The spectral weights of the wave-function sidebands for a quantum well in the presence of an inhomogeneous electromagnetic (EM) field are studied by introducing a wave function with the form of a Floquet state and then solving the time-dependent Schrödinger equation approximately. The two cases of radiation direction of the EM field parallel and perpendicular to the well axis are considered. We find that the inhomogeneity of the EM field may eliminate the sideband quenching. Based on the spectral weight, the transmission probability through the well is investigated. The energy-level splitting for a special case, the averaged vector potential equal to zero, is also studied.published_or_final_versio

Lin, TH

Sun, QF

Wang, J

The spectral weights of the wave-function sidebands for a quantum well in the presence of an inhomogeneous electromagnetic (EM) field are studied by introducing a wave function with the form of a Floquet state and then solving the time-dependent Schrodinger equation approximately. The two cases of radiation direction of the EM field parallel and perpendicular to the well axis are considered. We find that the inhomogeneity of the EM field may eliminate the sideband quenching. Based on the spectral weight, the transmission probability through the well is investigated. The energy-level splitting for a special case, the averaged vector potential equal to zero, is also studied.http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000075039700046&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=8e1609b174ce4e31116a60747a720701Physics, Condensed MatterSCI(E)8ARTICLE42008-20125

Institutional Repository of Peking University

Physical Review B

Lack of quenching for the resonant transmission through an   inhomogeneously oscillating quantum well

