Quantum Goos-Hänchen switch in graphene junctions

Goos-Hänchen (GH) shift, an interference phenomenon describing the lateral shift of the reflected beam along the interface during total internal reflection, has attracted great interests in the field of quantum transport in two-dimensional materials. In particular, the GH effect generates a novel pseudospin-dependent scattering effect in graphene, which in turn results in an 8e2/h conductance step in the bipolar junctions. Here, we reveal that a barrier region with effective barrier strength χ can greatly enrich the GH effect in graphene junctions. In contrast to the conventional case where the negative shift is allowed only in the p-n junction, the thin barrier enables both negative and positive spatial shifts in pIn and nIn junctions, where I represents the barrier region. More interestingly, the lowest channel degeneracy can be efficiently varied by tuning χ in both the symmetric pInIp and the asymmetric pnIp junctions, leading to a highly controllable switch that switches the conductance between 8e2/h and 4e2/h . These results advance the knowledge of GH effects in electronic materials and suggest experimental avenues for its observation and manipulation.


Introduction
Goos-Hänchen (GH) effect, a well known interference phenomenon in optics, describes the lateral shift of the reflected beam during total internal reflection at an interface [1].The analogies between optics and electronics have recently inspired the studies of the GH effect in a variety of electronic systems [2][3][4][5], including 2D Dirac materials [6,7] like graphene [8], as well as a number of topological phases, such as topological semimetals [4,9] and axion insulators [10].
The GH effect in graphene has drawn enormous attentions and was extensively studied.It displays rich reflection phenomena, which are summarized in figure 1.Specifically, two critical angles, α c and α * (with α * > α c ), are found to play key roles in the GH effect.In the n-n (p-p) junction, only positive GH shifts take place when the incident angle α satisfies α > α c .The positive GH shift exhibits a minimum at α = α * > α c .Then, it grows as α is further increased, displaying a non-monotonous feature in the spatial shift [6].In contrast to the n-n (p-p) junction, the refraction coefficient is negative for α < α c [11][12][13] in the n-p junction.This further leads to richer GH behaviors in the n-p case as α increases.In particular, for α * > α > α c , one observes that a negative GH shift can emerge, as demonstrated by figure 1(c).This negative GH shift is one of our main focuses in this work.
More interestingly, because of the negative GH shift in the p-n interface, the GH effect was found to induce rich transport behaviors in graphene [2,6,14].The shift of the beam displays a strong sublattice-dependence, in both its magnitude and sign.As a result, when two opposite graphene-based p-n interfaces form a p-n-p junction, the negative shift generates interesting propagating modes in the junction, which exhibit a remarkable two-fold degeneracy in the n-doped region [6].Consequently, exotic 8e 2 /h stepwise jumps in the conductance were predicted to show up when the channel width W is gradually increased [6].One should note that these predicted phenomena were absent at n-n or p-p interfaces, because of the absence of the negative shifts as discussed above.Although fundamental studies on the GH effect in graphene have been quite comprehensive, in order to observe the 8e 2 /h conductance step caused by the GH effect, proper electric doping is necessary to achieve the p-n-p junction.Moreover, since the exotic 8e 2 /h conductance step occurs with increasing channel width W (within the quantum regime U 0 W/hv F ∼ 1, where U 0 is the potential difference between the n and p regions), a number of devices need to be fabricated in order to confirm this novel feature.This not only brings about experimental difficulties, but also restricts further practical applications of the GH effect.Here, by introducing a barrier, we would like to explore whether graphene-based junctions can exhibit any new behaviors of the GH shift not categorized under any of the known phenomena depicted in figure 1, and whether there exists a more flexible and tunable scheme that promises a more direct observation of the GH-induced 8e 2 /h conductance step.
In this work, we reveal a highly tunable GH effect in graphene junctions with a barrier as shown in figure 2. We show that a thin voltage barrier at the interface introduces a new degree of freedom that greatly enriches the GH effect in graphene.More features of the GH shift beyond those illustrated in figure 1 are obtained.We show that the GH shift of electron states at the interface, σ, displays nontrivial dependence on the effective barrier strength χ.Since the barrier strength χ can be readily tuned by a gate voltage [15][16][17][18][19] and possibly by a carbon nanotube [20], it provides a much more controllable approach that produces more delicate features of the GH effect absent in conventional setups.Firstly, σ(χ) exhibits periodic oscillations with the period π, for both pIn (where I represents the barrier region) and nIn junctions.σ(χ) can be either negative or positive, depending on χ.This is in contrast with the conventional GH effect [6] where negative GH shifts can only take place in p-n junction.Secondly, we calculate the lowest channel modes associated with the multiple reflections with opposite pIn and nIn interfaces, and find that, for both cases, the GH-induced degeneracy of the channel modes is sensitive to χ.By tuning χ, the merging and re-splitting of the two spectrum extrema will take place successively, displaying a π period.This fact further results in a quantum switch for the GH conductance along the interface, changing it from 4e 2 /h to 8e 2 /h and vice versa.These results reveal that a thin barrier unexpectedly hosts new transport behaviors in graphene-based junctions, greatly enriching the physics exhibited by the conventional GH effect.The proposed setup is highly controllable and feasible in experiments, suggesting a new avenue to observe and manipulate the GH effect in 2D Dirac electronic systems.
The remaining part of this work is organized as follows.In section 2, we introduce our model and the formalism for obtaining the GH shift in pIn and nIn junctions during total internal reflection at an interface.The reflection coefficient in the presence of a local barrier region and the GH shift are derived and discussed in details.In section 3, we calculate the conductance caused by the GH effect in the pInIp and pnIp junctions.Based on the obtained conductance, the quantum GH switch displaying periodic jumps in the conductance between 8e 2 /h and 4e 2 /h is predicted and discussed.Section 4 includes a summary of our findings as well as further discussions.

