Choudhary, R B

Roy, P N

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Indian J. Phys. 72A (1), 23-30 (1998)IJP Aan international journalTunnelling current across a double barrierPNRoyDeportment of Physics, T N B College, Bhagalpur-812 007,Bihar, IndiaandR B ChoudharyDepartment of Physics, S S College, Mehush (Sheikhpura)-811 102, T M Bhagalpur University, Bhagalpur, Bihai, IndiaReceived I August 1997, accepted 16 September 1997Abstract : An analytical expression for tunnel current density across a double barrier has been obtained under non-resonant conditions. The derivation is based on the ideas of quantum measurement There is a good agreement with observed results in the nature of current-voltage and differential conductivity-voltage characteristics.Keywords : Tunnel current density, double barrier, quantum measurement PACS No. : 73 40.Gk1 . IntroductionTunnelling across a double barrier was studied experimentally by Chang ei al [1] in which they had observed resonant tunnelling under suitable conditions. Esaki and his coworkers [2-4] later applied the conventional theoretical models to explain their observations in such devices which find their application in superlattices. In recent times, Roy et al [5,6] have applied the ideas of quantum measurement model to study this problem.Double barrier tunnelling continues to evince interest in workers even now. Vanhoof and his coworkers [7] have studied spatially indirect transitions due to coupling between hole accumulation layer and a quantum well in resonant tunnelling diodes. Kuznesov et al [8] have studied the effect of electron-electron interactions on the resonant tunnelling spectroscopy of the localised states in a barrier. Silvestrini et al [9] have studied resonant macroscopic quantum tunnelling in SQUID system. Alonzo and his© 1998 IACS24 P N Roy and R B Choudharycoworkers 110] have presented a tunnelling spectroscopy of resonant interband tunnelling structures. Song [11] has presented a transition layer model and applied it to resonant tunnelling in hetero-structures.In the present study, the quantum measurement model of Roy and his coworkers [5,6J has been used to derive an analytical expression for tunnel current density across a double barrier. Since the earlier workers had observed negative differential conductivity (n.d.c.) under non-resonanl conditions in such systems, the main purpose of this work was to find a theoretical expression which could lead to n.d.c. effect.The quantum measurement model differs from the conventional model fundamentally The conventional model seems to rest upon the idea that the electron energy must remain unaltered throughout the tunnelling process. But tunnelling of particles is not a continually observable process. The tunnelling particle can be reckoned only after a definite time r measured from the instant of incidence of the particle upon the'barrier because of a finite time becoming necessary for potential energy estimation as required by Heisenberg’s uncertainty relation. The electron is then able to recover its wave or particle shape that it once lost while making tunnelling transition. In other words, we may regard tunnelling as a process ot quantum measurement being carried out by the barrier. Both energy and time being conjugate variables, simultaneous and accurate estimation of them is not possible because of Heisenberg’s uncertainty