Electric and magnetic waveguides in graphene: quantum and classical
David Barranco Şengül Kuru Javier Negro








	
	Electric and magnetic waveguides in graphene:

 
quantum and  classical
    
David Barranco^b[davidbarran95@gmail.com, ORCID: http://orcid.org/0009-0001-7405-1543
0009-0001-7405-1543], 
Şengül Kuru^a[sengul.kuru@science.ankara.edu.tr, ORCID: http://orcid.org/0000-0001-6380-280X0000-0001-6380-280X], 
Javier Negro^b[jnegro@uva.es,
ORCID: http://orcid.org/0000-0002-0847-64200000-0002-0847-6420]

 
^aDepartment of Physics, Faculty of Science, Ankara University, 06100 Ankara, Türkiye

^bDepartamento de Física Teórica, Atómica y Optica, Universidad de Valladolid,
 47011 
Valladolid, Spain
    
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Electric and magnetic waveguides  are considered in 
planar Dirac materials like graphene as well as their classical version for relativistic particles 
of zero mass and electric charge.
In order to solve the Dirac-Weyl equation analytically, we have assumed 
the displacement symmetry of the 
system along a direction. In these conditions we have examined the rest of symmetries 
relevant each type, magnetic or electric system, which will determine their similarities and differences. We have worked out waveguides 
with square profile in detail to show up some of the 
most interesting features also in quantum and classical complementary  contexts. All 
the results have been visualized along a series of representative graphics showing explicitly the main properties for both types of waveguides.




Introduction
Graphene and 2D Dirac materials have been a centre of attention beyond pure condensed matter researchers in the last years because of some of its peculiar electronic and optical properties in experimental and theoretical physics <cit.>. In particular, there has been an considerable number of works concerned about the effect of external electric and magnetic fields on the transport properties of Dirac electrons. However, electric and magnetic fields, give rise to quite different influences on Dirac electrons. One of the key points is the strong Klein tunnelling effect <cit.> which is most relevant in electric interactions. This leads to difficulties to achieve confinement under electric fields although it can be reached under some special conditions <cit.>; on the other hand, although magnetic fields seems more appropriate on this aspect, it happens that too intense magnetic fields may also have undesirable effects in this respect.

Many contributions have been devoted to the confining problem in bounded regions, that is in quantum dots, and in one dimensional regions like quantum waveguides, as well as to their  scattering properties on these field arrangements. This work is concerned with a detailed discussion about the differences that electric and magnetic waveguides may present.
In spite of the numerous previous papers on  waveguides, we thought it convenient to give a basic and global description about the fundamental and characteristic  aspects of electric and magnetic fields in the waveguide configuration.

We will start with a classical approach which is not usual, but it will prove to be quite clarifying afterwards, in the quantum frame.
We have taken three characteristic parameters determining both quantum and classical systems of massless particles: the energy E, the momentum k  in the y direction (symmetry direction) and the depth  or intensity of the potential (v_0 for electric  or a_0 for magnetic). We have discussed the influence of these parameters on the scattering and bounded motion of particles for both, scalar (electric) and vector (magnetic) potentials in the classical setup. We have taken in this work  simple rectangular profile shapes (mainly wells) which allow to obtain immediate results. For example, we have determined the regions of these parameters allowing for bounded  or scattering motion, which are essentially different for the electric and magnetic cases. Some diagrams show how is the motion of particles  with different values of parameters E, k and a_0 or v_0. The classical particles behave similarly to light when it passes from one region to another one with different refraction index by the Snell's law, as has been frequently mentioned in the literature. This refraction is quite different for electric or magnetics waveguides, showing asymmetric behaviour in the magnetic case.

Next, once introduced the classical viewpoint, we  investigated these questions in the quantum context by means of the two-dimensional Dirac-Weyl equation for both electric and magnetic interactions paying attention to scattering and bound states. In this program, the previous considerations on the classical  behaviour constitute a good reference to the quantum problem. For example; i) In the quantum case bound states are described by spectral curves, which in general are inside  the continuous region obtained in the classical study (they become a discrete quantization). ii) These spectral curves in the quantum problem for the vector potential have some especial bound states called edge states which are out of the classical region for this problem (they belong to a classically forbidden region). The edge states are missing in electric waveguides. iii) These two problems (scattering and bound states),  have also been connected by a join plot where spectral curves of the discrete spectrum are represented together with resonances of scattering states (T=1) for electric and magnetic potentials. These graphics give some suggestions about the capture of bound states “atomic collapse” into the hole sea.

We remark that this work contains many of the results that can be found scattered in the literature. However, we have made a unified presentation which we hope it will allow to appreciate the constrast of these two systems presented under the same approach. We have included many details that can only be fully displayed within our global program. We hope that this work will raise some interest in open questions which are present along our development. 

We wish to mention just a few representative papers close to this work which, in some way, have influenced our presentation. 
May be one of the first research which were aware of the importance of magnetic fields were  the works of De Martino et al <cit.>, see also <cit.>. An extensive study of different configurations with magnetic fields can be found in refs. <cit.>. The interesting paper by Gosh et al <cit.> is very close to our point of view.
Solutions of different inhomogeneous magnetic fields were obtained by SUSY methods in <cit.> (see also <cit.>). Some cases of electric waveguides are worked out in <cit.>. The join action of electric and magnetic fields were investigated in a series of contributions <cit.> and in particular a precedent paper <cit.> to this work.




Interactions with symmetry in the y-axis direction
	
We will restrict here to the stationary Dirac-Weyl equation in the plane for quasi-particles of zero mass and 1/2-spin with the electron charge. This equation corresponds to the continuous approximation to describe the electronic interaction in graphene for a range of energies near  the Dirac points (hereafter we will restrict to one of them, the point  K)
<cit.>. The equation for the interaction in external static fields, applying the minimal coupling rule, is

    v_F(  σ·  ( p+eÃ) )Ψ( x) = (ϵ - Ṽ( x)) Ψ( x)

where, as usual σ = (σ_x,σ_y) are sigma Pauli matrices,
p=-iħ∇ the two-component momentum operator in the xy-plane, v_F the Fermi velocity of graphene, e the electric charge and Ã( x), Ṽ( x)  magnetic and electric potentials defined on the plane; the wavefunction
Ψ( x)=Ψ(x,y) is a two-component pseudospinor and ϵ corresponds to energy. 

