Roy, P N

Singh, A P

Singh, B P

English

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Indian J. Phys. 6 6 A  (5 ) , 6 2 9 -6 3 7  (1 9 9 2 )Tunnelling in crossed electric and magnetic fieldsP N Roy, A P Singh and B P SinghDcpartm cm  of Physics, Bhaj;alpur Upivcrsiiy, Hhagarpur 812 ?K)7, Pihar, India R e c e iv e d  2 7  M a ri h 19^1 . a c c e p te d  3 O cio h er /Q9JAbstract : A new mcKicl ol lunnclhng in crossc’d electric and m agnetic fields proposed by Roy and Cjhosh |1 | has hi*cn iisc*il c’siablish an Imaly al cxprcssu)n b>r tunnel current density  CurretU-Volt age characlc'nstics in M-I-M junct ms and tunnel diodes have been com puted w hich are in gooil cpiiililativc agreement with ob .ervalions A m ethod of cf>mpuling tunnelling time has also been suggestedK eyw ords : riinn<-l cm rein clensiiy, tunnelling lime, crossed elcdric and magnetic fields P A C ’S N o . : 73 40 G k1. IntroductionThe study oi this problem started after Calawa ci al [21 reported their observations according to which the tunnel current was found to decrease with increasing magnetic fields m fnSb tunnel diodes placed in crossed electric and magnetic fields. Roth et al f'^ ] studied ihc influence of such fields on Ge tunnel diodes and their observations were identical with tliose of the former. Several attempts were made later on to understand the mechanism of lunnelling in crossed clcciric and magnetic fields, important among them being those by Roth ct al [3], Zawad/ki and Lax |4| and Aronov and Pikus |51.Roth and his co-workers [31 proposed a phenomenological theory assuming a quadratic dependence of Kane voluigc on the magnetic field. According to them, the shift in the Kane voltage V/, due to the magnetic field is defined asAV^ = a (*) where a  is a conslani and ^ is  the applied magnetic field. They obiatned an exprcs.sion for the tunnel current given byJd (?() = ( ^ )  exp {BV) [\ - -  (2)where A and B are constants and V' is the applied bias, 'they also obtained the change in the tunnel current due to the tran.svcrsc magnetic field given byAJa m  [2 -  -  V*)| for (V -  V* -  > 0= 0 for (V - V i, -  a9^) < 0 (3)© 1992 I ACS5A (7)where K IS an arbitrary parameter assumed to be dependent on the vollagePlotting A/^ against the magnetic field for several bias voltages, they have shown that their theoretical curves deviate from the quadratic behaviour for higher magnetic fields in case of lower biases whereas they claim good agreement for higher biases.Zawadzki and Lax 16] showed that tunnelling is not possible in the one-band effective mass approximation, A two-band model has been suggested by them Taking the energy-momentum relation for two interacting bands suggested by Kane [7],630 P N  R oy, A  P Singh an d  B P  Singhm  -  ± [ ( ! ) ' *1/2(4)and using the principles of classical relativistic mechanics, it has been shown that for {T. is the electric field) the motion is magnetic field type and for T > Pi, it is electric field type. They have written the Hamiltonian lor an electron in a periodic potential in the presence of a crossed I icld as+ C l . r  + \ ' { T ) (5)where e is the electronic charge and c is the velocity of light. Solving the Schroedinger’s equation after using W.K.B. approximaiion they have obtained the band picture which shows that the application of the magnetic field curves the allowed regions away from each other which effectively increases the hand gap which in lurn decreases the tunnel currcni His expression for the tunnel current is given byJ O f )  -   ^J i . 2  y p j  2/U''/ell V 2 )  J ^ 2 2  where l-df = 1 ------Imc 2(7)Thus we find that increasing the magnetic field decreases which has the effect ol decreasing the tunnel current./(///).Aronov and Pikus [5| have used a iwo-band equation which is similar in form to the Dirac equation and ireaimg ihc problem relaiivistically, they have obtained an expression for the tunnel current given by,iy2v rc (p. -  P i ) r?i6nh s L eh («)where 5 (E^ J  and Pl and PC arc effective electric and magnetic fields. Eq. (8) isfound to be similar in nature to eq. (6).In the present paper, a completely different approach has been adopted to compute tlic tunnel current in crossed electric and magnetic fields. Roy and his coworkers [8, 9] have pointed out the inadequacies ol the conventional theory of tunnelling and have proposed a completely new approach to the entire problem of tunnelling. The conventional models treatthe electron ns n psrticlc inside the poteniinl barrier which is untenable because one cannot know anything about the particle so long as it is negotiating the barrier. It is only its probability density which can be said to be continuous through the barrier and the new model is based upon this approach. In the next section, an expression for the tunnel current- density under the influence of crossed electric and magnetic fields has been derived based on ihc ideas of the new model.2. T u n n e l c u r re n t  d e n s ity  in  crossed e le c tr ic  and  m agne tic  fie ld s  based on the  new approachRoy and Ghosh fll have shown that when a tunnel device is placed in crossed electric and magnetic fields with the electric field along the jc direction and the magnetic fieldalong the Z-dircction, the barrier wave funclion for the electron m'ay he written asTunnelling in crossed  electric and magnetic fie ld s  6 3 1X!