Properties of the Landauer resistance of finite repeated structures

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Properties of the Landauer resistance of finite repeated structures
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  PHYSICALREVIEW B VOLUME 42, NUMBER 8 15 SEPTEMBER 1990-I Properties of the Landauer resistance of finite repeated structures M. Cahay Nanoelectronics Laboratoryand Department of Electrical and ComputerEngineering, University of Cincinnati,Cincinnati, Ohio 45221 S. Bandyopadhyay Department of Electrical and ComputerEngineering, University of Notre Dame, Notre Dame, Indiana 46556 (Received 25 January 1990; revised manuscriptreceived 12 April 1990) Several properties of the Landauerresistance of finite repeatedstructures are derived.Atheorem relatingtheenergies of unity transmission through a finite repeated structure to the band structure of aninfinite superlattice formed by periodicrepetition of the finite structure [Vezzetti and Cahay, J. Phys. D 19, L53 (1986)] is generalized to the case of structures with spatially varying effective mass. Wealso establish a sum rule for the Landauerresistances of periodic structures formed by periodicallyrepeatinga basic subunit. Finally, we derive an analyticalexpression for the  boundary resistance of a structure, as introduced by Azbel and Rubinstein in connection with pseudolocali-zation, and prove several properties of thisquantity. I. INTRODUCTION The Landauerformula' for calculatingtheresistance of a dissipationless mesoscopicstructure hasbeenusedquite widely in the study of quantum transport phenome- na.The formula relates in a simple way theresistance of a structure (in the linear-response regime) to theprobabil- ity of transmission of an electron throughthe structure. The usefulness of the formula lies in the fact that it reduces the problem of quantum mechanically calculat- ing resistance — rather difficult problem — o a muchsimplerproblem of calculating just the transmission probability. In this paper, we prove several interesting properties of theLandauerresistance (i. e. , theresistance in the linear-response regime) of a finite repeated struc- ture such asa semiconductorsuperlattice.Theseproper- ties are all derived fromthe properties of thetransmis- sion coefficient of an electron through aperiodic poten-tial of finite spatial extent. In Sec. II of this paper, we firstemploy atransfer-matrix technique to derive a general expression for the transmissionprobability of an electron through an arbi- trary potentialprofile. We then extend this result in Sec. III to calculatethe transmissionprobability ~ Ttt ~ of an electron through N subunits of a finite repeated structure. Using this expression, we extend an earlier result relat- ing the energies of unity transmissionthrough a finite re-peated structure to the energy — wave-vectorrelation for an infinite structure formed by periodically repeating the basic subunit of the finite structure. In Sec. IV, we prove a set of theorems that establishinteresting and useful re- lationships between the transmission probabilities (and hencethe Landauer resistances) associated with the sub- units of a finite repeated structure. These theorems are all illustrated with numericalexamples dealing with com- positional and effective-mass superlattices.In Sec. V, we establish a sum rule for theLandauerresistances of periodicstructures formed by successively repeating abasic subunit,and in Sec. VI, we derive an exact analyti- cal expression for the  boundary resistance of a struc- ture as introduced by Azbel and Rubinstein in connection with pseudolocalization. Finally, in Sec. VII, we summa- rize ourconclusions. II. TRANSMISSION OF AN ELECTRON THROUGH AN ARBITRARYPOTENTIAL g=(b(z)e (2) In this section, we first derive an expression for the transmissioncoefficient of an electron through an arbi-trary one-dimensiona/ potential of finite spatial extent. For the sake of generality, we allow for spatialvariation of the electron's eff'ective mass but assume it varies only in one direction.The time-independent Schrodinger equationdescribing thesteady-state (ballistic) motion of an electron throughsuch apotential is fi B g fi B g A' B 1 B1b 2m *(z) Bx 2m *(z) By 2 Bz m *(z) +E, (z) =Ef, (1) where E, (z) is the one-di. mensional potential that varies in the z direction