We will assume the symmetry of the fields along displacements in one direction (in the y-direction) and both fields to be perpendicular (the electric on the xy-plane, the magnetic in the z-direction perpendicular to the graphene plane). In fact, we will restrict to  potentials having the form

    Ã( x) = (0,Ã(x)),    Ṽ( x)=Ṽ(x)

Therefore, the magnetic field may depend only on x in this case and it is given by 

    B̃(x)= Ã'(x) ẑ=dÃ(x)/dxẑ

where ẑ is the unit vector of z-axis;
while the electric field 

    Ẽ(x)= -Ṽ'(x) x̂=-dṼ(x)/dxx̂
 
will be in the x-direction (x̂ is the unit x-vector) and it may also depend  on x. Hereafter, 
the “prime", will be used for derivatives with respect to the argument.

Let us make the following definitions in order to simplify the notation and determine the units for A(x), V(x) and ϵ:

    A(x)=-e Ã(x)/ħ,     V(x)= Ṽ(x)/ħ v_F=-eφ(x)/ħ v_F,     E=ϵ/ħ v_F

Thus, equation (<ref>) can be rewritten as

    H( x)Ψ( x)=(σ_x(-i ∂_x)+σ_y(-i ∂_y-A(x)) +V(x) I )Ψ( x) = E Ψ( x)

where I is for the unit matrix. This is the form that we will apply in waveguide configurations. H( x) is the Dirac-Weyl Hamiltonian under the external magnetic A(x) and electric V(x) potentials, but in this work we will discuss each field separately. In the following we will list some symmetries of H( x) depending on the parameters E, k and v_0/a_0 which enjoy the  potentials of our waveguides. They will greatly facilitate the finding and characterization of solutions.



 §.§ Symmetries



  (i) For both interactions: Translation in the y-direction

As we mentioned in the precedent section, the potentials A(x) and V(x) will depend only on x, so we can find solutions of the stationary equation (<ref>) that at the same time are eigenfunctions of the translation generator in the y-direction P_y= -i∂_y. If the eigenvalue of P_y is k then, these eigenfunctions will have the form

    Ψ(x,y) = e^i  k  y ψ(x)= e^i k  y (
    [ ψ_1(x); ψ_2(x) ])

where ψ_1(x), ψ_2(x) are components of the reduced spinor dependent only on x.


  (ii) For both interactions: Reflection in x

We will consider symmetric  potential (for both electric and magnetic) wells: A(x) = A(-x) and
V(x) = V(-x). In such conditions the stationary Dirac equation (<ref>) implements this symmetry by means of the operator

    R_x= σ_y R_x

where R_xψ(x) = ψ(-x) is the x-reflection operator and σ_y is the corresponding Pauli matrix. This means that if Ψ is a solution of eq. (<ref>), then R_xΨ will be another solution (may be the same if the symmetry is not broken):

    H( x)Ψ( x)=EΨ( x)   
    H( x)(σ_y R_xΨ( x))=E(σ_y R_xΨ( x))



  (iii) For pure magnetic interactions, V(x) = 0:
(a) Symmetry of positive and negative spectrum

If Ψ(x,y) is a solution with energy E, then σ_x R_x Ψ(x,y) will be another solution with energy -E. Therefore, the spectrum in the magnetic interaction will be symmetric in positive and negative energies:

    H( x)Ψ( x)=EΨ( x)   
    H( x) (σ_x R_xΨ( x))=-E (σ_x R_xΨ( x))



(b) Symmetry under k→ -k and A(x) → -A(x)

If Ψ(x,y) is a solution with energy E, momentum k and potential A(x), 
then σ_x  Ψ(x,y) will be another solution with the same energy E for
 momentum -k and potential -A(x) :

    H( x, k, A(x))Ψ( x)=EΨ( x)   
    H( x, -k, -A(x))) (σ_x  Ψ( x))=E (σ_x Ψ( x))



  (iv) For pure electric interactions, A(x) = 0: (a)
Symmetric spectrum of V(x) and -V(x)

If Ψ(x,y) is a solution with energy E, for the electric potential V(x) then σ_z Ψ(x,y) will be a solution with energy -E for the potential -V(x). Therefore, the spectrum of the potential well V(x) and the opposite barrier -V(x), will be symmetric:

    H( x, V(x))Ψ( x)=EΨ( x)   
    H( x, -V(x)) (σ_z  Ψ( x))=-E (σ_z  Ψ( x))



(b)  Symmetric spectrum for k and -k

If we reverse the sign of the conserved momentum k → -k, then the eigenfunctions 
transformed under σ_y will have the same eigenvalue:

    H( x,k)Ψ( x)=EΨ( x)   
    H( x,-k) (σ_y Ψ( x))= E (σ_y Ψ( x))





Classical waveguides

In this section, we will deal with classical waveguides  defined on a plane, which are produced by pure electric or magnetic potentials of square shape for zero mass particles with the electron charge. This is a limiting case of ultrarelativistic charges.

In all the cases, as mentioned earlier, we will have three constant values characteristic: (i) energy E, (ii) momentum in the symmetry direction k, and (iii) depth or intensity of the potential (v_0 for electric and a_0 for magnetic potentials). 
Although in the classical relativistic frame the energy always is positive, we will extend it to the negative sign for some situations explained below.



 §.§ The case of an electric well


Consider the case of an electric waveguide with potential of square profile given by 

    V(x,y)= 
    {[   - v_0, |x|<1  ,;       0,    |x|>1 ].

The width of the well has been fixed to simplify the discussion and fix units. We  will always assume that v_0>0, so the above potential will be a well (if v_0<0, this becomes a barrier. Trom the symmetries above mentioned, this situation is equivalent to a barrier, changing the sign of energy of the well).
The potential determine three regions: I for x<-1, II for -1<x<1 and III for x>1.





Let a classical relativistic charged particle with zero mass be incident from the left towards the well. Then, we have the following possible situations. Firstly we describe the situation of scattering states where a particle incident from the left (region I) into the well will cross it (through region II) to pass to region III.



