Localization in one-dimensional relativistic quantum mechanics
LLocalization in one-dimensional relativistic quantum mechanics
Abhay Mehta, Sandeep Joshi a ,
2, 3 and Sudhir R. Jain b2, 3, 4 St. Xavier’s College, Mumbai 400001, India Nuclear Physics Division, Bhabha Atomic Research Centre, Mumbai 400085, India Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400094, India UM-DAE Centre for Excellence in Basic Sciences,University of Mumbai, Vidynagari Campus, Mumbai 400098, India
We present the relativistic analogue of Anderson localization in one dimension. We use Diracequation to calculate the transmission probability for a spin- particle incident upon a rectangularbarrier. Using the transfer matrix formalism, we numerically compute the transmission probabilityfor the case of a large number of identical barriers spread randomly in one dimension. The particularcase when the incident particle has three component momentum and shows spin-flip phenomena isalso considered. Our calculations suggest that the incident relativistic particle shows localizationbehaviour similar to that of Anderson localization. A number of results which are generalizationsof the non-relativistic case are also obtained. Keywords:
Anderson localization, relativistic quantum mechanics, Dirac equation, transfer ma-trix.
I. INTRODUCTION
Anderson localization [1] has occupied centrepiece in the development of condensed matter physics.The beautiful state of the art argument by Anderson has been simplified a long time ago by Mott andTwose [2]. In addition, Anderson localization appears in quantum chaos as dynamical localization inkicked rotor [3] and Fermi-Ulam model [4, 5]. This phenomenon is also experimentally demonstratedin the physics of cold atoms [6, 7]. In the context of disordered solids, Anderson’s work promptedmany papers and experiments on localization [8][9][10][11].Mott and Twose simplified the Anderson’s model by considering an infinite array of rectangularpotential barriers with some disorder incorporated through the positions of the barriers or via theirheights. Anderson had introduced disorder in the structure by arbitrarily varying the energy ateach lattice site. Here, we consider the relativistic equivalent of Anderson model and consider thesolutions of the Dirac equation. We first calculate the transmission probability of an incident spinone-half particle over a finite rectangular barrier and subsequently, compute the same for a numberof identical barriers arranged randomly in space. Relativistic quantum mechanics presents somesituations which have no analogue in non-relativistic quantum mechanics. For instance, for a one-dimensional arrangement of barriers, if the momentum of the incident particle is taken with onecomponent along the arrangement of barriers, the spin of the particle does not flip. On the otherhand, if all momentum components are considered, then the transmitted particles would includethose also whose spin is flipped [12].Moreover, in the relativistic case, the tunneling for energy below the top of the barrier will beaccompanied by the complications arising due to pair production and Klein paradox [13, 14]. In thiswork, we do not address these situations. Thus, the energies considered are above the barrier.The layout of the paper is as follows: Section 2 defines and introduces the problem, Section3 presents the formalism and calculations for a single barrier followed by results on localization.Section 4 repeats the methodology followed in the former section but for the case of spin-flip.Section 5 concludes with the main results of the paper. a [email protected] b [email protected] a r X i v : . [ c ond - m a t . d i s - nn ] M a r II. DIRAC PARTICLE OVER A RECTANGULAR BARRIER
Consider a spin- particle, propagating along the z axis, incident upon a rectangular potential V of width a . The motion of the particle is described by the Dirac equation [13] : (cid:18) γ µ ∂∂x µ + mc (cid:126) (cid:19) ψ = 0 , (1)where ψ is a four -component wave-function and γ µ , with µ = 1 , , ,
4, are 4 × γ k = (cid:18) − iσ k iσ k (cid:19) and γ = (cid:18) I I (cid:19) , (2)where σ k are the three Pauli matrices and I is a 2 × a is defined as V ( z ) = , z < ,V , < z < a, , z > a (3) V E I Ψ III Ψ II Ψ FIG. 1. Pictorial representation of a Dirac particle incident on a potential barrier with
E > V . Thedarkened area represents the area over which the particle is above the barrier. In order to avoid the realm of the Klein Paradox [13, 14], we take the energy E of the incidentparticle, such that E > V − mc . We employ the transfer matrix approach to find the transmissionand reflection coefficients. We consider two cases for momentum of the incident particle :( a ) par-ticle carrying the momentum only along the incident direction and ( b ) momentum along all threedirections. For the second case the particle can undergo spin-flip upon incident on the barrier [12]. III. ONE-COMPONENT MOMENTUMA. Transfer matrix formalism