HKU Scholars Hub

Title Lack of quenching for the resonant transmission through aninhomogeneously oscillating quantum wellAuthor(s) Sun, QF; Wang, J; Lin, THCitation Physical Review B (Condensed Matter), 1998, v. 58 n. 4, p. 2008-2012Issued Date 1998URL http://hdl.handle.net/10722/43237Rights Creative Commons: Attribution 3.0 Hong Kong LicensePHYSICAL REVIEW B 15 JULY 1998-IIVOLUME 58, NUMBER 4Lack of quenching for the resonant transmission through an inhomogeneously oscillatingquantum wellQing-feng SunDepartment of Physics, Peking University, Beijing 100871, ChinaJian WangDepartment of Physics, The University of Hong Kong, Pokfulam Road, Hong Kong, ChinaTsung-han LinCCAST (World Laboratory), P.O. Box 8730, Beijing 100080, China;Institute of Theoretical Physics, Academia Sinica, Beijing 100080, China;and Department of Physics, Peking University, Beijing 100871, China*~Received 1 December 1997; revised manuscript received 17 February 1998!The spectral weights of the wave-function sidebands for a quantum well in the presence of an inhomoge-neous electromagnetic ~EM! field are studied by introducing a wave function with the form of a Floquet stateand then solving the time-dependent Schro¨dinger equation approximately. The two cases of radiation directionof the EM field parallel and perpendicular to the well axis are considered. We find that the inhomogeneity ofthe EM field may eliminate the sideband quenching. Based on the spectral weight, the transmission probabilitythrough the well is investigated. The energy-level splitting for a special case, the averaged vector potentialequal to zero, is also studied. @S0163-1829~98!04128-9#The electron tunneling through nanostructures in the pres-ence of a time-dependent electromagnetic ~EM! field hasbeen a subject of increasing interest in the past few years. Anumber of new effects have been observed, such as thephoton-assisted tunneling,1–3 the splitting of the Coulomboscillation peaks,1–3 the photon-electron pumping,4,5 etc.Since the pioneering work of Tien and Gordon,6 it hasbeen well known that an oscillating potential with frequencyv can change the energy of an electron state E into a set ofenergies E6n\v (n50,1,2,...), the so-called sideband en-ergies. All phenomena mentioned above are related to thespectral weights of the sidebands. Theoretically, there aretwo main approaches to study the sideband effects: one is totake the adiabatic approximation and use the Green’s-function technique,7–12 and the other is to solve the time-dependent Schro¨dinger equation directly.13,14 In the adiabaticapproximation approach, one assumes that the external po-tential, say, eV cos vt, only causes the single-electron en-ergy ek a rigid shift @ek!ek(t)5ek1eV cos vt#, but notransition between different electronic states takesplace.7,10–12 Then the obtained spectral weight of the nthsideband sn , which is proportional to Jn2(a) ~where a5eV/\v is a dimensionless variable for the effective fieldstrength!,7 will vanish at certain values of a, correspondingto the zeros of the Bessel function Jn . This result is usuallycalled sideband quenching.7 By using the Schro¨dinger equa-tion approach, sideband quenching has also beenobtained.7,13,14Since the spectral weights are related to the transmissionprobabilities, one can check the sideband effect by examin-ing the strengths of the side peak around the central reso-nance, or by measuring the heights of the side step of the I-Vcurves. Surprisingly, in the experiments by Drexler et al.1PRB 580163-1829/98/58~4!/2008~5!