  §.§.§ Scattering through an electric well
 




  * Regions I and III. (p_x>0)

The incident particle in region I (and in the outgoing region III) has energy E and momenta p_x and p_y. Since in the y-direction the potential is constant, we have a translation symmetry in that direction, the momentum p_y will be a constant of the motion which we  call k≡ p_y. The potential in the left and right regions is zero, so we have the relativistic zero mass relation
(we take c=1)

    E^2= p_x^2 + k^2   p_x=√(E^2-k^2)


We will suppose that p_x>0; in principle we also assume that k>0 and E>0, but we may as well include, without additional complications, negative values of k and E. So that once p_y=k is fixed, then from (<ref>) 

    |E| ≥ |k| ,     0≤ p_x < ∞








  * Region II. (p'_x> p_x)

Once the particle passes to region II, the effective energy E changes due to the potential of the well: E'=E+v_0, while the momenta will be p'_x and p'_y=k (which has the same constant value because the symmetry in the y direction persists).  The energy-momenta relation in region II is now

    (E+v_0)^2= (p'_x)^2 + k^2   p'_x=√((E+v_0)^2-k^2) > p_x

If E is positive, this means that inside II the x–momentum has increased: p'_x> p_x (Fig. <ref>, left).
Recall that the initial energy E is greater than k because the particle comes from region I where p_y=k.
Then the minimum value of p'_x  is reached if E=k (see Fig. <ref>, center) and therefore in that case, substituting in (<ref>), 

    p_x=0   p'_x=p'_ min = √((k+v_0)^2-k^2)


In conclusion, there will be classical scattering under an electric well for any incident particle characterized by the vertical momentum k and energy E^2> k^2.













  §.§.§ Incident and “refracted” directions


We may introduce the angles α for the incident particle in region I and α' for the particle entering in the well region II. In this subsection we are assuming that E is positive. 
These angles α and   α' are defined by 

    [                              sinα = k/E  ,                            sinα' = k/E+v_0;              tanα =  k/p_x=k/√(E^2-k^2)  , tanα'=  k/p'_x=k/√((E+v_0)^2-k^2) ,   α>α' ]

Therefore,

    p_x tanα = p_x' tanα' ,     (α≥α')

or

    E sinα = (E+v_0) sinα' ,     (E≥ k)

This is similar to Snell's law n_1sinθ_1 = n_2 sinθ_2 of refractive indexes and angles for light waves.
In particular for the minimum energy  E= k, α=π/2, there is a maximum angle α'_ max entering into the well: 

    sinα'_ max = k/k+v_0


Thus, we have the following ranges of  angles α and α':

    0≤α≤π/2 ,     0≤α' ≤α'_ max


The possible trajectories of a classical particle in an electric well are plotted
in Fig. <ref>, for the three cases: (left) scattering, (center) limiting case (α=π/2, α'= α_ max'),
(right) a bounded motion.







  §.§.§ Bounded motion inside the electric well
 




  * Region II. Bounded motion for |E|<k and |E+v_0|> |k|

In this subsection we describe the classical motion restricted to the waveguide, region II. The particle inside region II may be trapped in such region, if it is not coming from region I as in the above scattering process.
In this case, we have the following values of energy and momenta inside region II: 

i) Since the particle can not go out of region II (it is forbidden in I): |E|< |k|. 

ii) If the particle is in region II: (E+v_0)^2 = (p'_x)^2+ k^2,
 therefore |E+v_0|>|k|. 
 
Thus, E must satisfy the following inequalities (we allow also E and k to be negative) 

    [    i)   |E|< |k|, ii)   |E+v_0|>|k| ]

Then, the region of parameters satisfying these two inequalities for a particle trapped in region II, is shown in Fig. <ref>.

If k>v_0, then all the interval (k-v_0< E< k) is positive. If k<v_0, a
part of the interval (k-v_0, k) is negative.






Thus, as mentioned above, any particle coming from the left is going to pass, or will be scattered, because there is a well and there are no obstacles, see Fig. <ref>, left and center. 

It is also possible that a classical particle be inside the well without enough energy to go out, which means that there can be bound states. This behaviour is represented in Fig. <ref> (right). The bound states belong to the regions of Fig. <ref>. On the left it is shown the regions (in blue) of the E-k plane, for a fixed well intensity v_0, where the particle motion is bounded (in the x direction). The symmetry in the parameter k of the vertical constant momentum is explicit. On the right the regions of the plane E-v_0 where the motion is bounded are also displayed in blue color. In this graphic one can appreciate the symmetry under the change (E,v_0) → (-E,-v_0). The regions on the left and on the right correspond, respectively, to the two inequalities (<ref>).









 §.§ The case of a magnetic well


Next, let us consider the case of a magnetic well under the same point of view  with the following conditions.







The square magnetic well (barrier) is defined by the vector potential where A(x) is given by

    A(x)= 
    {[    -a_0, |x|<1  ,;       0,    |x|>1 ].

We have fixed the width again, leading to the same three regions: I for x<-1, II for -1<x<1 and III for x>1.
If a_0<0 it will be a barrier, while a_0>0 is for a well. 




  §.§.§ Magnetic scattering for k>0
 

We will consider a particle incoming from the left into the well of region II where a_0>0.
Then, we will see the different options for this particle. In this case the sign of the constant vertical momentum p_y=k is important due to the Lorentz force. If k>0 the potential acts like a barrier (the Lorentz force push the incident particle out of the well).



  * Regions I and III.

The particle initially has energy E, and momenta p_x and p_y. Since in the y-direction the potential does not change, the momentum p_y will be a constant, p_y=k, as in the electric case. Since the potential in the left region is zero A(x)=0, we have again the relativistic zero mass energy-momentum relation
(we take c=1)

    E^2= p_x^2 + k^2   p_x=√(E^2-k^2)


Besides k>0 we will also suppose that p_x>0 and E>0 (we may also consider negative values of E), so that once p_y=k is fixed, then from (<ref>) E ≥ k.
We have the initial ranges of values: 

    k≤ E <∞,     0 ≤ p_x <∞
 







  * Region II. k>0. (Fig. <ref>)



Once the particle passes to  region II the energy is the same as in region I, while the canonical momentum  in the y direction (p_y= k) has to adapt to the existing magnetic potential.
In this case p_y will change to p_y-A, according to the minimal coupling rule for electromagnetic interactions. Therefore, in region II we will have the value  p'_y=k+a_0.
Since a_0>0, then this value of  p'_y in the well is also positive and greater than p_y:

    p_y=k,     p'_y=k+a_0
 
The other component of the momentum (in region II)  p'_x is obtained from the energy-momenta relation. In this case p'_x  is less than p_x (momentum in region I), 