The general positive energy solutions of the Dirac equation in the three different regions in Fig.1 can be expressed as a superposition of plane waves:Ψ I ( z ) = Au ( p ) e ipzz (cid:126) + Bu ( p ) e ipzz (cid:126) + Cu ( − p ) e − ipzz (cid:126) + Du ( − p ) e − ipzz (cid:126) , Ψ II ( z ) = Eu ( q ) e iqzz (cid:126) + F u ( q ) e iqzz (cid:126) + Gu ( − q ) e − iqzz (cid:126) + Hu ( − q ) e − iqzz (cid:126) , (4)Ψ III ( z ) = P u ( p ) e ipzz (cid:126) + Qu ( p ) e ipzz (cid:126) + Ru ( − p ) e − ipzz (cid:126) + Su ( − p ) e − ipzz (cid:126) , where u and u are the usual positive energy spin-up and spin-down spinors [13]. Here p z is themomentum of the particle in free space and q z is the momentum of the particle over the barrier.Let us first consider the case when the particle carry only the momentum component in incidentdirection. For brevity, we shall henceforth write p z and q z as p and q respectively. It is well-knownthat there is no spin flip [14] in this case. This allows us to simplify (4) by ignoring the degeneraciesin spin. Eq. (4) is then rewritten as :Ψ I ( z ) = Au ( p ) e ipz (cid:126) + Cu ( − p ) e − ipz (cid:126) , Ψ II ( z ) = Eu ( q ) e iqz (cid:126) + Gu ( − q ) e − iqz (cid:126) , (5)Ψ III ( z ) = P u ( p ) e ipz (cid:126) + Ru ( − p ) e − ipz (cid:126) . According to the definition of the transfer matrix [15], the wave-function on both sides of thepotential are connected by the equation: (cid:32)
P e ipa (cid:126) Re − ipa (cid:126) (cid:33) = M (cid:18) AC (cid:19) . (6)It is now possible to solve for the continuity of the Dirac equation at z = 0 and z = a and obtainthe transfer matrix M . We can, however, simplify the calculation by calculating the transfer matrix M step of the potential step at z = 0. The transfer matrix M step is given by: (cid:18) EG (cid:19) = M step (cid:18) AC (cid:19) . (7)We solve (5) by evaluating the spinor u and obtain the matrix equation: (cid:18) p − p (cid:19) (cid:18) AC (cid:19) = (cid:18) rq − rq (cid:19) (cid:18) EG (cid:19) . (8)where r = E + mc E − V + mc = 11 − V E + mc (9)Solving (8) yields M step as: M step = (cid:18) + p rq − p rq − p rq + p rq (cid:19) (10)The transfer matrix M for the rectangular potential can now be found with : M = M − step .M .M step , (11)where M is the transfer matrix of propagation inside a barrier of length a , M = (cid:32) e iqa (cid:126) e − iqa (cid:126) (cid:33) . (12)From (11), the transfer matrix M is given by : M = (cid:18) u vv ∗ u ∗ (cid:19) ; (13)the elements u and v being u = cos (cid:16) aq (cid:126) (cid:17) + iα + sin (cid:16) aq (cid:126) (cid:17) (14) v = + iα − sin (cid:16) aq (cid:126) (cid:17) where α ± = 12 (cid:18) prq ± rqp (cid:19) . (15)The transfer matrix M maintains both time reversal symmetry and conservation of current density.This is verified by the calculation of the following relations [15]:Det M = 1 (time-reversal symmetry), (16) M † (cid:18) − (cid:19) M = (cid:18) − (cid:19) (conservation of current density). (17) B. Transmission over one barrier
Due to the degeneracy in spin, the transfer matrix for the relativistic case has the same formulationas that of the non-relativistic one. The transmission coefficient T for a particle transmitted througha single rectangular barrier [15] can simply be obtained from the transfer matrix M as: T = | t | = 1 | M | (18)Using the expression for M from (13) we obtain T as: T = (cid:20) (cid:18) prq − rqp (cid:19) sin (cid:20) aq (cid:126) (cid:21)(cid:21) − (19)where the initial wave-functions are described in (5). Equation (19) reduces to the non-relativisticlimit obtained from Schrodinger’s equation under two conditions. First, if r → V → E (cid:29) V , and second if r → ± p q . Thelatter being the ratio of kinetic energy in two regions in the non-relativistic limit E ≈ p / m . Wecompare results given by equation (19) to its non-relativistic counterpart for dependency on theincident momentum of the particle in Fig. 2. ���� (cid:1) �� ( MeV / c ) | T | RelativisticNon - Relativistic
FIG. 2. The plot shows the transmission probability of an electron over one barrier for increasing values ofincident momenta ’p’ of the particle calculated separately using the relativistic and non-relativistic formula.Here the height and width of the barrier are 0.10 MeV and 350 fm respectively. The transmission in therelativistic case is minimum when energy of the incident particle is only slightly greater than barrier potentialV.
