One-photon and two-photon spectroscopy and spin polarization

Transcription

1 J. Mater. Envron. Sc. 4 (3) (013) Mah ISSN : One-photon and two-photon spectroscopy and spn polarzaton M. Idrsh Mah 1 Department of Physcs, Unversty of Chttagong, Chttagong 4331, Bangladesh. Department of Mathematcs, Chttagong Unversty of Engneerng and Technology, Chttagong 4349, Bangladesh. Receved 1 Oct 01; Revsed 9 Dec 01; Accepted 9 Dec 01 *Correspondng Author Emal: m.mah@grffth.edu.au Abstract On the bass of a hgher-band Kane model for the energy-band structure and a mult-order perturbaton theory of the spn-polarzed electronc transport model a theoretcal nvestgaton has been carred out for the optcally generaton of spns va both one-photon and two-photon spectroscopy n materals wth a znc-blende structure (znc-blende semconductors). As the optcally excted hole spn relaxaton s extremely fast, the calculaton for the polarzaton gves effectvely the electronc spn polarzaton. It s found that the electron spn polarzaton depends on the mode of the spectroscopy and the exctaton photon energy. On comparson between two spectroscopc modes, t s clear that the multphoton spectroscopy enhances ~ 10% of the electron spn polarzaton. The reason for the possblty of enhancng the effect s dscussed. Keywords: Znc-blende semconductor; Nonlnear spectroscopy; Perturbaton theory; Spn polarzaton 1. Introducton Spn-polarzed electron transport or spn transport s an essental part of spn physcs, or spntroncs, where the electronc spn s utlzed n an equal footng as the electronc charge. Relable spn transport or spn transport wthout spn relaxaton, or the loss of the spn polarzaton, over dstances that are comparable to the devce dmensons s requred for practcal spntronc devces. However, once an electron spn mbalance s njected nto (or generated n) a semconductor, electrons experence spn-dependent nteractons wth the envronment,.e. wth mpurtes, defects and exctatons or phonons, whch causes spn relaxaton [1]. Spn relaxaton generally refers to the process whch brngs a nonequlbrum electronc spn populaton to a spn equlbrum state. Snce ths nonequlbrum electronc spn n metals and semconductors s used to carry the spn-encoded nformaton, whch s one of the mportant steps towards appled spntroncs and possble future quantum computaton, t s mportant to now how long the spn can travel wthout losng ts ntal spn orentaton, both n dstance as well as tme. The determnaton of spn-flp rate s extremely mportant for electronc applcatons, because f the spns relax too fast, the dstance travelled by the spn-polarzed current wll be too short to serve any practcal purpose. Proper understandng the spns of electrons and ther transport n semconductors s the cornerstone to developng the next generaton of quantum computers, spn-based mcro- and nano-technologes, and the spn transstor (the buldng bloc for future electronc devces). Consderable advances n understandng the transport of spns n sold state envronment and semconductors have been acheved n the recent years [-9]. However, the effcent spn njecton, or the generaton of hghly polarzed spns, for a relable transport stll remans a major problem n semconductor spntroncs. Because of the avalablty of hgh ntense tunably lasers, multphoton spectroscopy has been used to study the multphoton absorpton of the optcal nonlnear processes, partcularly n semconductors [10]. A two-photon spn-exctaton n lead chacogendes has been studed, where they predcted a hgh spn-polarzaton n these cubc materals [11]. For GaAs, the smlar results have been observed recently n an experment [5]. In the present nvestgaton, we study theoretcally the spn generaton n znc-blende structured semconductor GaAs va the both one-photon and two-photon optcal exctatons/spectroscopc technques. In the calculaton, a hgher-band Kane model for the energy-band structure [1] and a mult-order (frst- and second-order) perturbaton theory of the spn-polarzed electron transport are used, where the electro-electron (e-e) and electron-phonon (e-ph) nteractons are gnored. 46