relations. So, if a time r elapses in reckoning the tunnelling process, the electron energy at the conclusion of the process must be uncertain by f i / r. The electron energy is expected to undergo a fluctuation of (V{) -  E) energy around its original value E where V0 represents the height of the barrier. This idea of quantum measurement has been successfully applied by Roy and other workers [12,13) to different tunnel devices.2. Tunnelling across a double barrierThe electronic wave functions in various regions of a double barrier system (Figure 1) can be written asi//,(*) = a, etk]X + b{ e~tk'x; -°o < jr < x, 0)if/2(x) = a2e 'xi (x' x' } + b2e*iu"Ti); jr, <* <Jt2 (2)y {^x) = + b^ e‘,k'ix~X2)\ x2 <x< (3)\j/A{x) = + bAex*{i~*']; x^  <x<xA (4)Jt4 <*<» (5)where ,2 ,2 12 2m*Ek? = kl = kt = - h Y (6)and2 m‘ (V0 -E)x l = x l =  J (7)Tunnelling current across a double barrier 25Here m* is the effective mass of the electron in the double barrier system and V0 is the barrier height.Figure 1. Double barrier systemMatching the wave functions and their first derivatives at different boundaries, one finally obtains an expression for the tunnelling probability [14,15] asZ = V k W x U i° 1  ( * 2  + * , J ) U l  + * 3 ) ( * 4  +  * ? ) ( * 4  + * ? ) l  * 1  I 2( 8 )when-where1___1ri­ll ') sin2 (0 , + 04 - f t , w , )sin3 (0 , ~ 04 - ft, c u , )!~*4WJ sin2 (0 , - 04 + k l , u . )! ) sin 2 (0 , + 04 4 A, I ° M ^-2*., w, -  f  x : z cos 2 0 ^ .•sin(0 , + 04 M,) sm(0 , - 04 -  ft.+  2 s 2^ " - 1 cos 20 2 -Slll(0^ + 04 -A, w ,) sin(0 , — 04 +ft- 2cos[2(0 : + 0 ,)}.sin(0 , + 04 - ft,ai,) sin(0 , + 04 + * ,« , )  - 2 cos[2(03 -  0 , )}.sin(0 1 - 04 -  kx(Ox) sin(0 , - 04 + * ,w ,)+ 2e~x4co4 cos20 , .sin(0^ - 0 4 ~ A: to j ) sin(0 , + 0 4 +k^w^)- 2e 2*-w- cos20_s.sin (0 ,-04 +A,eu1)sin(0,+04 + A/u-,)j, (SI)tan _l tan-1*2_[ X a and s(10)26 P N Roy and RB ChoudharyIi is found that a resonance is obtained i.e. Z= 1 when the following conditions are satisfied simultaneously0) *, = X 7 = * j = Z 4 =nwhich leads to = 0 < = 04 -  Ps -  *(ii) X 2 <02 = X * a)Aand (lii) k (^0  ^ = n^ + jj7C wheren = 0, I, 2 . . . .3. Tunnel current densityRegions 2 and 4 of Figure I are barriers whereas region 3 is a potential well. The current in this system Hows by the quantum measurement mechanism through regions 2 and 4 an<4 by conventional mechanism through region 3. The principle of continuity suggests that fhe current densities through all these regions must be the same.The one-electron tunnel current density through region 4, generated by quantum measurement process, is given by [ 13]s in  to ,  r  , vm iq. ' lhXd  i t  •where V0I = ----- | u4 e -7*‘a ‘ . ( 12)mwhich on lurther simplification leads to•'«, * (* 5 + ^ ) z l« , I2- 03)X 4~^ r  2m*w/ r ------- ----- , r =  — — the tunnelling time.h hxlWhen a group of electrons having a random phase difference amongst themselves is incident upon the double barrier system, the differential tunnel current density is given by [12,131dJ(E) = p l ( E ) f , ( E ) d E ^sin to. r7 ~ + V  Sin K r+61) (14)where p/ (E)ft (E) dE is the density of the wave group at the incident end The minimum phase difference at ihe transmitted end isTunnelling current across a double barrierwhere € r is the difference in consecutive energy levels at that end. Thus, the summation of (14) can be converted into integration asdJ(E) = p l (E) f l (E)dE f - ^ - 1U rr  