and m*(z) is the spatially varying effective mass. In a semiconductor heterostructure, E, (z) is theconduction-band edgeprofilewhich incorporates any bandbending due to spacecharges, variations due to compositional inhomogeneity, and also variations due to any external electric field. Because theHamiltonian in Eq. (1) is invariant in the x and y directions, the transverse wave vector k, is a good quantum number. Furthermore, sincethe z component of the electron's motion is decoupled from the transversemotion in thex-y plane, the wave function P can bewrit- tenas 42 5100 1990 The AmericanPhysical Society  42 PROPERTIES OF THE LANDAUER RESISTANCE OF FINITE... 5101where k t = ( k, ky ) and p = ( x, y ). The z component of the wave function P(z) now satisfies the Schrodinger equation 2 pl dz y(z) dz + [E +E, [1 — y(z) ']   (. „-) y(z„) dz (t (z„) ~(n) pr(n) ' 11 12 pr( n) ~( n) 21 22 1 d(tt + ) y(z„+, ) dz P(z„+, ) (4) where 8'  ' are the elements of the transfermatrix, and z„+, and z„stand for z„, +e and z„— , respectively, with e being a vanishinglysmall positive quantity. Expli- cit expressions for the elements of the transfermatrixare givenin theAppendix. Assuming continuity of P(z) and [I/y(z)]/(dtttldz) everywhere in the structure, the overall transfer matrix W ' describing theentire region [O, L] (see Fig. 1) can be foundby simply cascading (multiplying) the individual I I IIII IIIIII III I II I I Ie I I I II contact I I I 0 z, z2 n n+1 '' contact I E FIG. 1. An arbitrarypotential profile approximated by a series of potential steps. Within each interval,the potential and e6'ectivemass are assumed to be spatially invariant. E, — (z) I tI)(z) =0, (3) where m,* is the effectivemass of the electrons in the  contacts sandwiching the region of interest (m, ' is spa- tially invariant within the contacts and isotropic), y(z)=m'(z)/m, *, E, =iri k, /2m, *, and E is the kinetic energy associated with the zcomponent of themotion in the contacts (E =Pi k, /2m, *). The above equation cannot be solved exactly for an ar- bitrary potential E, (z) How. ever, an approximate solu- tion can be found by approximating the potential profile by a series of potentialsteps (see Fig. 1) or byusing a piecewiselinear approximation for the potential. In the former scheme,the region over which thepotential variesis broken down into a finite number of intervals.Within each interval thepotential and the effective mass are as- sumed to be constant. In that case, the wave function and its first derivative at the left and right edges of an interval are related through aso-called  transfer matrix,   charac-teristic of that interval, whoseelements donot dependon the z coordinate and can bedetermined analytically. The transfer matrix for the nthinterval [z„„z„] sdefined according to transfer matrices for the individual intervals: gytot ~( X) ~( 1 ) (5) where 8 ' is thetransfer matrix for the nth interval as definedin Eq. (4). The overall transfer matrix 8 relatesthe wave func-tions and their first derivatives at the left and right con- tacts: dtt (L+) 1 d((t y(L+) dz y(0 — ) dz ttt(L+)  (0 ) In Eq. (6), ttt(0 ) and P(L+) are the electronic states inside the leftand right contacts. Theyare given by' Ekoz — ikoz e ' +Re  , z(0 iko(z — L) (7)Te ', z)L tt(z)= ' where ko [=(2m,  E /trt)' ] is the z component of the electron's wave vector in the contact and R and T are the overallreflection and transmission coeScients throughthe region [O, L]. Using these scatteringstatesfor the wave functions at z =0 andz =L+ and noting that, bydefinition, y(L+)=y(0 )=1, we obtainfrom Eq. (6)iko T ~tot 1 iko(1 — R) 1+R Equation (8) finally gives us the twoequations for the two unknowns T and R. From these twoequations T and R can be found by straightforward algebra. Eliminating R gives 0 1122 12 21 'I ( WtotWtot Wtot Wtot } ik ( W' + W ')+( W' king — W' ) (9) where 8',  ' are the elements of the matrix W' that are found from Eq. (5). Since 8 is aunimodular matrix, theterm within parentheses in thenumerator ofEq. (9) is unity. In addi-tion (see theAppendix), W~ is alwayspurely real.Therefore Eq. (9) gives [ T/'= 4ko k 2( Wtot + Wtot )2+ ( Wtotk 2 Wtot )2 0 ll 2221 0 12 (10) III. TRANSMISSION OF AN ELECTRON THROUGH A FINITE REPEATED STRUCTURE Havingfound a general expression for ~ T~, we now proceed to evaluate the transmissionprobability (and hencethe Landauer resistance) associated with a finite re-peated structure formed by the periodicrepetition of a structure with arbitrarily varying potential. Considerthe potential profile in Fig. 2 formed by theThe aboveequation givesus a general expression for the transmission probability of an electron through an ar- bitrarypotential. The transmissionprobability ~ T~ is, of course, related to the reflection probability ~R~ accord- ing to therelation ~T~ + ~R~ =1 as required by currentconservation.  