/$15.00and Blick et al.,2 no sideband quenching is observed. It isimportant to notice that almost all the theoretical works thatled to sideband quenching always considered the regionallyhomogeneous fields ~either in the quantum-well region or inthe lead region!. On the other hand, in experiments by Drex-ler et al.1 and Blick et al.,2 the broadband bowtie antenna isused to couple the microwave fields to the system, whichmay produce an inhomogeneous field in the quantum-wellregion. We expect that the inhomogeneity may play an es-sential role in eliminating sideband quenching.In order to check this idea, we consider a quantum wellapplied by a time-dependent field, propagating head-to-headalong the z axis, and forming a spatial inhomogeneouslystanding wave in the quantum-well region. Two specialcases, the radiation direction parallel and perpendicular tothe well axis, have been investigated ~see Fig. 1!. Taking theCoulomb gauge, the electric field E52(1/c)(]A/]t). Bysolving the time-dependent Schro¨dinger equation, the spec-tral weight of sidebands sl and the transmission probabilityT(e) are obtained. It turns out that lack of sideband quench-ing is found for these inhomogeneous field cases where bothFIG. 1. Schematic description of the model system. E is theexternal electric field. ~a! and ~b! correspond to the cases of theradiation direction parallel and perpendicular to the well axis, re-spectively.2008 © 1998 The American Physical SocietyPRB 58 2009LACK OF QUENCHING FOR THE RESONANT . . .E(r,t) and A(r,t) depend on the coordinates r. Besides, aslight asymmetry of the spectral weights of the sidebands isalso found, the same as the result by Wagner.141. The case with the radiation direction parallel to well axisIn order to calculate the spectral weights of the sidebands,we consider that the system is completely confined in the zdirection by assuming that the heights of the barriers areinfinite, i.e., the confining potential isV~r!5 H 1` ,0, uzu.a/2uzu<a/2. ~1!For the realistic system with finite barrier height, the tunnel-ing effect can be described by introducing the phenomeno-logical parameter G, which measures the decay width of theresonant state. To characterize the inhomogeneity of the ex-ternal field in the well region, we assume that the electricfield E is parallel to the x axis and choose A5(Ax,0,0)where Ax has the form of Ax5(E0c/v)@g(z)#cos vt. Ex-panding Ax into Fourier components, one has Ax(z)5(n Ax(qn)exp$iqnz%, whereAx~qn!5E0cav E2a/2a/2g~z !e2iqnzdz[E0cvgn , ~2!and qn52pn/a (n50,61,62,...). Neglecting the higher-order terms of Ax(qn) as in Refs. 6 and 15, the Hamiltonianof the system under consideration can be expressed asH5H082i\emcAx~q0!cos vtddx ,~3!H085H02i\emc (nÞ0 Ax~qn!eiqnzcos vtddx ,where H0 is the Hamiltonian without the external field, andthe wave function of H0 isf ~r,t !5ei~kxx1kyy !cos kzze2ie0t/\, ~4!with e05\2(kx21ky21kz2)/2m , and kz5(2n11)p/a .Next, we need to find the solution of the Hamiltonian H08 .The point here is to find a solution with a special form, theform of the Floquet state. The Floquet state is the analog to aBloch state when replacing a spatially periodic potential witha time periodic potential.14 Therefore, the wave function ofthe Hamiltonian H08, w(r,t), should be expressed asw~r,t !5e2iet/\u~r,t !5e2iet/\(lul~r!eilvt. ~5!ul(r) can be expanded into the series of Ax(qn):ul~r!5ul~0 !~r!1 (nÞ0ul ,qn~1 ! ~r!Ax~qn!1fl . ~6!Notice that w(r,t) should return to f (r,t) if Ax(qn)50; soonly one term in the set of $ul(0)(r)% is not zero, which isdenoted by u0(0)(r). In fact, which one we choose as thenonzero term does not affect the result.Substituting the expression of w(r,t) into the time-dependent Schro¨dinger equation of H08 Eq. ~3!, only keepingthe terms with the first order of Ax(qn), and by comparingthe coefficient of the terms with the same exp$ilvt%, one eas-ily finds~ l50 ! H0u0~r!5eu0~r!, ~7!~ l51 ! H0u1~r!2i\e2mc (nÞ0 Ax~qn!eiqnzddx u0~r!5~e2\v!u1~r!, ~8!~ l521 ! H0u21~r!2i\e2mc (nÞ0 Ax~qn!eiqnzddx u0~r!5~e1\v!u21~r!, ~9!