s in  CO, X .f — —J Cl), T '+J<n Jsin(<u|rT+0)rf(<BJi r)which finally leads to (12,131n hdJ(E) ^  I — “ \Jq\ P, (E)f,  (E)dE.But e t can be expressed in terms of the density of slates pr as—  = p r (£)[l - f r (E)]n,(16)(17)where i2 is the volume of the electrode at the transmitted end. Substituting (17) in (16), we geldJ{E) = - f r(E)]p, (E)pr (E)JE.For absolute zero temperature, f ( (E)[l -  f  r (E)] = 1 and hencedJ(E) = E M L j ^ Pi{E)Pr{E)dE(18)(19)Substituting for J0] from (13) and puttingT =2m* = hH i  = Vo ~ E, one finally obtainsdJ(E) =2 29/2 nLlqin**!2 1 a, |2 (V^0 — E)5?2 dEJC, | 5After certain simplifications, it is found that| K} |2 = 4 sin2 (202 -  k^co^ I ?4*?"2 t where x 2 a ) 2 =Thus, one gets£ 5(Vn - E ) ^ 1 e~KiV» 'E)',:dJ{E) = P ------- --------------------- ------ dE,sin2(202 -  )(20)(21)(22)28whereP N Hoy and H B Choudhary 2 19/2 JtQqm**12 | a { |2P =h r' V 1n v 0and K ^4V2m‘l/::w,(23)The lunne! current density can be expressed asH} {VU - £ )V 2 -£')'/= dEJ(E (24)sin: (20: -  k^0)?)where ^ is the applied bias. fcq. (24) has been obtained for non-resonant conditions.4. Results and conclusionsThe integration was done numerically with the help of a computer The ciirrcnt- vnliage characteristics lor AIGaAS-GaAS-AIGaAS as double barrier system, arc shi^wn inFiRure 2. Currcm-volt^e ihunictensucsg s 2 and 3 tor different ranges of biases. Negative resistance regions arc ly seen in these plots. Figure 4 shows the differential conductivity-voltagecharacteristics. One finds a good agreement so fai as the nature of these characteristics are concerned.(ftliun Ajdjiiqjo) f ^______ £ iSl ur. ^o-ngjo) 'P/fpTunnelling current across a double barrier 2 9fcure 3. C.iirrcnt-voltage characteristics.Figure 4. differential conductivity-voltage charactensticsP N Roy and R B ChoudharyAcknowledgmentThe authors are thankful to Dr. D K Roy of Indian Institute of Technology, Delhi forhis valuable suggestions.R efe re n c e s11 ] L L C hang , L Esaki and R Tsu Appl Phys Utt. 24 593 (1 974)[21 L Esaki and R Tsu HIM J Res Develop 14 61 (1970)|3J L  E sak i and I. L ( 'h a n g  Phys Rev Lett 33 495  (1 974)|4 |  R Tsu and L Esaki Appl Plm Leit 22 562  (1 973)[5] D K Roy. N S T  Sai and K N Rai Pramarui 19 231 (1 982)[6 | D K R o l a n d  A G h o sh  Proi IV Ini Workshop on Phys of Semicond De\ IIT  Madras (India) eds S C  Jam  and S R adh ak n sh n a  p 520  (1 987)[7J C  V anhoof. J G cnoc, J C Portal and  G B orghs Ph\s. Rev. B51 14745 (1 9 9 5 ) j[8 1 V V K uznesov  A K S avchenko , M  E  R aikh and L  J G lazem an  Phys Rev B 54  1502 (1 9 9 6 ) \M  S ilv cs in m , R R ug g iero  and V M O vch in n ik o v  Phys. Rev B 54  1246 (1 9 9 6 ) 1110| A ( A lonzo , D A C ollin s  and T  C M cgill Solid Stale Commun. 1 0 1 (B ) '60 7  ( 19 9 7 ) \| I J |  Y S ong Phys left A 216 1 8 3 (1 9 9 6 )112| I) K Knv Quantum Met humi til liumellmf; and its Applications (P h ila d e lp h ia  W orld  S c ien tif ic ) ( 19 8 6 )I H I P N R oy, P  N S ingh and D  K Roy Plm h it  A 63  8 1 (1 9 7 7 )114| hmellmx Phenomena in Soli,Is cd s  E R u tstc in  and  S L u n d q v is t (N ew  Y ork  . P lenum ) (1 9 6 9 )1151 I)  K Hoy and A S ingh Inr ./ Motl Plm 2 3 4 0 ) 9 ( 1 9 9 3 )30

Tunnelling Current Across a Double Barrier

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Tunnelling Current Across a Double Barrier

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