2 J. Mater. Envron. Sc. 4 (3) (013) Mah ISSN : Eght band model Band-structure calculatons for the drect gap (E g ) semconductors wth znc-blende structures show that the electron wave functons at the valence band (VB) maxmum and conducton band (CB) mnmum have p- and s-type symmetry respectvely [13]. There s a magnetc nteracton between the magnetc moments assocated wth the spn and the orbtal motons of the electrons called the spn-orbt nteracton whch removes the spn degeneracy, snce t produces an energy dfference between the states for whch the orbtal and spn magnetc moments are parallel and antparallel. For each value of j, the total angular momentum, there are j+1 possble values of m j. In case of a sngle electron there are always two values of j =l ± 1/ (orbtal angular momentum of electron plus/ mnus ts spn). For an s-electron (l=0) the only possble value of j s 1/, whch has the two projecton m j =1/, -1/, and for a p-electron (l=1) j can have the values j =l ± 1/ = 3/ or 1/. Thus, the CB (s-symmetry) s twofold degenerate at the centre of the Brlloun zone ( =0), correspondng to spn-up and spn-down electrons (m j = 1/) and the VB (p-symmetry) at =0 s splt by spn-orbt nteracton nto heavy-hole (HH) and lght-hole (LH) bands and a splt-off band (SO). The HH and LH bands wth large and small effectve masses are each twofold spn degenerate (m j = 3/, 1/). Therefore, spnorbt nteracton splts the sxfold degenerate p band at =0 nto a fourfold degenerate level and twofold degenerate level whch les (spn-orbt splttng). Fgure 1 shows the energy bands (eght bands) for GaAs at = 0. The bass sets for the eght band model are gven n Table 1. Fgure 1. Eght band model: A scheme showng the heavy-hole (HH), lght-hole (LH) and splt-off (SO) bands of GaAs. The eght bands or states (countng one for each spn; up or down, ) comprse sx p-le valence band (VB) states (the SO, HH and LH bands) and two s-le conducton band (CB) states. Table 1. Bands, states j,m j and ther bass sets. Band j,m j Bass set CB 1/,1/ S CB 1/,-1/ S HH 3/,3/ () -1/ (X+Y) LH 3/,1/ -(6) -1/ (X+Y) + (/3) 1/ Z LH 3/,-1/ (6) -1/ (X-Y) + (/3) 1/ Z HH 3/,-3/ () -1/ (X-Y) SO 1/,1/ -(3) -1/ (X+Y)- (3) -1/ Z SO 1/,-1/ (3) -1/ (X-Y)- (3) -1/ Z 3. Theory The spn polarzaton due to two-photon spn generaton has been calculated usng the eght band Kane model n the lmt of large spn-orbt splttng [13]. Here we estmate the electronc spn-polarzaton by calculatng both the one-photon and two-photon photo-generaton rates of electron spn denstes usng the perturbaton theory of the spn transport model n the long wavelength lmt, where the e-e and e-ph nteractons are gnored. An llustraton of the two-photon (one-photon) exctaton scheme showng the pump pulse ω (ω 1 ) couplng the ntal and fnal states n the valance band (VB) and conducton band (CB) and the probe pulse 47

3 J. Mater. Envron. Sc. 4 (3) (013) Mah ISSN : (ω Pr ) tuned to the band gap (E g ) exctaton resonance ( Pr Eg ) s shown n Fg. 1. For clarty, only the VB and CB are shown n Fg. Fgure. One-photon and two-photon spectroscopy: An llustraton of the one-photon (left) and two-photon (rght) exctaton schemes showng the exctaton pulse (ω 1 for one-photon and ω for two-photon) couplng the ntal and fnal states n the VB and CB bands and the probe pulse (ω Pr ) tuned to the band gap (E g ) exctaton resonance. Here E1 s the one photon energy and 1 1. The two-photon energy s E E wth a photon frequency of (1/ ) 1 and an energy of. 1 As optcally excted hole spn relaxaton s extremely fast, ther polarzaton s effectvely zero, and can be neglected. For an electrc feld : ( ) ( t t E t E e e ) (1) the two-photon spn generaton rate can be wrtten as: jlm j l* m* ds / dt E E E E, () jlm where s a ffth ran pseudotensor symmetrc on exchange of ndces j and, and on exchange ndces l and m. The two-photon spn generaton rate under the assumptons detaled above has been consdered earler for the doubly degenerate band case and s gven as [14]: 3 ˆ ()* () ds / dt ( / L ) c S c' C C { ( )}, (3) c',v, c,v, c, c', v, where Ŝ s the spn operator, L3 s a normalzed volume, n s a Bloch state wth energy ( ) and () Cc,v, s the two-photon ampltude gven by : () { E. vc, n( )}{ E. vn, v ( )} C c,v, ( e / ). (4) ( ) n nv Here v ˆ nm, ( ) n v m and ˆv s the velocty operator. In the Ferm s golden rule, the photo-generaton rate s tme-ndependent and can be smplfed f the spn-splt bands are well-separated [15]. If the spn-splt ()* () () bands are well-separated, Ferm s golden rule gves Cc',v, Cc,v, Cc,v, n Eq. (3). For GaAs, the spn-splt pars of bands should be treated as quas-degenerate n Ferm s golden rule as the splttng s at most a few mev [0]. Thus: 3 ()* () 1 ds / dt ( / L ) ˆ c S c ' Cc',v, C c,v, [ { cv( )} { c' v( )}]. [5] c, c', v, Smlarly, the optcal generaton rate of electron-hole pars can be obtaned as : 3 () c,v, c,, v, dn / dt ( / L ) C { ( )}. (6) The two-photon energy s ntroduced by the frequency term (=ω ). For the generocty, the subscrpt abccc from ω s omtted). Defnng A and Im aabac B, the component of the spn generaton rate along one of the cubc axes can be wrtten as : * ds / dt E E A ( B A) E E. (7) 48 cv cv n