    E^2= (p'_x)^2 + (k+a_0)^2   p'_x=√(E^2-(k+a_0)^2)<p_x











Recall that the initial energy E is greater than k: E≥ k (because the particle comes from region I). But due to the last inequality, if p'_x and p_x are positive,

    |E|≥ |k+a_0|

Then, if the particle pass to region II, the minimum value of the energy is  E= k+a_0 and therefore:

    p_ min = √((k+a_0)^2-k^2)

Then, in order to have a scattering crossing the magnetic well, the incident angle
α must be less than a maximum angle α_ max, 

    sinα = k/E≤k/k+a_0 = sinα_ max

This is the maximum incident angle for a scattering. Beyond this value of α_ max (for a fixed k), the incident particle will reflect on the boundary of the well.
In summary, there will be scattering (the particle will pass to region III) only if the incoming energy 
satifies the inequality (<ref>).





































Next, we will consider the case of opposite sign k= k^- (where k^- is negative), then the Lorentz force push the particle into region II, giving rise to different situations.



  §.§.§ Scattering and bound states for k^-<0
 
In this case there will be scattering, bound or reflection depending on the values of k^- + a_0.








If for a well a_0>0, the k-momentum is negative k^-<0 then p'_y=k^-+a_0= -|k^-|+|a_0|. The other component of the momentum is p'_x. There are three possibilities:




















































  (i)  k^-+a_0<0. (Fig. <ref>)

This means that k'_y= k^-+ a_0 < 0 and therefore p'_x> p_x.
If p_x=0, E=|k^-| then p'_x= √(k^2- (k^-+a_0)^2)= p'_ min is the minimum value of p'_x in the scattering process. The angle sinα'_ max=k^-+a_0/|k^-| is the maximum angle inside the well for scattering states. In conclusion there will always be scattering where the particle will be crossing the barrier (E^2>k^2).
The  trajectory is characterized by the negative incident angle sinα=k^-/E,  
and the negative refraction angle sinα'=k^-+a_0/E. The negative sign  means that trajectories in all the regions go downwards with 
|α|>|α'| (see Fig. <ref>).  
 


Bound states exist for 

    |k^-+a_0|<E<|k^-|,

and the angle of the bound trajectories are greater than  α'_ max:

    |sinα'| = |k^-+a_0|/|E|>|k^-+a_0|/|k^-|




  (ii)  0<k^-+a_0<|k^-|. (Fig. <ref>)

In this case p'_x> p_x.
If p_x=0, E=|k^-| then p'_x= √(k^2- (k^-+a_0)^2)= p'_ min is the minimum value of p'_x in the scattering process. In this cas α is negative, α' positive and |α|>|α'|. As in the previous case the angle sinα'_ max=k^-+a_0/|k^-| is the maximum angle inside the well for scattering states, but with positive sign.
Bound states exist for 

    k^-+a_0<E<|k^-|

characterized by the angle  sinα'=k^-+a_0/|E|, with opposite sign to the previous case.



  (iii)  |k^-| <k^-+a_0. (Fig. <ref>)

Then,  we have a very deep well without bound states, p'_x< p_x.
Only if E>k^-+a_0 there will be scattering, with angles:

    sinα =k^-/E<0,  sinα' =k^-+a_0/E>0,    |α'|>|α|

If |k^-|<E< k^-+a_0 then  there is a total reflection. The angle sinα_ max= k^-/(k^- + a_0) gives the maximum angle for scattering. Beyond that value, there is reflection.
No bound states  exist.



In summary (see also <cit.>), there are three cases for negative momentum k^-: i) for k^-<k^-+a_0<0 where the bound states propagate in the negative y-direction; ii) for |k^-|>k^-+a_0>0
where the bound (or refracted in region II) states propagate in the positive y-direction; and iii) 
for |k^-|<k^-+a_0, where the magnetic well is so intense that no bound states exist. This surprising behaviour of bound states as a function of magnetic intensity a_0 can be seen in Fig. <ref> (right) where the regions of bound states have a (symmetric) triangular shape. These regions correspond to inequalities (<ref>), where k^- is fixed. There is  a high contrast to the electric bound states as a function of the depth of the potential in Fig. <ref>, where one can find bound states for any high value of v_0. 

On the left graphic of Fig. <ref> it is shown the regions of bound states as a function of the vertical momentum k. This corresponds to inequalities (<ref>), once a_0 is fixed. In this graphic, we see that only for k<0 there are bound motions and there is a symmetry of bound states for positive and negative energies E. 













Quantum electric and magnetic wells

In the following we will examine the solutions of the quantum problem, in the context of graphene quasiparticles. We expect the following situations.

i) In the range of energies of total classical scattering we may find partial scattering with a non null reflection amplitude. Total scattering will take place for discrete values of the relevant parameter (here we will take the angle α or thee energy E).

ii) In the range of energies  of bound or trapped classical states there will be a set of discrete values of energy for true bound quantum states.

iii) In the case of energies with total reflection we may find some quantum states of partial tunnelling, allowing the transmission to the right region.

iv) There could even exist bound quantum states in forbidden classical regions: scattering, trapped or reflection states. This is the case of edge states. We will see that the bound magnetic states will allow two of them have an edge  character. This fact is quite important because edge states have special properties and  there is only a few situations allowing for such states.




 §.§ Quantum magnetic well


Using the vector  potential (<ref>) and the spinor form

    Ψ( x)= e^i k  y e^i k_x x([ ϕ_1; ϕ_2 ])

where ϕ_1, ϕ_2 are constants, in (<ref>), we can obtain the solutions of (<ref>) in the three regions: region I (x<-1), region II (-1<x<1), and region III (x>1).



