal excited states [5]. The structure around 1.5172 eV belongs to the electronic excited (D°,X)e[5] state. The spectrum has a noise limited dynamic range of eS9xlO3cm -! due to low intensity excitatlon and fluorescence background. The energy position of

264

Tit Hornung, R G Ulbrich / Faraday rotatton in the.lowest exctton resonance of GalhumArsemde

the lowest transverse exczton energy ET=I.5151 eV is indicated in Flg. l. Fig.2 shows the absorption spectra of the same GaAs sample at a magnetlc fleld of 3.4 T in LCP and RCP conflguratlon, antiparallel to the magnetlc fleld.

GoAs

' ~ ~ l ~

clton (low field case) we obtaln In llnear approxlmatlon that (n--n +) is equal to A E _ . . a n / a ~ . Thls leads in lowest order of a~/ak = Vg r wlthln the polariton framework to i c t ~B X = ~(~- -n(~,k))- ~ "geff'~- .B gr

(2)

The group velocity V r was calculated from the polarlton dlsperslon g c2k2/m2=e(ET,,~,k), where the transverse exclton energy E T, was taken to be (E~-E~)/2 i.e. the average position of the magnetic sublevels of the exclton. With the condltlon E T , > > I E T , - ~ I > 2 E L T whlch is fulfilled in the spectrum of Fig.3 the FR angle is than approximately ~ I ET' ELT t ~B X = ~ nB ( E T , _ ~ ) 2 ' ~ geff ° --E B

lS14

L~20 PHOTON ENERSY

Fig.

2

1~ h~ leVI

Left hand and right hand polarized (LCP and RCP) absorption spectra of the 5.7 ~m thick GaAs sample at magnetic fields of 3.4 T. The wave-vector of the incident white light xs antl-parallel to the dlrectzon of the magnetic field.

The energetlc difference of the center posztlons of the absorption peak in LCP resp. RCP configuration correspond to the magnetic splitting of the Is exclton state. They are shlfted to hlgher energles (1.516 eV - 1.517 eV) according to the diamagnetic shift. In the case of the absorption peaks of (D°,X), (D°,X) ~ and the n=2 exclton all allowed components are indeed well resolved in both polarizations. The peaks at 1.525 eV and 1.5265 eV are due to the lowest Landau level transltzons in the continuum [6]. Flg.3 shows the FR angle spectrum of the GaAs sample in the energy region of the two lowest exclton polarlton resonances n=l,2 at 1.O6 T. The spectrum is domlnated by a huge resonance around the is exclton level, denoted by E T, . The n=2 exclton resonance is much less pronounced and shows a more complicated structure, which will not be discussed here. In addltlon to the is resonance the FR spectrum shows rich structures at energies of (D°,X) and (Do,X)* We now turn to the description of FR in the exczton polarlton framework: The FR angle due to a simple oscillator in a magnetlc field is X = ~

(n- - n +) t

(I)

where t xs the sample thic_~ness,l the wavelength of incident light and n ,n the refractive indlces for light wlth LCP and _RCP+pelarizatlon [I ]. TO determine the term (n -n ) we used Hopfields expression for n (E~,~,k) and n(E~,~,k) where ~ , E ~ are the two transverse exczton energies split by ~Ema~=geffPB B in the magnetic field [7]. If the ~agnetlc splitting E~ - E~ = AEma_ of E T is small compared wlth the longltudlna~ transversal splitting ELT of the Is ex-

(3)

where n B denotes the optical refractive index due to the 5ackground of higher lying oscillators and geff the effective g-factor of the is exclton which describes the splitting of the dxpolactive Zeeman states. (n==12.56:Em,=Em+ bE__ ). This simple model is already an excellent ~escrlptzon of the shape of measured FR angle spectra (see the circles in Flg.3). The theoretical values calculated from equ.(2) and shown in Flg.3

T ISK j

I

GoAs B=I00T

,

-9

[~X) i

-2

n=1 i 0

i

i 2

(O°X]" i n=2

PHOTON ENERGY hw-h¢~r liner]

Flg.3: FR spectrum of the 5.7 ~ m thlck GaAs sample in the region of the two lowest exclton resonances. E T, denotes the center of the lowest transverse exciton magnetic sublevels.The diamagnetic shift relative to E T is O.15 meV at a field of 1.O6 T. The dashed llne marks the background caused by continuum Landau level transition. The circles are calculated values obtained from the model within the polariton framework (see equ.