5102 M. CAHAYAND S. BANDYOPADHYAY 42 Ec(z) I +sUBUNIT — N-1 ~~~ X h 1 2e' ~T~ ' h [sin (NO)] 2e kpW2, — W, 2kpsinO — +1- FIG. 2. The potential profile for a finite repeated structure formed by periodic repetition of a region with arbitrarily vary-ing potential. R4 h x 2e' /T, , ' =R h L where kp is the wave vector of the incident electron. periodic repetition of an arbitrarypotential. Every  period in this structure has the same transfer matrix (say W) characterizingthat period and the grandoverall transfer matrix 8 describing theentire structure is, as before, obtained by cascading the transfermatrices for the individual periods. It is easy to see that for a struc-ture with N periods with each periodidentical, Wtot — ( W)N As shown in Ref. 10, the elements of the matrix 8 can be expressed in terms of the elements of thematrix W IV. TRANSMISSION THEOREMS FOR A FINITE REPEATED STRUCTURE We now prove a set of theoremsrelated to transmission through finite repeated structures. First, we prove a theorem that relatesthe energies of unit transmission (i. e. , the values of theincident energy for which the transmissioncoefficientis exactly unity) through a finite repeated one-dimensional structure, to the band structure of the associated infinite lattice formed by periodic repeti-tion of the one-dimensional structure. This theorem was stated for the first time in Ref. 5. A more detailed proof isgiven here with generalization to the case of a structure with a variable effective mass. W«, W sin(NO) sin[(N — 1)O] (12) where I is a 2 X 2 identity matrix and 8 depends on theei- genvalues of the matrix W andis givenby exp(iO)=A, =A& '= — + r( W) Tr( W) 2 '2 1/2 (13) ~Tz~ = [sin (NO)] 2 '2 k p W2] W]2 2kpsinO — +1 (14) whichis our main result. The two- and four-probe (2-p and 4-p) Landauer resis- tances for astrictly one-dimensional repeated structurecan now befound easily by substituting Eq. (14) for the transmissionprobability ~ Tz~ in the single-channel Lan- dauerformula: where A, ] 2 are the eigenvalues of the 2 X 2 matrix W and thesecond equalityfollows from the fact that the matrix W is unimodular. Wecan now find the overalltransmissionprobability ~TN~ through a periodicstructure with Nperiods. For this, weuse Eq. (10) with the elements of W ' now givenby Eq.(12). This gives Theorem I. The transmissioncoefficient of a particle through a periodic structure, formed by N repetitions of a basic subunit, reaches unity atthe following energies: (a) energies at which thetransmissionthrough the basic sub- unitis unity,and (b) N — energies in each energy band of the lattice formed by infinite periodic repetition of the basic subunit,where these N — energies are given by E =E, (k =+nrrlNL)(n =1, 2, 3, ... , N — 1,and L is the length of a period). Here E, (k) is the energy — wave- vector relation (or the dispersion relation) for the ith band of the infinite lattice. l=~T, ~ = . [sin (O)] 22 k P W2] W]2 2kpsin0 — +1- (17) Part (a) of thetheorem is actually fairly obvious. All it states is that by connectingidenticalstructures of transmission unity, one always obtains unit transmission throughthecomposite structure. Although this is intui- tive, we proveitnevertheless for the sake of cornplete- ness. For this, we first note from Eq. (14)that the transmission ~ T~ ~ through N periods reaches unity when the term within the largesquare brackets vanishes. The term within the largesquare brackets vanishes when 2 2 k o W2] — W]2 (16) 2kpsinO We now show that this corresponds to the conditionthat ~ T, ~ (i. e. , the transmissionthrough one period, or the basic subunit) is unity. Substituting N =1 in Eq. (14), we get that the condition for unit transmission throughone subunit isgiven by  42 PROPERTIES OF THE LANDAUER RESISTANCEOF FINITE. . . 