Effect of coherent-light phase on tunneling process

Journal of University of Science and Technology BeijingMaterialsVolume 10, Number 4, August 2003, Page 61Effect of coherent-light phase on tunneling processPeng FengApplied Science School, University of Science and Technology Beiing, Bejing 100083, China(Reeived 2002-05-23)Abstract: The coherent-light-driven tunneling in double quantum wells has been studied. The electrons are coupled to a system ofphonons and subjected to the two beams of coherently optical waves. By adopting a gauge to both the extermal field and the phononfield, the phonon field operators in the Schrodinger equations are eliminated. In this way, an expression of the tunneling current isconveniently derived considering the relaxation efect. It is shown that under the intense laser field, the tunneling current oscillatesrapidly with time at low temperature. The duration of the oscillations is related to the temperature. By adjusting the phase differenceof the two light-beams, the osilation frequency can be modulated.Key words: tunneling curent; quantum wells; gauge transformation[This work was financially supported by the National Natural Science Foundation of China (No. 10074004.]1 Introduction2 Theoretical analysisAlthough the photon assisted tunneling processesLet two beams of the coherent lights irradiate twohave been studied extensively [1-6], it is not clear thatwells of the modulation-doped double quantum wellsthe phase of the coherent light-beams how to affectGaAs-AlGaAs-GaAs, respectively, as shown in figurethe tunneling processes. For the optical waveguides1. Since the external fields interact with the electronmade by the double quantum wells GaAs-AlGaAs-gasses, the electron occupations in the two-quantumGaAs [7], if the separation of the two waveguides iswells change with time, and generate the tunnelingvery closer, there would bring about the electron tun-current. In the follows, it is mainly discussed that howneling between the two waveguides. The changing-the phase difference of the coherent lights affects thetunneling current would affect the device performance.tunneling current.So it is necessary to find out the relation between thephase difference and the tunneling current so as toprovide useful information for the device design.In the tunneling process, the electron-phonon scat-tering would reduce the electron velocity and affectthe tunneling current. The tunneling structure is mod-eled on the basis of a single-particle picture consider-ing the electron-phonon interaction by proper ap-proximation [8]. Here a new approach is developed toGaAsAlGaAsGaAs .deal with the electron-phonon scattering. This approach is based on the local gauge to both the laserFigure 1 Two coherent light-beams 1 and 2 irradiate thefield and the phonon field, the effective Schrodingerdouble quantum wells GaAs-AIGaAs-GaAs.equations that do not involve the phonon operators areThe laser fields are described by the vector poten-constructed.tials of time-dependent:In this paper, the effctive Schrodinger equations of中国煤化工describing the tunneling process are derived, and theA= Aoe, cos(1coherent-light-driven tunneling current in doubleMYHCNMH Gwhich is the optical wave of irradiating on the left well,quantum wells is calculated.Corresponding author: Peng Feng, E-mail: fengpeng@ sas.ustb.edu.cn.62J. Univ. Sci. Technol. Bejing, Vol.10, No.4, Aug 2003and压= Are, cos(ot +δ)(2)-1φ2h2V2φ2+(Vφ2)-Ot2mwhich is the optical wave of irradiating on the rightehA2. Vφz +e'AR-iZv(age矿--+(f)(1)well. Here e is the unit vector, δ the phase differ-2mc .2mc2ence between two optical waves and A0 the ampli-tude of the vector potential. The optical waves are as-The phases 9r, 92 satisfying equations (8)-(11) aresumed to propagate along the z-direction. The spatial7e2 A7e2 Aξdependence of the optical waves is neglected (dipoleφ\=; sin(2ot)+16hmc'32homcapproximation). According to the general theory ofdiscussing the tunneling effect [9], the Schrodinger"Zv。(age师r -age)(12)equations for the electrons in two wells in the field ofthe laser light are=P--4出-7e2A.__ 7e2 A子sin(2ot + 28)+1 6hmc232homc2i江v。(ageir-aige-间F )4 + BY2(3)"Zv(ane4r -aeir)(13)Therefore, the Schrodinger equations are rewrittenin觉__p_°和x-asihar-2mV29+ Be(0-)3(14)i2v(aceir-aje-F )% +BW{(4)aq2.V22 + Bei(or- 0(15)where中，出are the electron wave functions on theleft well and on the right well, respectively; B thecoupling matrix element and ain , aig the annihilationFor the response of the electrons to the laser field isand the creation operators of the phonons in two wells,faster than that of the phonons, the electrons areseemed to move in the average phonon-potential field.respectively. The electron-phonon coupling constantTherefore the statistical average over the phonon op-Vq iserators，(aiq} and ain} (i= 1, 2) are used to elimi-nate phonon operators on both sides of each equation.