/ (x, t) = a‘i{t) a  exp1-■+ u,(t) /I exp+ A) I*+ + h  -Vexp ( u o , t )exp (~ib)rt) (9)2m*where A = (V',, -  E), being the maximum height of the barrier; rn*‘ is the elTcctivemass of the electron in the barrier, ui(t) and (iriD arc time-dependent coefficients and a  and /iare coefficients obuiined in the iime-mde|>eiuleni irealmcni ol the barrier problem.(Oj r =  , w'here E  is ihe energy ol the electron.Writing coi and cOy in the two terms on the right hand side of eq. (9) is one of the mam contentions of the new model (S, 9], The electron, which possesses a negative kinetic energy inside the barrier eannot he treated as a particle. Neither has it the character of a wave because the two solutions inside Ihc barrier arc exponentially growing and decaying lunctions. So the electron is treated as something intermediate between a particle and a wave. It interacts with the barrier potential and there is every chance that its incident energy undergoes change to energy E^  during the process of tunnelling.Taking a^{t) = 1 and a M  < < 1, wc may writei//(jc, i) -  a  exp V i {B,  + X) |j: exp (- ia>i t)«r(0 P exp +Vi^^ exp {-iCOrt) ( 10)The lime-dependent position probability density may be written as+ a*r(t) p*a  exp (-  itOri)ly^  = la f  exp + A ) pc+ a M  po(* e x p .( -  i(Oi^ ) + \ar(t)\^  r  exp r i n632where (OirE i - E rP  N  R oy, A P  Singh an d  B P  SinghUsing llie equation of continuity, the tunnel current clensiiy may be wriitcn asJ i -  € dx 0 2 )where r is the tunnelling time and X\ and X2 are the barrier extremeties. Evaluating the integral of eq. (12) one finally obutinswhereand, , sin coirTh  ~ ‘^ oi -A)2 6)n r, A e h w \ a ' ^ X ^J 02 = — ^  ( -  ^Xbm(13;(54)W = X2~Xi is the width of the barrier and t? is a phase factor. Also reBz 2m* ...(I5)Eq« (13) gives us a spectrum of tunnel current density. This expression suggests that the emerging particle will not necessarily have ihc samc energy at which it was incident. There is a definite probability for a particle incident with energy Ei emerging with a different energy Er. The problem of computing the tunnel current density becomes a bit complicated because of distribution of electronic states on both the sides of the barrier. Also the incident electrons are incoherent. To tide over these difficulties, eq. (13) is summed over all possible values of Er ranging from -  «> to + 00. The infinite range of Er has to be considered to lake into account the complete randomness in the incidence of electrons of particular energy. Due to the conjugate relationship between E and t, the randomness or uncertainty in time is transferred to the uncertainty in energy i.e. if we regard that all the electrons are incident simultaneously {At = 0), then the energy uncertainty becomes infinite and hence an infinite range for Er. Therefore, for electrons incident with a particular energy we may write [8, 9]7^ ; ^  W o ,  " l  sin (ft« .T + 0) (16)3 E^ = ~ooA (^ /)  = r zEr = - oTo convert the above summations into integrals, we write Ei~Er = n£ (17)where e  is the separation of consecutive energy levels £ , and /i is an integer having both positive and negative values. Since the levels are very closely spaced, one may conveniently write the above summations in the form of integreal.s as given below;Tunnelling in crossed electric and magnetic fields f>33+S°sin • —  'J,(E,) = /01 if)(t )dn + / 02 sin ^  dn ( 1 8 )which yieldsJ ,  (Et )  = CT -^ 01 (19)(20)The differential tunnel current density at low temperatures may be written as(B) ~  ( ^  Pt i E)  ^ Ewhere p, (£) is the density of states on the incident side, £, having been replaced by £.Writing |  = Pr (£)where Q  is the volume of the electrode to which tunnelling takes place, we may write [8 |(21)d J ,  (B ) -  X  P r ( £ )  ‘ IE1/2)  EHence the total tunnel current density may be written as J,(B) =After substituting for pi (E) and pr {E) one finally obtains./, (B) =  ^ ^  J Xe' c.