5103 1. C. , + 2m + 3m (N — 1)m W'e now have to prove that the above values of8 also cor- respond to the wave vectors k =+n~/NL where L is the period. For this, we firstapply the Bloch theorem to the infinite structure.The Bloch theorem gives P(z +L) =P(z}exp(ikL), where k satisfies the relation (20) det[ WJ 5„exp— ikL)] =0 . (21) In the above equation, 8', - is the ijth element of the transfer matrix W describing oneperiod and 5, is a Kronecker delta. From Eq. (21) we immediatelysee that exp(ikL) is the eigenvalue of the 2X2 unimodular matrix 8'and hence exp(ikL ) = I, , = A, 2 ' = + r(W) 2 Tr( W) 2 1/2 (22) The right-hand sides of Eqs.(13) and(22) are identical so that their left-hand sides must also be identical. Therefore exp(ikL) =exp(i8), (23) or which, after simplification, reduces exactly to Eq.(16). This proves the first part of the theorem, viz. , that the en- ergies of unit transmission throughoneperiod are alsothe energies of unit transmission through all thepcl iods. To prove the second part of the theorem, we note from Eq. (14) thatthe transmission ~ T~ ~ also reaches unity for those values of0 that satisfy the conditions sin(N8) =0, sin(8}WO; The locations of the band edges can be found directlyfromthe following property, which we prove: The states characterized by wave vectors k for which ~Tr( W}~ ) 2 are theevanescent states corresponding to the  stop band of a finite repeated structure.The states charac- terized by wave vectors k for which ~Tr( W)~ &2 are thepropagating states corresponding to the  passband of the finite repeated structure. To provethe property, we invoke Eq. (22). If ~Tr( W) ~ & 2, then the right-hand side ofEq. (22} is purely real and greater than unity. In that case, the wave vector k must be purely imaginary which means that the state is an evanescent state corresponding to the  stop band of the finite repeated structure. On the other hand, if ~Tr(W)~ &2, the right-handside of Eq. (22) becomes complex which permits k to be real.In the latter case, the state isa propagating state corresponding to the  passband of the structure.The values of wave vector k for which ~Tr[W]~=2 evidently correspond to the edgesbetween the pass bands and the stopbands. Theorem II. At the energies of unity transmissionthrough a finite repeated structure with N periods,the followingequality holds: ~ Ttv ~ = ~ Ttt ~ whenever 1 2 N, +Nz=N. Here ~Ttv ~ and ~Ttt ( are thetransmis- sion probabilitiesthrough two subsections with N& and N2 periodsrespectively. As stated in the proof of theorem I, the transmissionthrough N periods reaches unity under two condi-tions: (a)when thetransmission through each of the N periods is unity, and (b) when 8=+ (n =1, 2, 3,... , N — 1) . ~ (25) In case (a), the proof of theorem II is trivial. If the transmission through each period isunity, then, of course, the transmission through any arbitrary number of periods is also unity. In that case, obviously, (26) kL =8(mod2vr) . (24}Consequently whenever k =+n m/NL, the quantity 8 =+n m /N. Thusthe energies corresponding to k =+trlNL, +2m INL, +3m INL, ... , +[(N — 1)7r)/NL are the energies corresponding to 8=+m /N, +2m/N, +3ttlN, ... , +[(N — )m. ]IN, which, in turn, are the energies corresponding to unity transmissionthrough the finite repeated structure with N periods as previously noted. Stated in other words, thismeans that the energies associated with unity transmissionthrough an N-period structure are the bandenergies E(k„) corre- sponding to the wave vectors k, =+n ~/NL in an infinite repeated structure. This gives us the E(k„)-versus-k„re- lation and proves the theorem.The usefulness of theorem I lies in the fact that by evaluating theenergies of unit transmission througha finitestructure [which we can do from Eq. (14)], we cancalculate the band structure of an infinite superlatticeformed by theperiodicrepetition of the finite structure. regardless of what N, and N2 might be. This provesthetheorem forcase (a). The proof for case (b) proceeds as follows. We first note that sinN, 8 = sin(N Nz )8= sin(+ n n — N28 )— =( — 1)  +'sinN28, (27) where we used Eq. (25) to obtain the second equality. Us- ing the above equality in Eq. (14), we immediately see that which proves case(b). Theorem III. At the energies of unity transmission (~T&~ =1) through a finite repeated structure with N periods, the following equality holds: ~ Tz+M ~2 for