(2)).

include a smoothly varying FR background from higher Landau level transitions (dashed line in Fig. 3). ~he background was measured 5-90 meV below ET, and then extrapolated linearly into

Th Hornung, R G Ulbnch / Faraday rotanon ;n the lowest exctton resonance o f GalltumArsemde

the Is exclton reglon. For slmpliclty we did not include in the model the additional FR effect of (Do,X). A more precise descrlptlon should take into account the flne structure of the F 6 x F 8 exclton state in GaAs. Because of the exchange interactlon the exclton spllts in a 3 fold F 5 wlth total angular momentum F=I and a 3 fold F 4 and a 2 fold F 3 state wlth F=2 [8]. In low magnetlc flelds only the F=I, mF=±l states are dlpol allowed (in Faraday conflguratlon). Wlth Increaslng flelds there is a mlxlng between the four mF=±1 states of the F=I and F=2 exclton levels. Under the condltlon I E T , - ~ I > A E m a g the problem slmpllfies, however, and can agaln be treated wlth an effectlve two oscillator model to obtain the n- and n + refractlve indices : n + and n- are now the polarlton refractlve indlces correspondlng to two effectlve oselllators where E~, E~ zs taken to be the average (welghted wlth the approprlate osclllator strength) of the two sets of Zeeman states. The g-factor fo~ the is exclton is determlned from our LCP/RCP magneto absorptlon peak posltlons, and agrees reasonably well wlth the low fleld value of geff = -1.9 of Willman et al. [9]. The llne shape of the FR angle spectrum is descrlbed excellent by equ.(1). The effective two osclllator model differs only in the region I E T , - ~ I < O . 2 meV from the approach descrlbed in equs.(2) and (3). As a consequence the product ELT.gef f can be extracted wlth high preclslon and wlth no ad3ustable parameter from the experlmental data (see Flg.3) for the is exclton. Both factors depend in a speclfxc way on the magnetlc fleld B. In order to extract the fleld dependence of geff for the ls exclton, we have adopted the theoretlcal dependence of E ~ ( B ) from a varlatlonal calculatlon of Pollmann [10] wlth the zero fleld value E ~ ( B = O ) = O . O 8 meV and the parameters g~ven in [ii].~ In the fleld range O.5-1.O T there is an excellent agreement wlth the value geff=-l.9 reported by Willman et al. [9],see Fig.4. For lower fields (B

25 o° o

2O

oo

0

•

0

T10 05 0

l

265

Beyond thls dominating Is FR resonance rlch structures of (D°,X)~D°,X)~ and the n=2 exclton were found in the FR angle spectra (see Fig. 3). The (D°,X) states cause sharp structures in the FR spectrum 1 meV below ET,. Because of the relatlvely large spllttlng of the magnetlc sublevels of (D°,X) slngle levels can be resolved in the FR spectrum. The rotatlonal e x c l t e d states of (D°,X) gave only small humps on the low energy w~ng of the is resonance curve. (Do,X) ~ leads to the small structure around 2.3 meV above ET,. Probably the n=2 exclton induces structures ~n the FR spectrum in the region 3 meV above ET,. Here the sign of the FR angle of the energetically lowest component is opposlte to the slgn of the Is resonance, and xs ascrlbed tentatlvely to a posltlve geff for the n=2 exclton. IV CONCLUSION We have measured for the flrst time the giant resonance behavlour of PR around the lowest Is exclton in a high purlty semlconductor. The FR spectra in GaAs are described quantitaczvely withln the exclton polarlton framework. A preclslon determlnation of the product of the Is exciton g-factor, geff(F=l) and the LT spllt ELT, gives for B~O:geff.ELT = -O.184±O.O3 meV. REFERENCES [i] Mavroldes,J.G.,in: Optical Properties of Solids (ed.. Abel~s,F., North Holland Publlshing Comp., Amsterdam, 1972)Chap. 7. [2] Cho,K., in Excltons (ed. Cho,K., Springer Verlag, Berlln-Heldelberg- New York,1979) Chap. l,2. [3] Zv~ra,M., Prosser,V., phys. stat. sol.(b) 57, 773 (1973). [4] Hi--ll, D.E., Solzd State Comm. i__1,1187 (1972) [5] Dean, P.J., Herbert, D.C., in Bxcitons (ed. Cho,K., Sprlnger-Verlag, Berlin-Heldelberg-New York, 1979) Chap. 3. [6] Seisyan, R.P., Abdullaev, M.A., Zakharchenya,B.P., Sov. Phys. Semicond. 7, 649 (1973). [7] Hopfleld, J.J., Phys. Rev. 182, 945 (1969). [8] Cho, K., Suga, S., Dreybrodt, W., Willmann, F., Phys. Rev. B11, 1512 (1975). [9] Wlllmann, F., Suga, S., Dreyhrodt, W., Cho, K., Solld State Comm. 14, 783 (1974). [10] Pollmann, J., Verhandl. DPG (VI) ~, 472 (1976). [11] Ulbrlch, R.G., Welsbuch,C., Phys. Rev. Letters 38, 865 (1977).

i

0

MAGNETIC FIELD Fxg.4-

B [T]

Magnetlc field dependence of geff for F=I components of the Is-exciton in GaAs up to 2.6 T.

Reflectance spectroscopy in this region lacks sufflcient precision; FR spectra, on the other hand, are much more sensitive at low fields. F o r hlgher fields (B=I.O-2.6 T) the g-factor decreases linearly with the magnetic field (see Fig.4).

ACKNOWLEDGEMENT We thank Laboratolre d'Electronique de Physique Appllqu~e, Parls, France, for providing the GaAs samples.