^ P (- 2xb IV) £ ' dEFor large barriers we may neglect £  in compari.son with so that Xb (" Xxu'^)become independent of £  in the integrand of et). (24). Hence(22)(23)(24)---------- -3 - ^ -  exp ( - 2;Kb WO J dEeB, \  1 /2 (25)(26)J, (B) =(2m*It is the maximum height of the barrier which is innucncing the tunnelling process and not its shape. This is because in the new approach, the barrier is regarded to miroducc an uncertainty in the energy of the incident electron equal to ius maximum height. This simplifies the problem of compuUilion of the tunnel current density. Although the shape o the barrier in the case of an M-l-M junction is a rectangular one and that in the case ol a634 P  N  R oy, A P  S ingh a n d  B P  Singhtunnel diode is a triangular one, the maximum height of the barrier in the former case is taken as Ko <^ nd in the latter case as the band gap energy.For the sake of integration appearing in eq. (25), the zero of the energy scale is taken at the Fermi level of the electrode to which the tunnelling is taking place. Thus taking the limits to be 0 and cV in the case of M-l-M junctions and 0 and AE in the case of tunnel diodes, we have 110]andwhere.1-A ^I/, m lunncl diode ~  / I  ( A E ) ^ }A =2'exp (- 2x ,  HO(27)(28)and AE = £i + £ 2 " cV, £j and £ 0  being Fermi level degeneracies on the n and p  sides of the tunnel diode respectively.Eq. (24) may be applied to find out the lunncl current density for a barrier of any height but in that case Xb cannot be independent of £  and will have to be included in the integral. This yields a lengthy expression but it is qualitatively similar to eq. (27).R g u r c  1* J,(B) - V ( rev e rse )  cu rv e  fo r a G c  lu n n e l d io d e  in c ro sse d  c tc c in c  an d  m a g n e tic  f ie ld s  ( la rg e  b a rrie rs )The total tunnel current may be computed by taking the product of./, (6) with the area of the electrode to which the tunnelling is taking place.Considerations of the perpendicular component of energy may also be incorporated by splitting p (E) and E into one-dimensional and two-dimensional components. This too docs not change the results significantly.3. Conclusionsy, (B) -  V characteristics for a reverse biased Gc tunnel diode have been plotted in Figure 1 for different magnetic fields. As can be seen from the ligure the current rises monotonically with bias but decreases with increasing magnetic fields. Figure 2 shows the 7, (B) -  V characteristics for a forward biased InSb tunnel diode for different values of the magnetic fields. It is seen that the current dccrea.scs with increasing magnetic fields but the peak voltage docs not shift. Both the results pointed out above agree qualitatively with the observations of Roth ef al [3] and Calawa et al 12]. The characicristics for an M-I-M junction show that there is a small change even for high magnetic fields which may be possibly the reason for the early workers not having reported on such sysiems.Tunnelling in crossed electric and magnetic fields 635Figure 2. J f i B )  -  (lorward) curve for an InSb lunrlcl diode in crossed elecinc and magnetic fields (large barriers).The identity of the magnetic field dependent exponential obtained in this work and that in Zawadzki's work may be shown in the following nuMmereB  Iexp ( -  2xb W) = expI I U W I I l l U l C ' 1 \ / 1= exp ^ e B / i2ehe 2 m * V o636 P N  Roy, A P Singh and B P Singh= exp - 2 e f i e ^ ] (2 9 )V„where = -;? has been substituted, e lThe effective electric field appearing in eq. (29) may be expressed as e B ^  T 1/2and‘Z'cfl -  ^«-2 ^  effeBJhr 1 —1after binomial expansion (since eB^h « -2m *Substituting for a tunnel diode, we haveeBJi(30)■E^ eff = ( 1 -( )(31)(32)Hence, exp {- 2zb = exp y -  26 1^1:^We find that (Ecff in our case is different from the one appearing in eq. (7) obtained by Zawadzki and Lax [6]. The magnetic field dependent term is found to be proportional to m our case whereas it is proportional to in Zawadzki’s treatment.From eq. (31) we find the value of cut-off magnetic fi^ld( S )  -v^zJcui-off “ (33)at which l r^r = 0 which makes the tunnel current vani.sh.Defining the critical cyclotron frequency 0)^  (at which the tunnel current ceases) asCO,e  (Bz).cui-oirIt can be shown from eq. (33) that2 £ .  2 2(Or = h/E,( 34 )(35)because tunnelling time r is defined as_h________ hET = in the new approach.barrier heightThis conclusion is exactly the same as has been shown by Roy and Ghosh [1]. Thus a theoretical method for computing the tunnelling time is obtained.References11) D K Roy and Amitabh Ghosh 1988 Pramana J Pkys 31 45312] A R Calawa.R H Redilccr , B Lax and A L McWhorter 1960 Phys. Rev. U ti  5 55(31 11 Roth, W Bernard and W D Siraub 1966 Phys. Rev 145 667(4J W Zawadzki 1969 Tunnelling Phenomena in Solids eds. E Burstein and S Lundqvist (Plenum : New York) p219|5j A G Aronov and G E Pikus 1967 .SVjv. Phys Jin'P  24 18H [61 W Zaw adzki and B Lax 1 9 6 6  Phys R e v  L eU  16 1(X)1 1 7 ] E O Kane 1957 J. Phys. Chem. Solids ] 249(81 D K Roy 1986 Quantum Mechanical Tunnelling and lU A ppU calioni (I>hiladelph.a World Scicnlilie) (91 P N Roy P N Singh and D K Roy 1977 Phys U u  63A 81 [10] A P Singh 1988 PhD 'Iliesis (Bhagalpur University, India)Tunnelling in crossed electric and magnetic fields 637SA (8)

Tunnelling in Crossed Electric and Magnetic Fields

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Tunnelling in Crossed Electric and Magnetic Fields

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