GH shift in pIn and nIn junctions
We first consider a pIn junction in a graphene sheet in the xy plane, namely, the right half of the junction shown in the upper panel of figure 2. The electrically doped regions are from x = −∞ to x = −d and x ⩾ 0 for all y.A barrier is located in the region from x = −d to x = 0 with a barrier potential V 0 .Such a local barrier can be implemented by either using the local chemical doping or electric field effect [15][16][17][18][19], or possibly by a carbon nanotube [20].We assume that the width of local barrier satisfies d ≪ λ F = 2π/k F , where k F (λ F ) is the Fermi wave vector (length) for graphene [15].The graphene system is well described by the two-dimensional Dirac equation as [6,15] ( where v F denotes the Fermi velocity, σ x and σ y are the Pauli matrices acting on sublattices.The spinor wave function the relative shifts of the Fermi energies in the two regions, where θ(x) is the Heaviside step function.Here we adopt the sharp boundaries for simplicity, due to the boundary smoothness at least does not disturb the conductance step qualitatively [6,21,22].The barrier is characterized by a dimensionless parameter Here, we focus on a local thin barrier with V 0 → ∞ and d → 0 such that χ remains finite.We emphasize that this condition is assumed only to facilitate calculations.The results remain qualitatively the same for finite V 0 and non-zero d (see appendix B).In particular, when χ = 0 our setup reduces to the graphene junctions investigated in [6].In the following, we first demonstrate our results based on the pIn case.The nIn case will be discussed afterwards.
The electron wavefunction Ψ(r) in the n-doped, barrier and p-doped regions can be straightforwardly obtained by solving equation (1).Since the interface has translational invariance along the y direction, we firstly consider a solution describing a beam with transverse wave vector q, incident on the n-I interface from the n-doped region x < −d with U(r) = 0. q and k are the transverse and longitudinal wave vectors respectively, and α is the incident angle.Since E = E F for x < −d, the dependence of k and α on q can be obtained as f(q − q) in equation ( 3) is the distribution function for the beam with the incident angle ᾱ = arcsin(q/E F ) ∈ (0, π/2), whose maximum is located at q ∈ (0, E F /hv F ). Since none of our results rely on the shape of f(q − q), for concreteness, we can take it to be a Gaussian function with width ∆ q .For ∆ q ≪ k F = E F /hv F , we can expand k(q) and α(q) to the first order around q. Substituting equation ( 5) into equation ( 3), the Gaussian integral can be performed and the spatial profile of the incident beam is obtained as in [6,23].
The reflected wave function Ψ − (x, y) can be straightforwardly obtained from the incident wave (3) by the replacements k → −k, α → π − α and the multiplication by the reflection amplitude r(q) = |r(q)|e iϕ (q) .The center of the wave packets can be obtained from the incident and reflected wave functions above, from which a pseudospin-dependent average displacement is derived as where the primes indicate derivatives with respect to q. Equation ( 6) describes the lateral shift of the reflected beam along the interface caused by interference during total reflection-the GH shift.We further consider the scattering at the other interface I-p of the pIn junction.In the p-doped region, the wave function is described by the evanescent mode as This is a solution of the equation (1) (with U(r) = U 0 and E = E F ) that decays into the ±x directions near the interface for hv F |q| > |E F − U 0 |.Furthermore, The wave functions in the barrier region Ψ I can be obtained in a similar manner as those in the n-doped region, with the replacement of This describes the right/left-moving electrons in the ±x directions.