C. Localization in one dimension
We now consider an infinite one-dimensional (1D) array of identical barriers separated by a meandistance d (Fig. 3). The distances between the barriers is random, i.e., the distance between twoconsecutive barriers is d + δ where δ ( (cid:28) d ) is random. We will now study localization in this 1Darray due to presence of the randomness in position of the barriers.As the barriers are identical, their transfer matrices are equal, given by (13). The free propagationmatrix differs between each barrier because of randomness. It retains the same form as (12) butwith p in place of q as the propagation is now over free space. The presence of randomness in thearray is reflected in the propagation matrix in the form of the distance between two consecutivebarriers: d + δ . FIG. 3. A finite portion of an infinite array of identical 1D barriers. M is the transfer matrix for the barrierand is the same for all. P is the propagation matrix over free space and differs due to randomness in thesystem.
The transfer matrix approach reduces the problem of infinite barriers to one involving product ofrandom matrices. As transfer matrices follow the composition law [15], the total transfer matrix Mof a system of N barriers can be written as : M T otal = M P N − M...P M P M (20)where M is given by (13) and P i has the form: P i = (cid:32) e ip ( d + δi ) (cid:126) e − ip ( d + δi ) (cid:126) (cid:33) . (21)The first diagonal element represents a particle traveling towards the right and the second a particletraveling towards the left. The matrix M T otal is the transfer matrix for an array of N identicalbarriers with disorder in position. It is easy to see that M T otal is given by the product of randommatrices and for a large value of N it would model the situation in Fig. 4 suitably.We began by placing a constraint on the energy of the incident particle E > V . With this it isexpected that for E (cid:29) V , the localization length tends to infinity. However, the same cannot besaid for energies much closer to V .The product given in (20) was computed using MATHEMATICA for 1000 identical barriers andthe transmission coefficient was calculated using (18). The term δ was generated randomly froma normal distribution with mean zero and standard deviation W . The parameters were given innuclear units with the barrier height and width as 10 MeV and 400 fm respectively, mean barrierdistance d as 150 fm, incident particle momentum as 11 MeV/c and disorder strength W as 2. Theplot shown in Figure 5 is the averaged result of 100 iterations.As the number of barriers increase, the transmission coefficient tends to zero exponentially. Thisexponential localization is also confirmed by an exponential curve fitted to the data. The strengthof this localization is dependent on the strength of the disorder, namely W for weak disorder. Aswe increase the disorder strength, we attain a constant value. ( fm ) | T | - - - - - - -
10 System Length ( fm ) Log | T | FIG. 4. Exponential localization of the wave function of an electron. Here, the barrier height and width werechosen to be 10 MeV and 400 fm respectively, mean barrier distance d as 150 fm, incident particle momentumas 11 MeV/c and disorder strength W as 2. The plot above is the averaged result of 100 iterations.
In Fig. 4, the transmission probability was calculated after every successive barrier and averagedover 100 iterations. This transmission probability was then plotted alongside the system length. Aswe are varying the inter-barrier distances to introduce randomness into the system, they subsequentlyhad minor differences in each iteration. Hence, to plot the transmission probability calculated afterevery barrier with system length, the inter-barrier distances were averaged as well.Essentially, this means that both quantities on the X and Y axis are averaged results for a largenumber of iterations (100 in this case).We can further discuss the Lyapunov exponent γ and the localization length ξ for the abovesystem. The two are related and defined as: γ = 1 ξ = − lim L →∞ < log e | T | >L (22)where L is the total length of the chain L = N ∗ ( a + d ). ( Disorder Strength ) L ya puno v E x pon e n t × × × × W ( Disorder Strength ) Lo ca li z a t i onL e ng t h ( f m ) FIG. 5. The Lyapunov exponent (left) and the localisation length (right) as a function of disorder strengthfor an electron. The incident energy of the electron here is 11 MeV/c and the system parameters are thesame as in Fig 4. Each point on the two plots here is the calculated result averaged over 50 iterations.