4 J. Mater. Envron. Sc. 4 (3) (013) Mah ISSN : Here the ndces a, b, and c denote components along the standard cubc axes [100], [010], and [001]. For a cubc sotropc materal, the spn generaton rate can be descrbed by only one real parameter ( B A ) and thus : * ds / dt E E E A. (8) It should be noted that the cubc ansotropy means that the two-photon spn generaton from crcularly polarzed lght (CPL) depends on the angle of ncdence of the lght relatve to the cubc axes [15]. For CPL ncdent along ẑ specfed by polar angles θ and relatve to the cubc axes, 4 B A ds / dt zˆ A E 1 (, ), (9) 4 A where the upper, (lower, +) sgn s the exctaton for rght, σ + (left, σ - ) crcularly lght: σ + ( xˆ yˆ) / σ - ( xˆ yˆ) / and 4 (, ) sn ( ) sn ( )sn ( ). (10) The (, ) term s an mportant functon whch shows the cubc ansotropy n pumpng,.e. the angular dependence of pumpng n spn-exctaton n the cubc system. The term (, ) as a functon of θ and φ s plotted n Fg x x Fgure 3. The term (, ) as a functon of θ and φ. Ths feature can be seen n the fgure. Owng to the cubc ansotropy, the net generated spn s not always parallel to ẑ.for example, for lght along a <001> drecton, ds / dt A E along a <111> drecton, 4 ds / dt (4/3)( AB) E 49 4, whle for lght ncdent. Snce hole spn polarzaton n bul semconductors s nown to relax very fast, on a longer tme scale one typcally obtans only the electron spn polarzaton. The electronc spn polarzaton (P) can be estmated as: ds/ dt zˆ P, (11) dn / dt where P s defned as P ( n n ) /( n n ). Here the spn s opposte to the photon angular momentum (generated wth a σ + lght beam) and n ( n ) s the densty of spn-down (spn-up) electrons. The electronc spn-polarzaton due to one-photon exctaton can also be obtaned by replacng ((=ω ) by (=ω 1 ) and

5 J. Mater. Envron. Sc. 4 (3) (013) Mah ISSN : the two-photon ampltude class of semconductors. C by the one-photon ampltude D. The approach can be appled for a wde () c,v, (1) c,v, 4. Results and dscusson Spn polarzaton as a functon of one-photon and two-photon exctaton energes was calculated n the long wavelength lmt of the mult-order perturbaton theory usng values of the parameters for GaAs [16]. The assumpton of a long wavelength lmt s chosen to neglect both the e-e and e-ph nteractons. Calculated results (sold lnes) are shown n Fgs. 4 and 5, where the spn polarzatons are plotted as a functon of the excess exctaton photon energy for the one-photon ( E ) and two-photon ( E ) exctatons: E 1 Pr 1 Eg E1 Eg (1) E Pr Eg 1 Eg E1 Eg. (13) Here E1 1 and 1. Symbols n the fgures show the data taen from one-photon [17] and two-- photon [5] spectroscopc experments. On comparson, over the whole range of the exctaton photon energy, a good agreement between experment and theory s acheved P (%) Excess photon energy, E (mev) Fgure 4. Spn-polarzaton as a functon of one-photon energy (excess photon energy, E Eg E1 Eg): expermental (symbol square) and theoretcal values (blac sold lne). Fgure 5. Spn-polarzaton as a functon of two-photon energy (excess photon energy, E Eg E1 Eg): expermental (down trangle) and theoretcal values (blue sold lne). Here E1 1 and 1.* As expected, the spn polarzaton ncreases wth exctaton photon energy for low excess energy and remans almost constant up to ~00 mev, and then decreases. However, the maxmum s obtaned for the excess energy consderably less than the SO splttng energy of ev for GaAs. The maxmum value s ~50% for two-photon exctaton. For the hgher excess energy, the polarzaton decreases rapdly due to the mxture of LH and HH states wth the SO valence band states whch have an opposte sgn. The LH and SO band transtons create the same electron spn orentaton and the sum of ther nter-band matrx elements s equal to the HH nter-band dpole-transton matrx element. There s stll a szeable degree of electron spn polarzaton even at excess energy hgher than the spn-orbt splttng energy. Although the maxmum optcal spn-polarzaton for an unstraned bul sample s expected to be 50% n theory [15], the maxmum has expermentally been observed to be less, as obtaned spectroscopcally wth a sngle-photon exctaton. In a bul sample there mght have some bacground unpolarzed electrons, whch mght not be excted by sngle-photon exctaton [18]. For an optcally generated electron densty n(0)=n (0)+n (0), there s a bacground densty of unpolarzed electrons n bul materals. On a comparson between Fgs. 4 and 5, t can be seen that a two-photon exctaton enhances the spn polarzaton. Ths s because t taes the advantages over one-photon spn spectroscopy due to a much longer absorpton depth, whch allows spn exctaton n the deep level,.e. throughout the volume of a thn bul sample. 430

