Solid State Commumcatlons, Vol 79, No 9, pp 749-754, 1991 Prmted m Great Bntam 0038-1098/91 $3 00 + 00 Pergamon Press plc QUANTUM MAGNETOTUNNELING TIME OF ELECTRONS IN Ga, _ ,Al,As-GaAs SUPERLATTICES H Cruz, A Hernandez-Cabrera and A Muiioz Dpto Fislca Fundamental y Experimental, Umversldad de La Laguna, 38204 La Laguna, Tenenfe, Spam zyxwvut (Received 8 January 1991 by D Van Dy ck) In this work, we have calculated magnetotunnelmg times of electrons m Ga,_,Al,As-GaAs multlquantum well heterostructures under an external electric field through analytical solutions Effective-mass equation of electrons m crossed electric and magnetic fields has been analytically solved by means of Confluent Hypergeometrlc Functions finding the existence of mdlvlduahzed tunneling channels connected with antlcrossmgs m the dlsperslon relation It ISfound that the quantum tunnehng time between two coupled quantum wells decreases if B IS increased I INTRODUCTION 2 METHOD OF CALCULATION RESONANT tunneling through quantum well hetero- structures has attracted considerable attention recently because of its possible apphcatlon to ultrahighspeed electronic devices [l-6] The effect of a magnetic field B on tunnehng m smgle barrier [7-181 or multlbarrler [19-251 heterostructures had been reported m a num- ber of recent papers The effect of a longltudmal mag- netlc field (J 11 B) on the tunneling current has already been used to probe the difference between sequential and resonant tunnehng [26] The Hamlltoman describes a spmless particle, of effective mass m* and charge e, subject to a constant and uniform transverse magnetic field B and a perpendicular electric field F, I e , zyxwvutsrqponmlkjihg +eFz+ V&+,(z), (1) Recently, the escape time of electrons photo- excited mto the quantum well of a double-bamer resonant tunnehng structure was measured by tlme- resolved lummescence [27] Mamly static features of the tunneling phenomena have been investigated, while dynamic aspects have not been studled thoroughly because of the characterlstlc time scale of the process 1s of the order of plcoseconds or less, which IS shorter than the measurable time by commonly available methods Nevertheless, dynamics should be mvestlgated since It not only determines the ultimate speed of tunneling devices, but It ehcldates fundamental aspects of electromc waves m sohds, particularly their behavior m the time domam [28, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA 291 where p = -fiV, A 1s the vector potential and VMFoW (z) represents the multiple fimte quantum well (MFQW) potential Let us consider an electron, of energy E and effective-mass rnticb, in the well (barner), m a MFQW system under an external applied crossed electric and magnetic fields With B 11 x and z the heterostructure growth du-ectlon, from equation (1) the Hamlltoman of the system m the well 1s given by [14, 171 H” = _ g V2 + $ lefiBz d - - - + eFz, ” H m,c ay and m the barriers 1s ii2 Hh = --VI+ e2B2z2 IehBz a 2m, 2m,cZ ---+ V,+eFz, mbc ay In this paper, we report a theoretical study of the electnc and magnetic field effect on the electron energy levels m Ga, _ ,Al,As-GaAs superlattlces The method of calculation used m this work 1s based on the transfer matrix approach and evidence for mdlvlduahzed tunneling channels m the dlsperslon relation where the gauge A = (0, - Bz, 0) has been chosen By using plane waves m the x and y direction (and neglecting spm effects), one find (see Appendix) that the envelope functions $“(z) and t/i’(z) descrlbmg the motion of the electron along the z direction satrsfy the equations H,” b(~)t,h” b(z) = E$” b(z), (3) (24 (2b) 740