  §.§.§ Scattering



  * Region I. (x<-1)

    ([        0 (-i+k_x); (ik+k_x)        0 ])([ ϕ_1; ϕ_2 ])=E
    ([ ϕ_1; ϕ_2 ])


From this matrix equation, we have two equations for ψ_1 and ψ_2:

    (-i+k_x)ϕ_2=E ϕ_1,      (ik+k_x)ϕ_1=E ϕ_2

where k_x and k are constants.  The consistency of both equations give

    E^2 = k_x^2 + k^2 ,     E_± = ±√(k_x^2 + k^2)

The two signs of E_± correspond to positive or negative energies.
If we take ϕ_1=1, 

    ϕ_2=(k_x+ik)/E_±

Since for each energy E_± we have k_x=±√(E^2-k^2), then we get the solutions

    Ψ_I^±(E_±, x)= e^i k y e^± i k_x x([              1; (± k_x+ik)/E_± ])=
     e^i k y e^± i k_x x([         1; e^iα_ i/o ]),
           
     k_x=√(E^2-k^2)

where x = (x,y) and the incoming angle is α_ i=arctan k/k_x and outgoing angle α_ o=arctan- k/k_x. Thus, the general solution is

 

    Ψ_I(E_±, x)= α_1 Ψ_I^+(E_±, x)
    + α_2 Ψ_I^-(E_±, x)

where α_1, α_2 are arbitrary constants.



  * Region II. (-1<x<1)

For this region from the matrix equation taking into account that, due to the magnetic potential, -i∂_y is replaced by -i∂_y+a_0, we get the equations for ϕ_1 and ϕ_2:

    (-i(k+a_0)+k'_x)ϕ_2=E ϕ_1,      (i(k+a_0)+k'_x)ϕ_1=E ϕ_2

Solving them together, we obtain ϕ_1 and ϕ_2 and we can write the independent solutions 
Ψ_II^±(E_±, x)

    Ψ_II^±(E_±, x)= e^i k y e^± i k'_x x([                     1; (± k'_x+i(k+a_0))/E_± ]) ,    
     k'_x=√(E^2-(k+a_0)^2)

The general solution is

    Ψ_II(E_±, x)= β_1 Ψ_II^+(E_±, x) +
    β_2 Ψ_II^-(E_±, x)

where β_1,β_2 are arbitrary constants.



  * Region III.  (x>1)

Following the same procedure as above, for this region we have the same solutions of region I:

    Ψ_III^±( x)= e^i k y e^± i k_x x([              1; (± k_x+ik)/E_± ]) ,
           
     k_x=±√(E^2-k^2)


    Ψ_III(E_±, x)= γ_1 Ψ_III^+(E_±, x)
    + γ_2 Ψ_III^-(E_±, x)


where γ_1,  γ_2 are  the constants.


Remark 1. In order to solve the scattering  problem, we must assume that k_x is real  
    k_x^2=(E_±)^2 - (k)^2>0,     |E_±| > |k|

 to have  incoming  and outgoing plane waves, in the first and third region, respectively.

Remark 2. For second region we must take into account that
 
    k'_x^2=(E_±)^2 - (k+a_0)^2


∙ If k'_x^2=(E_±)^2 - (k+a_0)^2>0 or
 
    |E_±|> |k+a_0|

then the momentum k'_x in region II will be real. Classically this means that
the electron will pass the magnetic strip.  

∙ If k'_x^2=(E_±)^2 - (k+a_0)^2<0 or 
 
    |E_±|< |k+a_0|

then k'_x will be imaginary. This is going to happen if 
 
    |k|<|E_±|< |k+a_0|

This is the case where classically there will be total reflection. However, in the quantum case there may be transmission due to quantum tunnelling. In this case, we must take the solutions:

    Ψ_II^±(E_±, x)= e^i k y e^± k'_x x([                       1; (∓ i k'_x+i(k+a_0))/E_± ]) ,    
     k'_x=√((k+a_0)^2- E^2)

For the condition (<ref>), the general solution is

    Ψ_II(E_±, x)= β_1 Ψ_II^+(E_±, x) +
    β_2 Ψ_II^-(E_±, x)

where β_1,β_2 are arbitrary constants.


Then, we consider the following type of scattering solutions:

    Ψ_I( x)= Ψ_I^+( x) + r Ψ_I^-( x),    Ψ_II( x)= A Ψ_II^+( x) + B Ψ_II^-( x),    Ψ_III( x)= t Ψ_III^+( x)

where r  and t are the reflection and transmission amplitudes, respectively. Depending on the values of the E, k and a_0, we can use for the solutions Ψ_II, the formulas  (<ref>) or (<ref>). These solutions must satisfy the matching conditions at x=-1 and x=1:

    Ψ_I(-1)=Ψ_II(-1),    Ψ_II(1)=Ψ_III(1)



If we have a particle with momenta k=(k_x, k) with 
k_x= √(E^2-k^2) corresponding to energy E, then the incident angle (the angle of the incident direction with the normal to the magnetic barrier/well) will be

    sinα = k/|E|  |E|= k α

In this way, we can compute the reflection and transmission coefficients depending on the incident angle (for fixed k and a_0)

    R(α,k,a_0)= |r(α,k,a_0)| ,      T(α,k,a_0)= |t(α,k,a_0)|

They depend on the values k of the y-momentum, a_0 the depth of the well and α the incident angle (or equivalently the energy E). For example the transmission coefficient, obtained by finding the solution (<ref>) by solving the matching conditions (<ref>) in  terms of energy (for the allowed classical scattering) is given by

    t(E, k, a_0) = 
    exp[-2 i k_x]  k_x k'_x/k_x k'_x
           cos(2 k'_x) - 
         i (E^2 - k (a_0 + k)) sin(2 k'_x)

and for the forbiden classical scattering is given by

    t(E, k, a_0) = 
    exp[-2 i k_x] |k'_x|^2k_x / |k'_x|^2 k_x
           cosh(2 |k'_x|) - 
         i (E^2 - k (a_0 + k)) sinh (2 k'_x)


Figs. <ref>-<ref> show the transmission coefficient T= |t| of the scattering states for the magnetic well. The resonances T= |t|=1 will take place for
the values sin(2 k'_x)=0, i.e., when the phase acquired by the wave in crossing  the well must be equal to nπ, n∈ℤ. This is shown in Fig. <ref>. 