We now look for a relation between the localization length and the disorder strength of the system.We expect that the relation between the two quantities can be expressed as a power law. This canbe confirmed if we plot a Log-Log plot of the two quantities and obtain a straight line. This straightline would then correspond to a power law relation which would give us- ξ ∝ W c (23)Log[ ξ ] ∝ c Log[ w ] (24)In a Log-Log plot as described above (Fig. 6), we see a linear relationship when the disorderstrength is small, which is broken at large disorder strengths. The region where the slope of thegraph approaches zero is where the disorder strength is too large and the approximation of smallperturbations in the inter-barrier distance no longer holds.To verify this reasoning, we plot (Fig. 7) the maximum percentage difference between the idealmean inter barrier distance and the new value obtained after introducing disorder. We see that evenfor W=10, the maximum difference is upto 20% for a mean inter-barrier distance of 150 fm. Foreven higher values of disorder strength, the condition of small perturbations breaks completely. - [ W ] ( Disorder Strength ) Log [ Lo ca li z a t i onL e ng t h ] FIG. 6. The Log-Log plot of the localisation length vs disorder strength plot in Fig. 5 (right). A straightline in the Log-Log plot corresponds to a power law relation between the two quantities. Here, we see thatfor small disorder strengths there is a linear relationship which does not hold for larger values of the same. M ax i m u m P e r ce n t a g e D i ff e r e n ce FIG. 7. The plot shows the maximum percentage difference between ideal mean inter-barrier distance andthe perturbed value in the presence of disorder in relation to the disorder in the system. The ideal inter-barrier distance corresponds to 150 fm (d) here and the perturbed value is (d+ δ ) where δ is generatedrandomly in each iteration. For high disorder strengths, we see that the is difference is very large and cannotbe classified as “small” perturbations. In order to examine any variation in the linear behaviour at small disorder strengths with incidentenergy of the particle (electron), we plot the Log-Log plots (Fig. 8) for different values of incidentmomentum.The slope (c) for successive values of incident momentum was found to be as:-1.96253 (11 MeV/c), -2.08447 (12 MeV/c), -2.00154 (13 MeV/c), -1.5271 (14 MeV/c), -1.95633 (15MeV/c), -2.02576 (16 MeV/c), -1.90843 (17 MeV/c),-1.98469 (18 MeV/c), -1.98425 (19 MeV/c),-2.02876 (20 MeV/c).An average value of the constant c in (23) was calculated to be − . ± . p =
11 MeV / c LogofLocalizationLength - Log [ W ] ( Disorder Strength ) p =
12 MeV / c LogofLocalizationLength - Log [ W ] ( Disorder Strength ) p =
13 MeV / c LogofLocalizationLength - Log [ W ] ( Disorder Strength ) p =
14 MeV / c LogofLocalizationLength - Log [ W ] ( Disorder Strength ) p =
15 MeV / c LogofLocalizationLength - Log [ W ] ( Disorder Strength ) p =
16 MeV / c LogofLocalizationLength - Log [ W ] ( Disorder Strength ) p =
17 MeV / c LogofLocalizationLength - Log [ W ] ( Disorder Strength ) p =
18 MeV / c LogofLocalizationLength - Log [ W ] ( Disorder Strength ) p =
19 MeV / c LogofLocalizationLength - Log [ W ] ( Disorder Strength ) p =
20 MeV / c LogofLocalizationLength - Log [ W ] ( Disorder Strength ) FIG. 8. The Log-Log plots of the localisation length vs disorder strength plot for different values of incidentmomentum of an electron. The height of the barrier is 10 MeV in all cases and the remaining systemparameters are the same as described in Fig. 4. The straight line fit in each case fits the first 16 points inthe linear region going from a disorder strength of 0.25 to 4.
IV. THREE COMPONENT MOMENTUMA. Transfer matrix in presence of spin flip
In certain situations, a 1D Dirac scattering over a potential barrier gives rise to spin-flips intransmission of the particles [12]. The spin-flips are a result of the transverse momentum pos-sessed by the particle and disappear when these components vanish. This is visualized in Fig. 9 andFig. 10. We extend this for a rectangular barrier, and subsequently to a disordered chain of barriers. V E Incident Reflected Transmitted z
FIG. 9. Pictorial representation of spin-flip of an incident Dirac particle in 1D as described in [12]. Thegreen arrow represents spin in the ’up’ direction and the red arrow represents spin in the ’down’ direction.