6 J. Mater. Envron. Sc. 4 (3) (013) Mah ISSN : It thus can be concluded that due to a much longer absorpton depth hghly spn-polarzed electrons can be produced optcally by the two-photon exctaton of the bul semconductors. However, n a znc-blende semconductor, le GaAs, the spn polarzaton would decay wth tme due to the randomzaton of the ntal spn polarzaton by the Dyaonov-Perel (DP) spn relaxaton mechansm [1]. The DP spn relaxaton occurs n semconductors lacng nverson symmetry due to the spn precesson about an ntrnsc magnetc feld nduced by the presence of the spn-orbt nteractons n a znc-blende structure. However, the observed twophoton absorpton s mght be related to the excted e-ph nteracton whch s gnored n the present calculaton. Concluson An nvestgaton for the optcal spn polarzaton va both one-photon and two-photon spectroscopy n zncblende structured GaAs was carred out usng a hgher-band Kane model for the energy-band structure and a mult-order perturbaton theory of the spn-polarzed electronc transport. The electronc spn polarzaton for both spectroscopc modes was calculated as a functon of crcularly polarzed excted photon energy. The polarzaton was found to be preserved wthn the excess photon energy of ~100 mev. However, t depolarzed for the exctaton photon energy equals to or larger than the energy gap of the SO band to the CB. On a comparson between two spectroscopc modes, t can be concluded that a two-photon spectroscopy enhances (~10%) the electron spn polarzaton over a sngle- or one-photon. The results n comparson between two spectroscopc modes are dscussed. The results, however, suggest a possble way of enhancng the electronc spn polarzaton n semconductors. References 1. Dyaonov M. I., Perel V. I., Sov. Phys. JETP 33 (1971) Zese M. Thornton M. J., Spn Electroncs, Eds., Vol. 569, Sprnger Verlag, Hedelberg, Awschalom, D. D., Loss D., Samarth, N.,Semconductor Spntroncs and Quantum Computaton, Eds (Sprnger, Berln, 00). 4. Mah, M. Idrsh, Appl. Phys. Lett. 9 (008) Mah, M. Idrsh, J. Phys. Chem. B 113 (009) Mah, M. Idrsh, J. Phys. D: Appl. Phys. 40 (007) Dyaonov M. I., Khaets, A. V., Spn Hall Effect (Spn Physcs n Semconductors, Dyaonov, M. I., Ed., Sprnger-Verlag, Berln, 008). 8. Mah, M. Idrsh, Appl. Phys. Lett. 94 (009) Mah, M. Idrsh, Mater. Sc. Engneer. B 176 (011) Mah, M. Idrsh, Opt. Mater. 18 (001) Ivcheno, E. L., Sov. Phys. Sold State 14 (1973) Kane, E. O., J. Phys. Chem. Solds 1 (1957) Lampel, G., Wesbuch, C., Sold State Comm. 16 (1975) Danshevs, A. M., Ivcheno, E. L., Kochegarov, S. F., Stepanova, M. I., Sov. Phys. JETP 16 (197) Perce, D. T., Meer, F., Phys. Rev. B 13 (1976) Adach, S., GaAs and Related Materals: Bul Semconductng and Superlattce Propertes, World Scentfc, Sngapore, Mah, M. Idrsh, Gray, E. MacA., Curr. Opn. Sold State Mater. Sc. 14 (010) Mah, M. Idrsh, J. Appl. Phys. 103 (008) (013) 431