~ lÞ0,61 ! H0ul~r!5~e2l\v!ul~r!. ~10!Notice that the boundary condition w(r,t)uz56a/250 corre-sponds to ul(r)uz56a/250.From Eq. ~7!, which is an eigenequation, one easily ob-tains u0(r)5eikxxeikyycos kzz, and e5\2(kx21ky21kz2)/2m5e0 with kz5(2n11)p/a . It should be pointed out that e isindependent with Ax(qn) in the first-order approximation.Then we solve Eq. ~8!. Substituting u0(r) and u1(r), Eq.~6!, into Eq. ~8!, and noticing that u10(r)50, one easily ob-tainsH0u1,qn~1 ! ~r!1\kxe2mc eiqnzcos kzzeikxxeikyy5~e02\v!u1,qn~1 ! ~r!. ~11!Then let u1,qn(1) (r)5u1,qn(1) (z)eikxxeikyy; one has2\22md2dz2 u1,qn~1 ! ~z !1\kxe2mc eiqnzcos kzz5S e02 \2~kx21ky2!2m 2\v D u1,qn~1 ! ~z !. ~12!Noticing that the above differential equation is not aneigenequation, and the general solution isu1,qn~1 ! ~z !5Cqneikz8z1Dqne2ikz8z2\kxe4mc F ei~qn1kz!zeqn1kz2e01\v 1 ei~qn2kz!zeqn2kz2e01\vG[Cqneikz8z1Dqne2ikz8z1 f qn~z !, ~13!where kz85Akz222mv/\ , eqn6kz5\2@kx21ky21(qn6kz)2#/2m , and f qn(z) is a compact notation of the last termon the right-hand side of u1,qn(1) (z), and Cqn and Dqn are ar-bitrary constant that should be determined by the boundarycondition, u1,qn(1) (6a/2)50. From the boundary condition,one has the following equations:2010 PRB 58QING-FENG SUN, JIAN WANG, AND TSUNG-HAN LINCqneikz8a/21Dqne2ikz8a/21 f qn~a/2!50,~14!Cqne2ikz8a/21Dqneikz8a/21 f qn~2a/2!50.Then the coefficients Cqn and Dqn are determined, and u1(r)is obtained immediately. Notice that f qn(2a/2)5 f q2n(a/2)52 f qn(a/2), one easily finds Cq2n1Cqn50 and Dqn1Dq2n50. If one only considers the case of Ax(qn)5Ax(q2n) @i.e., Ax(z)5Ax(2z)#, ul(r) will reduce tou1~r!5 (nÞ02\kxe2mccos~kz1qn!zeqn1kz2e01\veikxxeikyyAx~qn!.~15!By using the same procedure, we can solve Eqs. ~9! and~10! asu21~r!5 (nÞ02\kxe2mccos~kz2qn!zeqn2kz2e01\veikxxeikyyAx~qn!,~16!ul~r!50 ~ lÞ0,61 !. ~17!Then substituting ul(r) into Eq. ~5!, and the wave func-tion of the Hamiltonian H08 , w(r,t), can be obtained asw~r,t !5ei~kxx1kyy !e2ie0t/\H cos kzz2 \ekx2mc3 (nÞ0 F eivt cos~qn1kz!zeqn1kz2e01\v1e2ivtcos~qn2kz!zeqn2kz2e02\vGAx~qn!J . ~18!Referring to the Tien-Gordon theory,6 we finally find thewave function of the total Hamiltonian H ,c~r,t !5w~r,t !(lJ l~a!e2ilvt, ~19!wherea5ekxAx~q0!mcv5E0ekxq0amv2is a dimensionless variable for the effective field strength,and c(r,t) can be expressed in the following form:c~r,t !5(l(k˜zBl~k˜z!ei~kxx1kyy !cos k˜zze2i~e01l\v!t/\,~20!where k˜z5(2n11)p/a . Since the set of the functionscos(k˜zz) with different k˜z is orthonormal, the spectral weightof the sidebands can be expressed as Sl5Sk˜zuBl(k˜z)u2. Ifonly the ground state (kz5p/a) of the electron in the well istaken into account, then Sl is given bySl~e0!5UJl~a!1 \ekx2mc @h2~1 !Jl21~a!1h1~1 !Jl11~a!#U21S \ekx2mc D2(n.0uh1~n !Jl11~a!1h2~n11 !Jl21~a!1h1~2n21 !Jl11~a!1h2~2n !Jl21~a!u2, ~21!whereh6~n ![Ax~qn!eqn1kz2e06\v. ~22!Let sl denote the normalized spectral weight of the side-bands, sl5Sl(e0)/( lSl(e0).The dimensionless variable obtained in this work isa5ekxAx~q0!mcv5E0ekxq0amv2,which is scaled as v22, the same as the result obtained byWagner,14 but different from the scaling of v21 obtained bythe adiabatic approximation approach. Moreover, now a de-pends on the transverse momentum kx . From Eq. ~21! oneeasily sees that when kx50, sl (lÞ0) is identically equal tozero for any amplitude of the external field E0 , which meansthat the energy level e0 does not split into sidebands.Now let us consider the more interesting case withkxÞ0. From Eq. ~21!, the spectral weight of the sidebands slis highly nonlinear with E0 . Since the vector potentialA(r,t) depends on r, the spectral weight of the sidebands slwill be related to the Bessel functions Jl21(a), Jl(a), andJl11(a). It is impossible to find a nonzero value of a mak-ing all three Bessel functions @Jl21(a), Jl(a), and