The general positive energy solutions of the Dirac equation can again be written as (4). In thiscase however, we cannot reduce our 4 × × p x and p y .0 V E I Ψ III Ψ II Ψ ( ) C ↑ D ↓ ( ) A ↑ B ↓ ( ) P ↑ Q ↓ ( ) R ↑ S ↓ FIG. 10. Pictorial representation of a Dirac particle incident on a potential barrier with
E > V with spin-flip incorporated. The particle here posses transverse momentum as well. A and B are the incident spin-upand spin-down amplitudes from the left respectively, while C and D are the reflected spin-up and spin-downamplitudes respectively. Similarly, P and Q are the transmitted spin-up and spin-down amplitudes movingtowards the right and R and S are the spin-up and spin-down incident amplitudes for a particle incidentfrom the right. Amplitudes in region II are present but not marked for clarity. We follow our previous approach to work out the transfer matrix for this case. P ↑ e ipza (cid:126) Q ↓ e ipza (cid:126) R ↑ e − ipza (cid:126) S ↓ e − ipza (cid:126) = M A ↑ B ↓ C ↑ D ↓ . (25)Here A and B are the incident spin-up and spin-down amplitudes from the left respectively, while C and D are the reflected spin-up and spin-down amplitudes respectively. Similarly, P and Q arethe transmitted spin-up and spin-down amplitudes moving towards the right and R and S are thespin-up and spin-down incident amplitudes for a particle incident from the right (if any).To simplify we find the transfer matrix M step for the potential step at z = 0 satisfying theequivalent relation (7) for the spin-flip case- E ↑ F ↓ G ↑ H ↓ = M step A ↑ B ↓ C ↑ D ↓ . (26)Where E , F , G and H are amplitudes of the wavefunction within the barrier region. Note thatby conservation of momentum p x = q x and p y = q y . Evaluating the spinors, we obtain the matrixequation : p z p − − p z p − p + − p z p + p z A ↑ B ↓ C ↑ D ↓ = rq z rp − − rq z rp − rp + − rq z rp + rq z E ↑ F ↓ G ↑ H ↓ , (27)where p ± = p x ± ip y . (28)1 M step is then given by M step = (cid:16) pq z r + 1 (cid:17) − p − ( r − q z r − p q z r − p − ( r − q z rp + ( r − q z r (cid:16) p z q z r + 1 (cid:17) p + ( r − q z r − p q z r − p z q z r p − ( r − q z r (cid:16) p z q z r + 1 (cid:17) p − ( r − q z r − p + ( r − q z r − p z q z r − p + ( r − q z r (cid:16) p z q z r + 1 (cid:17) . (29)The transfer matrix M step in (29) reduces to the one in (10) in the absence of spin flip( p x = p y = 0). The transfer matrix M for the rectangular potential in the presence of spinflip can now be worked out with (11) as before. The free propagation transfer matrix M , however,is now a 4 × M = e iqza (cid:126) e iqza (cid:126) e − iqza (cid:126)
00 0 0 e − iqza (cid:126) (30)The transfer matrix M : M = u v ∗ w ∗ u w v ∗ v w ∗ u ∗ w v u ∗ , (31)with u , v , w as u = cos (cid:16) aq z (cid:126) (cid:17) + iα + sin (cid:16) aq z (cid:126) (cid:17) ,v = + iα − sin (cid:16) aq z (cid:126) (cid:17) , (32) w = iµ ± sin (cid:16) aq z (cid:126) (cid:17) . Here, α ± = 12 (cid:34)(cid:18) p z rq z ± rq z p z (cid:19) ± ( p x + p y )( r − p z q z r (cid:35) , (33) µ ± = p ± ( r − rq z , and p ± are given by (28). Note that µ − is conjugate to µ + . The matrix element w disappears inthe absence of transverse momentum components and α ± reduce back to (15). The 4 × × M maintains both time reversal symmetry and con-servation of current density. This is again verified [15] by the calculation of the following relations:Det M = 1 for time-reversal symmetry (34) M † (cid:18) I − I (cid:19) M = (cid:18) I − I (cid:19) for conservation of current density (35)2where I is the 2 × T z (transmission in the z direction) is now given by the matrix ele-ments M and M . These elements correspond to the transmission of spin-up (down) componentswhen the incident particle is spin-up (down). Since these are independent of each other, for a unitincidence they are equal. T z = | t | = 1 | M | = 1 | M | (36)Using the expression for M from (31) we obtain T z as: T z = (cid:20) (cid:40)(cid:18) p z rq z − rq z p z (cid:19) + 2 η ( p z + rq z ) + η p z q z r (cid:41) sin (cid:20) aq z (cid:126) (cid:21)(cid:21) − (37)where η = ( p x + p y )( r − = p + p − ( r − . Note that in absence of spin-flip, η becomes zero and(37) reduces to (19). The presence of η here also indicates that the transmission coefficient is lowerwhen spin-flips are possible. Thus, we would expect localization to occur at much shorter lengthsas compared to the earlier case. B. Localization in one dimension