all M such that 1 &M & N.  5104 M. CAHAY AND S. BANDYOPADHYAY 42 The proof of this theorem is very similar to that of theorem II andis thereforenotpresented. A. Numericalexamples To illustratetheorem I, we show in Fig. 3 the construc- tion of the energy-band diagram of an infinitely repeated structure whose basic subunitis shown in the inset. The points Q, Q' are the twolowestenergies at which the transmission through two subunits is unity,whereas thepoints P, P' and R, R ' are thetwo lowestenergies for which the transmission through three subunitsis unity. These points are on the two lowest-energy bands. Other points on the energy-band diagram can be found similarly by steadily increasing the number of periods and search- ing for the energies of unit transmission.Finally,thepoints PI, P~ and Q„Q2 correspond to theband edgesand are found from thecondition ~Tr( W) =2. To illustrate theorems II and III, we provide thefol- lowing numerical examples. E xample 1. We have calculatedthe transmission ~ T~ ~ [using Eq. (14)] through a compositional superlattice con- P2 C 0. 3ev l soA 0. 3 ) ~ 0. 2— CC LLI z k 0. 1— IIIIII 'k IIIIII I = m/4L r I I IIIII II IIII III I = 7tj2L I IIII , ' k I IIII t I I I III 370)'4L ', Q, p , Q ', R I I II I 0. 05 0. 0079 0. 0157 0. 0236 0.0314 k = 70'L BLOCH WAVE NUMBER (A ' } FIG. 3. Ener gy-band diagram of an infinitely repeated struc- ture whose basic subunit is shown in the inset. The conduction band is constructed by numericallyevaluating theenergies at w ich the transmission throughincreasing number of periods go to unity. The points Q, Q' correspondto the two lowest ener- gies at which transmissionthrough two bo su units is unity, whereas thepoints P, P' and R, R' correspond to the lowesten- ergies for which transmissionthrough three subunits is unity. The pomts P, , P2 and Q, ,Q2 correspondto theband edgesand are found fromthe condition Tr[ W] =2. Theorem IV. If the Fermi energy of a finite repeated one-dimensional structure lies at the boundary between a  passband anda  stop band,   then thefour-probe Lan- dauerresistance of N periods of the structure is equal to N times the four-probe Landauer resistance of one period.This means that the four-probeLandauer resis- tance increases with the structure's length as L instead 0. 15 LU 0 0. 10— U LU 0 O Z', 0 (0 CO 5 005- (0 Z K I— I I II I I II II III I I II III I I I IT3)~ III I I I II I 0 3eV I 50A II I I I II 0 105 0. 00 0. 075 0. 081 0. 087 0.093 0. 099 ENERGY (ev) FIG. 4. Transmissioncoefficientsthrough a periodicstruc- tureformed by repeatingthe subunitshown in theinset. The subunit consists of a GaAs well andan Al, Ga As barrier both 50 A thick. The barrier heightis 0. 3 eV and the effectivemassisassumed to be 0. 067mo everywhere. Notethat when IT31'=1, T, '=IT, /'. Also whenever [T /'=1, /T f'=/IT2/I' These illustrate theorems II and III, respectively. s&sting of rectangular wells and barriers in which the bar- rier and well thicknesses are 50 A. The effective mass was assumed to be 0. 067mo everywhereand the barrier height was taken to be 0. 3 eV. Figure 4 shows the transmissioncoefficient through one, two, and three bar- riers in the vicinity of the lowest resonant energy through two barriers. (Resonant transmissionthrough two bar- riers has been studied extensively in connection with thedouble-barrierresonant tunneling diode. ' ) Figure 4 is a clear illustration of theorem III. It shows that when the transmissionthrough two barriers is unity, the transmissionthrough threebarriers is equal to the transmissionthrough one barrier, i. e. T, I2 ~ with N = 2and M = 1. Figure 4 also shows that whenever ~T3~ =1, ~TI ~ =~T2 illustrating theorem II for the case N = 1, N =2. ~ 2 Exam le 2. . In Fig. 5 we show the transmissionthrough an effective-mass superlattice' in which theconduction-band edges in the differentlayers are assumed to be aligned but the effective masses are different. We assumeeffectivemasses of 0. 039mo and 0. 073mo, respec- tively, in two alternating layers.(These correspond to thetransmissionsthrough one,two, and three layers were calculated from Eq. (14) at theresonant energy through threelayers.Clearly, when ~ T3 ~ = 1, ~ TI ~ = ~ T2 This illustrates theorem II. Also when ~T ~ =1, T ' =(T3( ~ 1 3 as stated in theorem III.
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