  §.§.§ Bound states



  * Region I. (x<-1) 

In this region, in order to have bound states, we have the condition: k^2>E and q_x=±√(k^2- E^2). Then the solution (<ref>) has the form:

    Ψ_I( x)= e^i k y e^q_x x([ ϕ_1; ϕ_2 ])



    ([ (-ik-iq_x)ϕ_2; (ik-i q_x)ϕ_1 ])=E
    ([ ϕ_1; ϕ_2 ])

From these equations, E^2=k^2- q_x^2


    Ψ_I^±(E_±, x)= e^i k y e^± q_x x([                1; (∓ i q_x+ik)/E_± ]),
           
     q_x= √(k^2-E^2)

 The general solution in region I is

    Ψ_I(E_±, x)= α_1 Ψ_I^+(E_±, x)
    + α_2 Ψ_I^-(E_±, x)

We also have the inequality

    |E|<|k|



  * Region II. (-1<x<1)

For this region, we get two equations for ϕ_1 and ϕ_2:

    (-i(k+a_0)+k'_x)ϕ_2=E ϕ_1,      (i(k+a_0)+k'_x)ϕ_1=E ϕ_2

Solving them together, we obtain ϕ_1 and ϕ_2 and we can write the independent solutions 
Ψ_II^±(E_±, x)

    Ψ_II^±(E_±, x)= e^i k y e^± i k'_x x([                   1; ± k'_x+i(k+a_0)/E_± ]) ,    
     k'_x=±√(E^2-(k+a_0)^2)

Then, we have the inequality

    |E|>|k+a_0|


Thus, the general solution is

    Ψ_II(E_±, x)= β_1 Ψ_II^+(E_±, x) +
    β_2 Ψ_II^-(E_±, x)

where β_1,β_2 are arbitrary constants.


  * Region III.  (x>1)

Following the same procedure as above, for this region we have the same solutions of region I:

    Ψ_III^±( x)= e^i k y e^± q_x x([                1; (∓ i q_x+ik)/E_± ]) ,
           
     q_x=±√(k^2-e^2)


    Ψ_III(E_±, x)= γ_1 Ψ_III^+(E_±, x)
    + γ_2 Ψ_III^-(E_±, x)


where γ_1,  γ_2 are  the constants.



In order to get the bound state  solution, we must assume that q_x for the region I is real  
    q_x^2=k^2 - (E_±)^2>0,     |E_±| < |k|


Next, in  region II we have also two possibilities:
 
    k'_x^2=(E_±)^2 - (k+a_0)^2>0,   or  q'_x^2=(E_±)^2 - (k+a_0)^2<0
 
The first possibility gives us the standard bound states and the second one leads to edge states. 
Then, we consider the following solutions for the bound problem:

    Ψ_I( x)= A Ψ_I^+( x),    Ψ_II( x)= b Ψ_II^+( x) + C Ψ_II^-( x),    Ψ_III( x)= D Ψ_III^-( x)

where A, B, C, D  are constants. Depending on the values of the E, k and a_0, we can use for the solutions Ψ_II standard bound  (<ref>) or edge states:

    Ψ_II^±(E_±, x)= e^i k y e^± |q'_x| x([                       1; ∓ i |q'_x|+i(k+a_0)/E_± ]) ,    
    | q'_x|=±√((k+a_0)^2-E^2)


 These solutions must satisfy the matching conditions at x=-1 and x=1:

    Ψ_I(-1)=Ψ_II(-1),    Ψ_II(1)=Ψ_III(1)


The resulting spectrum of bound states is shown in Fig. <ref> (see also the discussion in <cit.>). In the left graphic the parameter of the magnetic potential a_0=4 has been fixed. This graphic represent the energy eigenvalues as a function of the vertical momentum k. The spectrum are in blue curves inside the classical sector, and outside the classical region bound spectral curves in red correspond to ground edge states. 


If k is fixed, so that the energy depends on the magnetic intensity a_0 the corresponding spectral curves are shown in the right of Fig. <ref>. This type of magnetic spectrum is completely different to the corresponding electric case, as we will see later. The spectral curves belong to a triangular region, except for the edge states (in red) that are in the exterior of the classical region of  Fig. <ref> (right).


In Fig. <ref> it is shown the magnetic potential with the bound energy levels, where two of them are standard an the other two correspond to edge states. Their spectrum are symmetric with respect to the origin.  This potential is not the effective potential of an Schrödinger equation, so that some of the energy levels are above the potential.

Next, in figure Fig. <ref> it is shown  two types of curves: the spectral curves of bound states together with the curves of resonant states whose transmission probability is T=1. These two types of curves match so that one may interpret that the well captures resonances to turn them into bound states.

Fig. <ref> shows the components of a standard ground state, while Fig. <ref> (left) are the graphics of an edge ground state. We see the difference, each component of edge states  has a high peak at each of the edges of the magnetic strip. In Fig. <ref> (right) it is shown the components of an excites states. In all the graphics Fig. <ref>- <ref> the spinor wave functions satisfy the reflection symmetry in x given in (ii) of sec. 2.1.


















 §.§ Quantum electric well

In this case we have an electric potential V(x) given in (<ref>), therefore
the Dirac–Weyl equation is

    (σ_x(-i ∂_x)+σ_y(-i ∂_y))Ψ( x) = (E - V(x))Ψ( x)

The spinor form is as in (<ref>)








where ϕ_1, ϕ_2 are constants, in (<ref>). We can obtain the solutions of (<ref>) in the three regions: region I (x<-1), region II (-1<x<1), and region III (x>1).



  §.§.§ Scattering




  * Region I. (x<-1)

    Ψ_I^±(E_±, x)= e^i k y e^± i k_x x([              1; (± k_x+ik)/E_± ]),
           
     k_x=√(E^2-k^2)

where x = (x,y). The general solution in region I is

    Ψ_I(E_±, x)= α_1 Ψ_I^+(E_±, x)
    + α_2 Ψ_I^-(E_±, x)




  * Region II. (-1<x<1)

We obtain ϕ_1 and ϕ_2 and we can write the independent solutions 
Ψ_II^±(E_±, x)

    Ψ_II^±(E_±, x)= e^i k y e^± i k'_x x([                1; (± k'_x+i k)/E_± ]) ,    
     k'_x=√((E_±+v_0)^2-k^2)

The general solution is

    Ψ_II(E_±, x)= β_1 Ψ_II^+(E_±, x) +
    β_2 Ψ_II^-(E_±, x)

where β_1,β_2 are arbitrary constants.