Then, we match the wave functions at the interfaces by the continuity and obtain the reflection coefficient where Γ = i(E F − U 0 ) + q + κ and η = E F − U 0 + i(q + κ).We obtain |r| = 1, at which total internal reflection takes place with r = e iϕ , when the incident angle α satisfies We note that equation (10) imposes a restriction on the ratio of U 0 /E F , i.e. 0 < U 0 /E F < 2. From equation ( 9  The GH shift can then be readily obtained by substituting equation ( 9) into equation ( 6), leading to Equations ( 9) and ( 11) constitute the key results of this work, which describe the reflection amplitude and the GH shift respectively.They are applicable for both the pIn and nIn junctions, i.e.E F < U 0 and E F > U 0 , respectively.It can be clearly seen from equations ( 9) and ( 11) that both the reflection coefficient r (including its phase ϕ) and the GH shift σ are oscillatory functions of the effective barrier potential χ with a period π.This is in sharp contrast with results obtained from conventional non-relativistic systems [2] and graphene-based bipolar junctions [6,7].We note that in the special case when χ = nπ for any integer n, r and σ reproduce the results of [6].
We plot the GH shifts σ as a function of α in figure 4, with different barrier strength χ for both the pIn (E F < U 0 ) and nIn (E F > U 0 ) cases.As shown, an interesting oscillatory behavior with a period of π is observed.Moreover, σ undergoes sign changes (regardless of the relative magnitude of E F and U 0 ) in each period, thus the GH shift can be either positive or negative, depending on χ.We also mention in passing that the obtained periodic feature of σ is a general result, which does not rely on the values of U 0 /E F .
Here, it is worthwhile to note that the negative shift can now take place in both the pIn and nIn junctions.This is a new feature compared to the cases without the barrier, where the negative shift is absent in the n-n  11) for U0/EF = 1.5 (pIn junction) and for U0/EF = 0.5 (nIn junction).The critical angle αc for total internal reflection to occur (σ = 0 when α < αc), given by equation (10), equals αc = 30 • in both cases.The sign-change angle α * satisfying the equation (12).Lower panel: σ as a function of χ for α = π/3.The other parameters are chosen to take the same values as those in the upper panel.and p-p junctions [6].Thus, for both the pIn and nIn cases, one can determine a particular angle α * , for which the net GH shift is zero.By requiring σ = 0, one obtains ( where the sign ± applies for the U 0 > E F and U 0 < E F cases respectively.We can solve for α * in equation (12).Its exact expression is complicated but unilluminating for our discussions.The important fact to note is that it is found to be an oscillatory function of χ with the period π.In the following, we discuss some special values of χ at which α * takes simple but illuminating forms.Several conclusions can be drawn from equations (11) and (12).Firstly, we consider the special case where χ = nπ.One can find from equations ( 11) and ( 12) that Thus, for E F < U 0 , a critical angle α * = arcsin √ sin α c emerges, at which the GH shift σ changes its sign.In contrast, this critical angle is absent for E F > U 0 .This is in accordance with the previous observation that negative GH shifts can only occur in p-n junction [6].Secondly, in the special case χ = (n + 1/2)π, one finds that σ Accordingly, the critical angle α * now takes place for U 0 < E F .Thus, the negative GH shift now can emerge in the n-n or p-p junctions as well.Lastly, in the special case U 0 = E F such that α c = 0, one obtains This gives rise to α * = 0.It then becomes clear that for the special values of χ = nπ, the GH shift σ reduces back to that obtained in [6].Besides, the additional χ-dependence brings about oscillatory behaviors in σ and admits negative shifts in both the n-n (p-p) and the n-p junctions, making it a general effect with high tunability.The GH behavior studied here depends on the pseudospin (sublattice in the honeycomb lattice of the graphene) degree of freedom of the massless Dirac fermions, hence it is a novel interference phenomenon related to the total internal reflection at an interface and the coherence as well as the sublattice degree of freedom of the massless Dirac fermions.In fact, it can also be inferred from equation ( 12) that the GH shift will undergo a sign change, and at the critical incident angle α * the GH shift vanishes and there maybe a novel specular reflection of the total internal reflection at an interface.Thus in general, there is a negative shift parameter region, which can in turn bring about the GH induced conductance behavior during the multiple reflection process as proposed by [6].