、Exq(5)It is convenient to evaluate the statistical average inthe holomorphic representation [10-13]. The averageLetof any operator B({ay },{ain }) is expressed as甲= φeimn(6)1por Sign-fIdanda"; B({aw }{i})}(B),=不(16)-MoL Sainaim(7)2-t[IandaieSubstitute forthe中and 4 in equations (3) and (4).In this way, the effective Schrodinger equations areThe following transformations are adopted:obtained by the average to equations (14) and (15):ihvq,=ihe(8)2。n'2 mci9V2虫+ B((0-0), ,4(17)'aihvo,_ihea,(9), 84n、m2mcot_V23 + B(0-0)，(18)V2ρn+h (Vφ)- -hIn calculation, the phonons in two wells are consid-2nered to be two.中国煤化工orrespondingetoperators elLu-raged respec-本.Vφ+iZr(a,@的-ae-F )(0)tively, i.e.YHCNM HG'.P. Feng, Effect of coherent-light phase on tunneling process63((1onea=(=()(en),(19)anda027e2 ABOn evaluating, there havehtn2k= εkn2k + Bexp0t32h omc'(0n),=exp -it7e6t+i7e2A- sin(2ot)-[sin(2ot + 28)- sin(28)- sin(2or)]-16hmc232homcSDior(20)2Phiori(O2x - Ohk >{nuen2x)2(29)The any one of the two equations is divided into the(i*); =exp|? 16tmeczt+'7e2A?7e2 A子real part and the imaginary part. Its real part isonk7e2A .[in(2on+ 28)-sin(20)]-_-+1-(21)32homc23h°or」[n(nr)sin()in(ot + 8)]- (Ox -0 )(numx)2The expansion formula(30)4=> CheiF(22)Considering that when the phase difference δ iszero，there is no tunneling current， one derivesθzx-01k=0. For the two wells are identical, thereshould be n2k ≈nk . Therefore the tunneling current3=2Czeki(23)is expressed asare substituted into equations (17) and (18), there haveI'=-e2(eBZsin{t aC1k=ε:Cik + (B(i0-01>),C2k .(24)\0t/。九At[sin(ot )sin(δ )sin(ot + 8)]}(31)=8qC2k + B(-10.-n),2Gikr(25)where ... is the statistical average to the electronsystem. Show the summation for > ，the relativewhere Ek =h2k几is the energy of the electron.current becomesAlthough the phonon operators are eliminated byeB7e2 Aζthe statistical average to the phonon system, the effectI=I'/P=e () sin{' 32h omcof the electron-phonon interaction on the electron tun-neling exhibits in the coupling term of each equation[sin(or)sin(δ )sin(ot + 6)}(32)(the second term on the right side of each equation).whereThe wave functions in k space is expressed asCik =(nmx)"2ihk(26)t。≈1/、2e2kT(_ 1_ 1 )。9max(33)πh2 (EoCx=(nx)"2ei02s(27)is the damping time of the tunneling current, here9max is the wave vector in the boundary of the firstwhere Mk， n2k are the particle numbers with theBrillouin zone.wave vector k in well 1, 2, respectively.日k， 02xTake the double quantum wells GaAs-AlGaAs-are the electron phases of time-dependent in the twoGaAs as an example, the material parameters of thewells. Equations (26) and (27) are substituted intoGaAs are chosen as &=12.91, 8=10.91, m=0.067moequations (24) and (25), there have(mo for the static mass of the electron), and the latticemk. himux00=&pMx + Bexp{i[. 7e2 A子spacing is 0.565 nm. It is found that the damping timeatto of the tunneling current is related to the temperature.The higher the temperature is, the shorter the damping[sin(2ot + 28)- sin(28)- sin(20r)]-time will be.中国煤化工ing time is2”+i(O2x -0k >{nmn)2(28)t.=5.72x10MHCN MH ais efee. thePiriontemperature musl b 1wi uiail +.2 i "On the other.6eJ. Univ. Sci. Technol. Bejjing, Vol.10, No.4, Aug 2003hand, in order to achieve the larger tunneling current,the CO2 laser (wavelength λ = 10.6 um) is applied tothe laser field strength Eo should obeyirradiate the two wells, the laser field strength shouldnot be less than the order of 105 V/cm.7c2E > T8mhoF22，ie.As shown in figure 2, in ultra-short time, the rela-tive tunneling oscillates rapidly with time. The 'oscil-12E。z4rmto(34)lation frequency' of the tunneling current induced bye2the two light-beams with phase difference π/2 ishigher than that with phase difference π/4. The dura-From equation (32) it is found that as long as theretion time of the tunneling current changes with tem-is one function of zero value in the three functionsperature, and the 'oscillation frequency' of the tunnel-sinδ，sin(ot), and sin(ot +8), the current is zero.ing current changes with the phase difference betweenFor the double quantum wells GaAs-AlGaAs-GaAs, ifthe two beams of coherent lights.a)]b)]三0W5010051/(