  * Region III.  (x>1)

Following the same procedure as above, for this region we have the same solutions of region I:

    Ψ_III^±( x)= e^i k y e^± i k_x x([              1; (± k_x+ik)/E_± ]) ,
           
     k_x=±√(E^2-k^2)


    Ψ_III(E_±, x)= γ_1 Ψ_III^+(E_±, x)
    + γ_2 Ψ_III^-(E_±, x)


where γ_1,  γ_2 are  the constants. The scattering state is characterized by:
α_1 = 1, α_2 =r, γ_1= t, γ_0=0, where r is the reflection amplitude and t is the transmission amplitude.


Remark 1. In order to solve the scattering  problem, we must assume that k_x is real   and |E_±| > |k|, to have  ingoing and outgoing plane waves in the first and third regions, respectively.

Remark 2. Next, for second region we must take into account
 
    k'_x^2=(E_± +v_0)^2 - k^2


∙ If 
|E_±+ v_0|> |k| 

Then, the momentum k'_x in region II will be real. Classically, this means that
the electron will pass the electric waveguide.  

∙ If |E_±+ v_0| < |k| 

Then, k'_x will be imaginary. This may happen, if 
 
    |E_± + v_0|< |k| < |E_±|

This situation can not happen for positive energies E_+ and positive v_0. So, we will not discuss it in greater detail. In the  classical context there will be total reflection. However, in the quantum case there may be transmission due to quantum tunnelling. In this case, we must take the solutions:

    Ψ_II^±(E_±, x)= e^i k y e^± k'_x x([                1; ∓ i k'_x+i k/E_± ])

For the condition (<ref>), the general solution is

    Ψ_II(E_±, x)= β_1 Ψ_II^+(E_±, x) +
    β_2 Ψ_II^-(E_±, x)

where β_1,β_2 are arbitrary constants.



Once obtained the general solutions in the three regions, the scattering solution
can be found in the same way as the magnetic case, taking into account the
formulas (<ref>), (<ref>) and (<ref>). 
After this computation with the help of Mathematica, the transmission amplitude, in terms of energy  

takes the form

    t(E, k, a_0) = 
     2exp[-2 i k_x]  k_x k'_x/2 k_x k'_x
           cos(2 k'_x) - 
         i (k_x k'_x)^2 sin(2 k'_x)


Some examples of the transmission coefficients for  scattering by an electric well  are given in Fig. <ref> as functions of the incident angle. In this figure one can appreciate that deeper wells give rise to higher interferences and less transmission.
Fig. <ref> shows the amplitude as function of the incident energy. The resonances with T=|t|=1 take place for the values sin(2 k'_x)=0, i.e., 2 k'_x= nπ,n∈ℤ <cit.>, as in the magnetic case.









  §.§.§ Bound states




  * Region I. (x<-1) 

In this region, in order to have bound states, we have the  same condition
as in the magnetic case: k^2>E^2 and q_x=±√(k^2- E^2). 
Thus, we have

    |k|>|E|

Then, the solution (<ref>) has the form:

    Ψ_I( x)= e^i k y e^q_x x([ ϕ_1; ϕ_2 ])

From these equations, E^2=k^2- q_x^2

    Ψ_I^±(E_±, x)= e^i k y e^± q_x x([                1; (∓ i q_x+ik)/E_± ]),
           
     q_x=  √(k^2-E^2)

 The general solution in region I is

    Ψ_I(E_±, x)= α_1 Ψ_I^+(E_±, x)
    + α_2 Ψ_I^-(E_±, x)



  * Region II. (-1<x<1)

Next, in the second region we have also two possibilities:
 
    k'_x^2=(E_±+ v_0)^2 - k^2>0,       -q'_x^2=(E_±+ v_0)^2 - k^2<0
 
Therefore, we get the following inequality:

    |E_±+ v_0|< |k|

The first possibility gives us standard bound states and the second one leads to edge states which are not present in the electric well. 

For the first possibility  we obtain ϕ_1 and ϕ_2 and we can write the independent solutions 
Ψ_II^±(E_±, x)

    Ψ_II^±(E_±, x)= e^i k y e^± i k'_x x([                  1; ± k'_x+i k/E_±+v_0 ]) ,    
     k'_x= √(k^2-(E_± +v_0)^2)

The general solution is

    Ψ_II(E_±, x)= β_1 Ψ_II^+(E_±, x) +
    β_2 Ψ_II^-(E_±, x)

 For the second option of edge states:

    Ψ_II^±(E_±, x)= e^i k y e^± |q'_x| x([                     1; ∓ i  q'_x+i k/E_±+v_0 ]) ,    
    | q'_x|= √(k^2-(E+v_0)^2)


But we will see later that in the electric case there will be no solution of this type.



  * Region III.  (x>1)



    Ψ_III^±( x)= e^i k y e^± q_x x([                1; (∓ i q_x+ik)/E_± ]) ,
           
     q_x= √(k^2-e^2)


    Ψ_III(E_±, x)= γ_1 Ψ_III^+(E_±, x)
    + γ_2 Ψ_III^-(E_±, x)






Fig. <ref> includes the spectral curves in the plane energy-depth (v_0) of the well (left) and corresponding to the plane energy-vertical momentum, k (right). We have selected the spectrum for a particular configuration: k=2.5, v_0=4. We have computed the energy values (see Fig. <ref>, where it is represented the electric well together with the energy levels) and plotted the corresponding spinor eigenfunctions in Figs. <ref>-<ref>. All of them have x-reflection symmetry. It happens that the two lowest eigenvalues are negative and their corresponding eigenfunctions have a partial edge character (a cusp at the border of the well). The two upper eigenvalues are positive and they have the standard form. It is clear the difference with the true edge state of Fig. <ref>, for the magnetic waveguide.

Finally, Fig. <ref> shows the spectral curves of bound states and the curves of the resonant scattering states which have the highest transmission coefficient T=1 (see also <cit.>). These two kinds of curves match continuously where the resonant states turn into bound states.
This is a representation of the phenomenon  of capture and collapse processes which is produced for electrons and holes.













§ CONCLUSIONS



In this work,  we have dealt with Dirac particles
(relativistic massless, spin-1/2) of planar Dirac materials like graphene, 
under external electric and magnetic waveguide potentials in a quite comprehensive way including a classical relativistic model as introduction.
We have paid attention to the symmetries that are the main ingredient of our program. They are made explicit through the graphics and are responsible of qualitative differences. 
This presentation contains three main complementary comparisons which allow for a better understanding: i) Quantum and classical 
approaches to the waveguide systems. ii) Bounded and scattering states in both quantum and classical 
problems. iii) The effects of  electric and magnetic 
potentials. All the results, as we mentioned along this work were presented 
not only through describing  boundary conditions and explicit formulas but also in a series of appropriate representative graphics. Thus, we hope that
all the different features can be easily appreciated from these graphs. In particular,  we want to remark the importance of those corresponding to the energy-momentum (E-k) and energy-intensity (E-v_0/a_0) which are complementary and exhibit the different symmetries of electric and magnetic systems. In the near future, we plan to end this program for waveguides of simultaneous electric and magnetic potential interactions.