Conductance in the pInIp and pnIp junctions
We now consider the multiple reflection process in the pInIp and pnIp junctions.We let U(r) = 0 in the middle n-doped region, i.e. x ∈ (−d − W, −d) with the width W; U(r) = U 0 for the left/right n-doped for the pInIp junction is shown in figure 2 and for the pnIp junction can be defined similarly.
The GH shift accumulates upon multiple reflections in the channel between two evanescent regions.If the separation W is large compared to the wavelength λ F , the motion between reflections may be treated semiclassically.The time between two subsequent reflections of evanescent regions is τ = W/v F cos α (d → 0) for α c < α < π/2.As shown above, the GH shift σ can be either positive or negative, depending on the barrier strength χ, the incident angle α as well as the ratio of U 0 /E F .This can be understood by evaluating the velocity of the propagating electrons along the interfaces, which is readily obtained as v ∥ = v F sin α + (σ/W)v F cos α.We plot v ∥ as a function of the barrier strength χ in figure 5, with different values of U 0 /E F and α.We find that the velocity v ∥ is an oscillatory function of the effective barrier potential χ with a period of π.Moreover, it can also undergo sign changes within a period, reflecting the fact that the GH shift σ can be negative.We note that the observed periodic behavior of v ∥ is a general result, which does not rely on the values of U 0 /E F and α.Similar to the pIn and nIn junctions discussed above, where a critical angle α * at which the GH shift at the interface changes its sign is observed, another critical angle α 0 can also be defined for the pInIp junction, at which the motion of the electrons along the junction changes its sign.The critical angle α 0 for the pInIp junction can be deduced by requiring v ∥ = 0.Then, combining the derived v ∥ with equation ( 11), we obtain where the sign ∓ applies for U 0 > E F and U 0 < E F , respectively.For χ = nπ, equation ( 16) reduces to sin 2 α 0 = U0/EF−1 κW+1 , in consistent with the results for the p-n-p junction [6].We now examine the GH conductance during the multiple reflection process.The GH conductance is closely related to the degeneracy of the low energy propagating modes [6].To clearly examine the energy spectrum of the propagating modes, we solve the Dirac equation (1) with potential profile U(r).Matching of the propagating waves to the decaying waves at the corresponding boundaries produces the following relations between E and q as cot (kW where equations ( 17) and ( 18) are applicable for the pInIp and the pnIp junctions, respectively.Here we choose E F = 0 as the reference energy for equation (1).For χ = nπ, both equation (17) (pInIp) and equation ( 18) (pnIp) reduce to [q 2 + E(U 0 − E)] sin kW + kκ cos kW = 0 [6], while for χ = (n + 1/2)π, they are cast into [q 2 − E(U 0 − E)] sin kW − kκ cos kW = 0 and k(U 0 − E) cos kW − Eκ sin kW = 0, respectively.From equations ( 17) and ( 18), the dispersion relations E(q, χ) can be derived, which describe the propagating modes along the junctions for both the pInIp and pnIp junctions.Clearly, E(q, χ) has an additional tunable parameter χ, which reveals more information and enriches the graphene-based GH effect.The velocity v ∥ (q, χ) analyzed in the semiclassical picture above can now be derived rigorously from the spectrum E(q, χ), namely, v ∥ (q, χ) = dE/hdq.Clearly, v ∥ (q, χ) = 0 implies the energy extremum of the propagating modes.The minima, as was proposed by [6], mimic an effective valley degeneracy, thus doubles the conductance-a significant prediction of the graphene-based GH effect.
We plot several lowest modes for the pInIp junction in figure 6.The minima with v ∥ = 0 are clearly visible for E ≲ U 0 .The loci of the extrema, obtained from equation ( 16), are shown by the red dashed curve.For E ≳ U 0 the GH effect increases the transverse velocity, which is visible in E(q, χ) as a local increase in the slope of the dispersion relation.The blue solid curves in figure 6 represent the dispersion relations of the modes that are confined to the narrow channel between two evanescent regions.At the dotted lines hv F |q| = |E − U 0 | these channel modes are joined to the modes in the wide regions.Interestingly, the effective barrier potential χ plays a key role in bringing about qualitative changes to E(q), by modifying the number of extrema.
For narrow channel width with U 0 W/hv F becomes of order unity, the full quantum mechanical regime is reached.The minima of the dispersion become very pronounced for the lowest channel mode, as shown in the right panels of figure 6. Importantly, we find that the two minima can merge with each other at q = 0 and vice versa, as we tune χ.Importantly, it is known that each of the minima corresponds to a value of e 2 /h in the conductance per spin and per valley degree of freedom [6].Thus, the total conductance from the lowest channel mode is switched from 8e 2 /h to the usual amount of 4e 2 /h during the merging process.This is a key effect introduced by the inserted barriers.
It is now clear that, although the GH effect doubles the degeneracy of the lowest propagating mode and introduces a twofold degeneracy on top of the usual spin and valley degeneracies, such degeneracy is sensitive to and highly tunable by the effective barrier strength χ.This is evident from figure 7. The calculated conductance G for the pInIp case with different channel widths is shown in the upper panel, and that for the pnIp case is shown in the lower panel.We observe a switching in the values of the conductance between 8e 2 /h and 4e 2 /h as χ is varied within a period of π, implying a quantum switch of GH-induced conductance in graphene junctions.