Our approach here follows that of Section 3.2. The transfer matrix for the barrier is now given by(31) and randomness in the system is reflected in the propagation matrix. The propagation matrixis now a 4 × P i = e ip ( d + δi ) (cid:126) e ip ( d + δi ) (cid:126) e − ip ( d + δi ) (cid:126)
00 0 0 e − ip ( d + δi ) (cid:126) . (38)The first and second diagonal elements represent a spin-up and spin-down particle respectively, trav-eling towards the right. Similarly the last two represent a spin-up and spin-down particle travelingtowards the left. The transfer matrix M total for an array of N identical barriers is given by (20). Asmentioned before, the presence of the additional term involving η in the transmission probabilitygreatly affects the transmission of the particle.Our constraints on the system are the same as in Section 3.3. The product given in (20) wascomputed employing MATHEMATICA for 500 identical barriers and the transmission coefficientwas calculated using (37). Here, the barrier height and width were chosen to be 10 MeV and 400 fmrespectively, mean barrier distance d as 150 fm and disorder strength W as 2 for both the spin-flipand no spin-flip case. An important difference here is to specify the incident angle of the particleon the barrier as the incident momentum is three dimensional. For this case, the particle is chosento be incident at an angle of 45 ◦ with respect to the z-axis.Fig. 11 shows the presence of an exponential localization of the wave function in the z directionfor the spin-flip case. As the number of barriers increase, the transmission coefficient tends to zeroexponentially. This exponential localization is also confirmed by an exponential curve-fit on thedata.3 ���� (cid:1) �� | T | Spin - flipNo Spin - flip ���� (cid:1) �� - - - - -
20 Number of barriers | T | Spin - flipNo Spin - flip FIG. 11. Exponential localization of the wave function of electron when possessing transverse momentumcomponents. Here, the barrier height and width were chosen to be 10 MeV and 400 fm respectively, meanbarrier distance d as 150 fm and disorder strength W as 2 for both the spin-flip and no spin-flip case. Forthe spin-flip case, the particle was incident on the x-z plane at an angle 45 ◦ with a total energy of 28.2889MeV and 20.0065 MeV for the no-spin flip particle possessing momentum only in the z direction. Theseenergies were taken such that the incident momentum of both the particles in the z direction were equal to20 MeV/c. The plot above is the averaged result of 100 iterations Fig. 11 also shows a key distinction between the two cases of spin-flip and no spin-flip. Thelocalization is much faster in the case of spin-flip because of the additional η term in the denominator.However, in this case the localization is only along the z direction, the x and y components areunaffected. V. CONCLUSIONS
We have established localisation in 1D relativistic systems for a spin particle through a mixof theoretical and numerical methods. We considered two different relativistic systems. The first“ordinary” case where the particle is moving in 1D and is incident on a 1D barrier and the secondwhere the particle is moving in 3D yet is incident on a barrier that is 1D. We see that localisation inthe second case is much “quicker” than in the first case. We also explored the Lyapunov exponentand the localisation length of an electron for the former case. On the basis of a numerical computa-tion, we also determined the dependence of localisation length on the disorder strength arriving ata relation of ξ ∝ W ( − . ± . .In the realm of future studies, we see many openings following a similar line of work. An interestingexample would be the extension of this work into the energy regime of E + mc < V where-in theeffect of the Klein paradox has to be considered. Pair-production in such a scenario could lead tounexpected results. Furthermore, an exhaustive analysis of the spin-flip case for localisation lengthand density of states is also a very exciting prospect. VI. AUTHOR CONTRIBUTION STATEMENT
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