§ ACKNOWLEDGMENTS


We appreciate the support of the QCAYLE project, funded by the European Union–NextGenerationEU, and PID2020-113406GB-I0 project funded by the MCIN of Spain.
Ş. K. thanks Ankara University and the warm hospitality of the Department of Theoretical Physics of the University of Valladolid, where part of this work has been carried out, and to the support of its GIR of Mathematical Physics.

99

Neto10 A. H. C. Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, A. K. Geim,  The electronic properties of graphene, Rev. Mod. Phys., 81,  109-163 (2009).


Katsnelson06 M. I. Katsnelson, K. S. Novoselov, A. K. Geim,  Chiral tunneling and the Klein paradox in graphene,  Nat. Phys., 2,  620-625 (2006).


superKlein20 A. Contreras-Astorga, F. Correa, and V.  Jakubský, Super-Klein tunneling of Dirac fermions through electrostatic gratings in graphene,  Phys. Rev. B, 102, 115429 (2020).

vit22 V. Jakubsky, Ş. Kuru, J. Negro, Dirac fermions in armchair graphene 
nanoribbons trapped by electric quantum dots, Phys. Rev. B, 105, 165404 (2022).


deMartino07 
A. De Martino, L. Dell-Anna, R. Egger, Magnetic barriers and confinement of Dirac-Weyl quasiparticles in graphene, Solid State Commun., 144, 547-550 (2007).  

deMartino07b 
A. De Martino, L. Dell-Anna, and R. Egger,
Magnetic confinement of massless Dirac fermions in graphene,
Phys. Rev. Lett., 98, 066802 (2007).

deMartino09
L.  Dell-Anna and A. De Martino,
Multiple magnetic barriers in graphene,
Phys. Rev. B, 79, 045420 (2009); Erratum Phys. Rev. B, 80, 089901 (2009).


Goerbig11	M. O. Goerbig, Electronic properties of graphene in a strong magnetic field, Rev.  Mod. Phys., 83,  1193–1243 (2011).

peeters09 M. Ramezani Masir, P. Vasilopoulos and F. M. Peeters, Magnetic Kronig-Penney model for Dirac electrons in single-layer graphene, New J. Phys., 11, 095009 (2009).


peeters10 M. Ramezani Masir, P. Vasilopoulos and F. M. Peeters, Fabry-Perot resonances in graphene microstructures: Influence of a magnetic field, Phys. Rev. B, 82, 115417 (2010).


Moldovan18 D. Moldovan, M. Ramezani Masir and F. M. Peeters,   Magnetic field dependence of the atomic collapse state in graphene,  2D Materials, 5, 01501 (2018).


gosh09 S. Ghosh and M. Sharma, Electron optics with magnetic vector
potential barriers in graphene, J. Phys.: Condens. Matter, 21,  292204 (2009).


kuru09 Ş. Kuru, J. Negro and L. M. Nieto,  Exact analytic solutions for a Dirac electron moving in graphene under magnetic fields , J. Phys.: Condens. Matter, 21, 455305 (2009).

Milpas11	E. Milpas, M. Torres, G. Murguia,  Magnetic field barriers in graphene: an analytically solvable models, J. Phys.: Condens. Matter 23,   245304 (2011).

peeters06 J. M. Pereira Jr, V. Mlinar, F. M. Peeters, P. Vasilopoulos, Confined states and direction-dependent transmission in graphene quantum wells, Phys. Rev. B, 74, 045424 (2006).

peeters10b J. M. Pereira Jr, F. M. Peeters, A. Chaves and G. A. Farias, Klein tunneling in single and multiple barriers in graphene, Semicond. Sci. Technol., 25, 033002 (2010).


portnoi14 R. R. Hartmann and M. E. Portnoi, Quasi-exact solution to the Dirac equation for the hyperbolic-secant potential, Phys. Rev. A, 89, 033002  (2014).


portnoi17 R. R. Hartmann and M. E. Portnoi, Two-dimensional Dirac particles in a Pöchl-Teller waveguide, Scientific Rep., 7, 11599 (2017).

lukose07 V. Lukose, R. Shankar, and G. Baskaran, Novel electric field effects on Landau levels in graphene, Phys. Rev. Lett., 98, 116802 (2007).


Ghosh19 P. Ghosh and P. Roy, Collapse of Landau levels in graphene under uniaxial strain, Mater. Res. Express, 6, 125603 (2019).


 Tan10 L. Z. Tan, C.-H. Park and S. G. Louie, Graphene Dirac fermions in one-dimensional inhomogeneous field profiles: Transforming magnetic to electric field, Phys. Rev. B, 81, 195426 (2010).


roy20 D.-N. Le, V.-H. Le, P. Roy, Graphene under uniaxial inhomogeneous strain and an external electric field: Landau levels, electronic, magnetic and optical properties, Eur. Phys. J., 93, 158 (2020).
 
Bautista20 M. Castillo-Celeita, E. Díaz-Bautista, M. Oliva-Leyva, Coherent states for graphene under the interaction of crossed electric and magnetic fields, Ann. Phys.,  421, 168287 (2020). 


Do21 T. H. Do, P. H. Shih, G. Gumbs, D. Huang,  Influence of electric and magnetic fields and σ-edge bands on the electronic and optical spectra of graphene nanoribbons,  Phys. Rev. B, 103, 115408 (2021).

Afshari17 F. Afshari, M. Ghaffarian,   Electronic properties of zigzag and armchair graphene nanoribbons in the external electric and magnetic fields,  Physica E Low Dimens. Syst. Nanostruct., 89, 86-92 (2017).


kuru22  I. B. Ateş, Ş. Kuru and J. Negro,  Graphene Dirac fermions in symmetric electric and magnetic fields: the case of an electric square well, Phys. Scr., 98, 015816 (2022).