Conclusion and discussion
To experimentally measure the stepwise changes of conductance, the setup proposed by [6,24] is useful.As shown by the shaded inset in figure 2, the electrostatic potential landscape produces a quantum point contact.In the channel region of length L, the barrier potential V is tuned so that electron transport through the barrier takes place at the Dirac point.In contrast to previous proposals [6], where the width W has to be adjusted for the observation of the stepwise conductance, it can now be achieved simply by tuning the barrier parameter χ.
Here we note in addition that, although we assumed d → 0 and V 0 → ∞ while keeping χ = V 0 d/hv F finite to simplify calculations, the results obtained are not restricted to cases with thin barriers with high voltage gating.In fact, the effects of V 0 and d on the GH shift can be analyzed separately by treating both of them as tunable parameters.As shown in appendix B, we found that the oscillatory behavior of the GH shift (and thus that of the conductance) persists for the whole parameter range of V 0 and d.Such behaviors are in accordance with [25], which studied the normal-insulator-superconductor graphene junctions.Thus, the barriers with finite lower voltage V 0 and non-zero larger d will also work and lead to the conductance steps shown in figure 7.
In experiments, the local barrier can be readily implemented and manipulated [15,17,20].In terms of experimental parameters, the typical Fermi energy is E F ≃ 80 meV and the corresponding Fermi-wavelength is λ F = 2π/k F ≃ 50 nm [15,17].The effective barrier with potentials up to 500-1000 meV and widths of d ≃ 20 − 10 nm have been achieved in the experiments [15][16][17].To observe the oscillatory behavior of the GH conductance, one can then tune χ by gradually increasing V 0 .For a local barrier of a fixed width of d/λ F = 0.1 and strength of V 0 /E F = 10, a stepwise change of V 0 by δV 0 = 12.5 meV will be sufficient to clearly detect the GH-induced conductance jump.We also would like to mention that although the unusual 8e 2 /h GH conductance has long been theoretically predicted in the graphene-based bipolar junctions [6], its experimental observation is still lacking, despite remarkable progress has been made in recent years [20-22, 26, 27].Notably, instead of the 8e 2 /h GH conductance, the 4e 2 /h conductance have been reported by these works.It should be emphasized that the key ingredient of the 8e 2 /h conductance is the GH induced degeneracy as shown by figure 6, which doubles the 4e 2 /h conductance from the valley and spin degeneracy.Because the GH induced two-fold degeneracy is purely a quantum interference effect of electrons, it is more significant in narrower junctions and would be obscured as the channel width W increases.Additional calculations with increasing W is included in appendix A, which clearly demonstrates this point.Therefore, our results suggest that it is would be an interesting experimental direction to further explore the GH induced degeneracy by applying the barrier region.
To summarize, this work reveals that a voltage barrier can unexpectedly bring about interesting features of quantum interference in graphene.As a result, the energy dispersions of the channel modes of the junctions acquire a new 'resolution' with respect to the barrier strength χ.This greatly enriches the GH effect in graphene.The resulting conductance displays a novel π periodic oscillation controlled by χ, generating a conductance switch from 8e 2 /h to 4e 2 /h.These results indicate an experimental scheme to identify the GH effect in graphene.Furthermore, they open possibilities towards achieving highly controllable platforms for investigating novel quantum interference effects in 2D materials.