Jl11(a)#be zero. Therefore, with increasing the amplitude of the ex-ternal field E0 , the spectral weight of the sidebands sl varies,but never vanishes, i.e., the sideband quenching is eliminatedunder the inhomogeneity of the EM field. This behavior isshown in Fig. 2~a! for Ex5E0 cos qz sin vt. One can alsosee a slight asymmetry for the sidebands s6l . For compari-FIG. 2. The spectral weights sl vs a for the case of the radiationdirection parallel to the well axis, where v51, a50.1, kx5ky5kz5p/a , and in units of e5\5c5m51. ~a! Ex5E0 cos qz sin vt with q5p/a; ~b! the homogeneous case withEx5E0 sin vt for comparison.PRB 58 2011LACK OF QUENCHING FOR THE RESONANT . . .son, Fig. 2~b! presents the case of a homogeneous field bytaking Ex5E0 sin vt. In this case one simply has sl5Jl2(a), exhibiting the sideband quenching.2. The case with the radiation direction perpendicularto the well axisIn this case the system is confined in the y direction @seeFig. 1~b!#, and the confining potential isV~r!5 H 1` , uy u.a/20, uy u<a/2 . ~23!Now the vector potential is set to be A5(Ax,0,0) with Ax5(E0c/v)g(z)cos vt. Considering the standing wave as aperiodic function, i.e., Ax(z1b)5Ax(z), and taking the Fou-rier expansion, one has Ax(z)5SnAx(qn) exp(iqnz) withAx~qn!5E0cbv E2b/2b/2g~z !e2iqnzdz[E0cvgn , ~24!where qn52pn/b (n50,61,...). By neglecting the higher-order terms of Ax(qn) as in case 1, the Hamiltonian can bewritten as the same form as in Eq. ~3!. Then one obtains thewave function asc~r,t !5e2ie0t/\ei~kxx1kzz ! cos kyy3H 12 \ekx2mc (nÞ0 eiqnz@h1~n !eivt1h2~n !e2ivt#J3 (l52`1`Jl~a!e2ilvt, ~25!where ky only takes the values of (2n11)p/a (n50,61,...). Different from case 1, there is no restriction ofAx(z)5Ax(2z) @or Ax(qn)5Ax(q2n)# in this case. Then thespectral weight of the sideband obtained issl~e0!5BF Jl2~a!1S \ekx2mc D 2 (nÞ0 uh1~n !Jl11~a!1h2~n !Jl21~a!u2G , ~26!where B is the normalization factor.Figure 3~a! shows sl vs E0 ~or a!. Again, the spectralweight of the sidebands sl has a lack of quenching at anyvalues of E0 and exhibits a slight asymmetry ~not shown inthe figure!. These features originated from the inhomogene-ity of the electric field in the quantum-well region. We alsosee from Fig. 3~a! that sl saturates at large a. Figure 3~b!presents the spectral weight of the sidebands for the casewith homogeneous field for comparison, which clearlyshows the sideband quenching and the symmetric spectralweights.Based on the spectral weight sl , one can calculate theelectron transmission probability through the quantum wellin the presence of the external field T(e).14 Let GL (GR)denote the half width of the resonant energy level due to thetunneling through the left ~right! barrier to the left ~right!lead without the external field. Since the states of differentsidebands are orthonormal, the transmission probability T(e)can be obtained by the Breit-Wigner formula asT~e!5(ls lGLGR~e2e01l\v!21~G/2!2, ~27!where G5GL1GR . Figure 4 shows T vs e for case 2. In Fig.4~a!, we take Ax(q0)5E0c/v , Ax(q61)50.5E0c/v , andAx(qn)50 for all other qn’s. The sideband peaks do notvanish at any a ~except a50! with a slight asymmetry ~toosmall to see!. With the increasing of E0 ~or a!, more andmore sideband peaks emerge. These features are in agree-ment with the experiments by Drexler et al.1 For compari-son, Fig. 4~b! presents the result of homogeneous field withAx(qn)50 (nÞ0), showing a strong quenching for the mainpeak at a52.4.Finally, we study a special case with Ax(q0)50, i.e., theaverage of vector potential A(r,t) on the coordinates r equalFIG. 