Appendix B. The effects of V 0 and d of the barrier on σ separately
The barrier strength χ is related to the voltage V 0 and the thickness d, via χ = V 0 d/hv F .In the main text, V 0 → ∞ and d → 0 are assumed in order to facilitate calculations.However, we note that this condition is not necessary; similar GH conductance can be also obtained for finite V 0 and non-zero d.In figure 4, the GH shift σ is shown only as a function of χ.Here, we consider the effects of V 0 and d separately, and plot the GH shift σ as a function of either V 0 or d.As shown in figure B2, it is found that the oscillatory behavior of σ observed in figure 4 also occurs for the whole parameter range of V 0 and d.Importantly, the oscillatory behavior already becomes quite significant for relatively low values of V 0 around 5E F ≃ 0.4 eV.This clearly indicates that, in order to observe the barrier-induced conductance jumps shown in figure 7, V 0 → ∞ is not necessary.As a result, the value of the gate voltage in the barrier region can be achieved within experimental reach, without destroying the electronic properties of the graphene sample.

Figure 1 .
Figure 1.The electron optics in graphene-based junctions.(a) When the incident angle α satisfies α < αc, only normal reflections and refractions occur in the p-p and n-n junctions.In comparison, the refraction angle is negative in the n-p junction (bottom half of the figure below the black dashed line).(b) At α = αc, total internal reflection begins to take place.(c) For αc < α < α * , the GH shift σ is positive for the p-p and n-n junctions, but is negative for the n-p junction.(d) For α = α * > αc, the GH shift σ is positive but attains its minimum value for the p-p and n-n junctions, while it is zero for the n-p junction.(e) For α > α * > αc, σ is positive for both the p-p (n-n) and the n-p junctions.(f) When the incident angle is in parallel with the interface, the GH shift σ becomes positive and infinite for all junctions.

Figure 2 .
Figure 2. Upper panel: potential profile of an n-doped channel between p-doped regions with additional local barrier regions.Lower panel: top view of the channel in the graphene sheet.The red solid/blue dashed line depicts the center of a beam on the A/B sublattices.The two centers have a relative displacement δ0.Upon reflection, each pseudospin component experiences large and small shifts σ ± in an alternating manner, which results in a average displacement σ between the centers of the incident and reflected beams.The shaded inset shows a schematic experimental setup for measuring the GH's conductance in channel: a quantum point contact (QPC) of width W and length Lc.
), one can calculate the phase ϕ of the reflection coefficient as a function of the incident angle α.The result is shown in the upper panel of figure3for different values of U 0 /E F .The dependence of ϕ on χ is also shown in the lower panel of figure3for different values of U 0 /E F .As shown, ϕ can be efficiently tuned by varying the barrier strength, which will consequently generate a χ-sensitive GH shift.Meanwhile, it can be readily confirmed that |ϕ(−π/2) − ϕ(π/2)| = π is always satisfied for all values of U 0 /E F and χ.

Figure 3 .
Figure 3. Upper panel: the phase ϕ of the reflection coefficient as a function of the incident angle α for different values of U0/EF with χ = nπ.The results for both the pIn junction (U0/EF > 1) and the nIn junction (U0/EF < 1) are shown.The inset shows the results for χ = nπ/2.Lower panel: The calculated phase ϕ as a function of χ for different U0/EF.The incident angle is fixed to be α = π/3.

Figure 4 .
Figure 4. Upper panel: the GH shift σ as a function of the angle of incidence α for different values of χ, calculated from equation(11) for U0/EF = 1.5 (pIn junction) and for U0/EF = 0.5 (nIn junction).The critical angle αc for total internal reflection to occur (σ = 0 when α < αc), given by equation(10), equals αc = 30 • in both cases.The sign-change angle α * satisfying the equation(12).Lower panel: σ as a function of χ for α = π/3.The other parameters are chosen to take the same values as those in the upper panel.

Figure 5 .
Figure 5.The calculated velocity v ∥ as a function of χ for different values of U0/EF and the incident angles α.The width W of the junction is chosen to be W = 1.

Figure 6 .
Figure 6.Energy of the waves propagating with wave vector q in the y-direction, confined in the channel −(W + 2d) < x < 0 by the potential profile U(r) (in the pInIp junction).Left panel: The blue solid lines correspond to different modes in the semiclassical regime.The red dashed curve corresponds to the loci of v ∥ = 0, the extrema of the dispersion relations.Right panels: Same as the left panel, but now showing the lowest channel modes in the full quantum mechanical regime for different values of χ.It clearly shows how two extrema merge into a single extremum at q = 0, upon changing χ.All parameters are normalized by the width of the junction W.

Figure 7 .
Figure 7.The GH-induced conductance G vs χ in one period of π, for the symmetric pInIp (upper panel) and asymmetric pnIp junctions (lower panel).To demonstrate the quantum GH effects, only the contribution from the lowest channel mode in figure 6 is shown here.

Figure A1 .
Figure A1.(a) The energy spectrum of confined modes in the channel −(W + 2d) < x < 0 in the pInIp junction.The results for W = 6a, 20a, 100a are shown, where a is the lattice constant.(b) The lowest channel mode's contribution to the conductance.Results for W = 6a and W = 100a are shown.We see that the GH induced conductance steps are obscured in wide channels.

Figure
Figure The effects of V0 and d on the GH shift σ separately.(a) σ as a function of V0, where d = 0.1λF is fixed.(b) σ versus d with V0 = 10EF being fixed.(c) σ as a function of both d and V0.The incident angle is α = π/3 for all figures.