3. The spectral weights sl vs a for the case of the radiationdirection perpendicular to the well axis, where v51, a5b50.5,and kx5ky5kz5p/a . ~a! Ex5E0 cos qz sin vt ~for uzu<b/2) withq51.5p/a; ~b! the homogeneous case with Ex5E0sin vt for com-parison.FIG. 4. T(e) vs e for the case of the radiation direction perpen-dicular to the well axis, where v51, a5b51, kx5ky5kz5p/a ,a52.4, and GL5GR50.1. ~a! Ax(q0)5E0c/v , Ax(q61)50.5E0c/v , and all other Ax(qn) are equal to zero; ~b! all Ax(qn)are equal to zero except Ax(q0)5E0c/v .2012 PRB 58QING-FENG SUN, JIAN WANG, AND TSUNG-HAN LINto zero, leading to a50. In this particular case the mainresonant peak will mainly be splitted into three side-band peaks ~see Fig. 5!. From Eq. ~21! or Eq. ~26!, onecan easily find that only s0 and s61 have significant ampli-tudes; the others are zero in the first-order approximation.If one considers the second-order terms, Ax2(qn), the side-bands s62 will emerge, but the ratio s62 /s61 is aboutFIG. 5. T(e) vs e for the case of the radiation direction perpen-dicular to the well axis and with Ax(q0)50, where v51, a5b50.4, kx5ky5kz5p/a , E050.1, GL5GR50.1, Ex5E0sin qz sin vt ~at uzu<z/2) with q51.5p/a .eAx /cPx'eE0 /v\kx'0.01 ~all parameters are the same asin Fig. 5, and in the units of e5\5c51).In summary, we have studied the sideband effect of theelectron transmission through a quantum well in the presenceof a time-dependent field. We find that whether the sidebandquenching appears is critically dependent on the homogene-ity of the field in the quantum-well region. Lack of sidebandquenching is found for the inhomogeneous field case, whichis in agreement with the experiments by Drexler et al.1 andBlick et al.2 Recently, we have noticed that Oosterkampet al. reported the electron tunneling through a quantum dotin which one can see a sideband quenching.3 We guess itprobably comes from the homogeneity of the field in the dotregion. In addition, two characters are predicted in this work:no splitting of the resonant energy level for kx50; andmainly splitted into three sidebands for Ax(q0)50.This work was partially supported by the National NaturalScience Foundation of China and the Doctoral ProgramFoundation of the Institution of Higher Education. J.W. wassupported by a CRCG grant from the University of HongKong.*Mailing address.1 H. Drexler, J. S. Scott, S. J. Allen, K. L. Campman, and A. C.Gossard, Appl. Phys. Lett. 67, 2816 ~1995!.2 R. H. Blick, R. J. Haug, D. W. van der Weide, K. von Klitzing,and K. Eberl, Appl. Phys. Lett. 67, 3924 ~1995!.3 T. H. Oosterkamp, L. P. Kouwenhoven, A. E. A. Koolen, N. C.van der Vaart, and C. J. P. M. Harmans, Phys. Rev. Lett. 78,1536 ~1997!.4 L. P. Kouwenhoven, S. Jauhar, J. Orenstein, P. L. McEuen, Y.Nagamune, J. Motohisa, and H. Sakaki, Phys. Rev. Lett. 73,3443 ~1994!.5 L. P. Kouwenhoven, S. Jauhar, K. McCormick, D. Dixon, P. L.McEuen, Yu. V. Nazarov, N. C. van der Vaart, and C. T. Foxon,Phys. Rev. B 50, 2019 ~1994!.6 P. K. Tien and J. P. Gordon, Phys. Rev. 129, 647 ~1963!.7 M. Wagner, Phys. Rev. B 49, 16 544 ~1994!; Phys. Rev. A 51,798 ~1995!.8 R. Aguado and G. Platero, Phys. Rev. B 55, 12 860 ~1997!.9 Y. Goldin and Y. Avishai, Phys. Rev. B 55, 16 359 ~1997!.10 C. A. Stafford and Ned S. Wingreen, Phys. Rev. Lett. 76, 1916~1996!.11 Antti-Pekka Jauho, N. S. Wingreen, and Y. Meir, Phys. Rev. B50, 5528 ~1994!.12 Qing-feng Sun and Tsung-han Lin, J. Phys.: Condens. Matter 9,4875 ~1997!; Phys. Rev. B 56, 3591 ~1997!.13 D. Sokolovski, Phys. Rev. B 37, 4201 ~1988!.14 M. Wagner, Phys. Rev. Lett. 76, 4010 ~1996!.15 J. In˜arrea, G. Platero, and C. Tejedor, Phys. Rev. B 50, 4581~1994!.

Lack of quenching for the resonant transmission through an inhomogeneously oscillating quantum well

http://hub.hku.hk/bitstream/10722/43237/1/38886.pdf

Lack of quenching for the resonant transmission through an inhomogeneously oscillating quantum well

Abstract

Similar works

Full text

Available Versions

Institutional Repository of Peking University

HKU Scholars Hub