Scattering in one-dimensional heterostructures described by the Dirac equation

0.24 MB·English

by  N. M. R. Peres

9
S
attering in one-dimensional heterostru
tures
0
0
des
ribed by the Dira
equation
2

n
a
N. M. R. Peres
J

0 Center of Physi
s and Department of Physi
s, University of Minho, P-4710-057,
3 Braga,Portugal

]
l
l Abstra
t. We
onsiderele
troni
transporta

rossone-dimensionalheterostru
tures
a
des
ribed by the Dira
equation. We dis
uss the
ases where both the velo
ity and
h
- themassarepositiondependent. WeshowhowtogeneralizetheDira
Hamiltonianin
s
e order to obtain a Hermitian problem for spatial dependent velo
ity. We solve exa
tly
m the
ase where the position dependen
e of both velo
ity and mass is linear. In the
.
ase of velo
ity pro(cid:28)les, it is shown that there is no ba
ks
atteringof Dira
ele
trons.
t
a In the
ase of the mass pro(cid:28)le ba
ks
attering exists. In this
ase, it is shown that the
m
linear mass pro(cid:28)le indu
es less ba
ks
attering than the abrupt step-like pro(cid:28)le. Our
-
results are a (cid:28)rst step to the study of similar problems in graphene.
d
n
o
c
[ PACS numbers: 72.10.-d, 73.21.Hb, 73.23.Ad

1
v
1. Introdu
tion
7
5
8
Most of the a

umulated knowledge about the physi
s of heterostru
tures assumes that
4
. the ele
trons in these materials are e(cid:27)e
tively des
ribed by the Shrödinger equation,
1
0 with a position dependent mass. The
ommon belief is that Dira
equation is of no use
9
in
ondensed matter physi
s (spin-orbit
oupling
an be treated using the Pauli version
0
: of the S
hrödinger equation). The dis
overy of graphene [1, 2℄
hanged drasti
ally this
v
Xi perspe
tive. Condensed matter physi
ists are now fa
ing a
ondensed matter system
r where the e(cid:27)e
tive low-energy model for the quasiparti
les is that of a ultrarelativisti
,
a
i. e. massless, Dira
equation, albeit with an e(cid:27)e
tive Fermi velo
ity that is mu
h
v = c/300 c
F
lower than the velo
ity of light. In graphene the Fermi velo
ity is , with
the velo
ity of light. In fa
t, sin
e the isolation of graphene
rystallites [1, 2, 3℄ that a
renovated interest in the properties of the massless Dira
equation in 2+1 dimensions
started to emerge [3, 4℄. Although our interest in this paper is on the ba
ks
attering
properties of ele
trons e(cid:27)e
tively des
ribed by the Dira
equation in 1+1 dimensions,
as an e(cid:27)e
tive low-energy theory of the ele
troni
properties of a quasi-one-dimensional
solid, we shallrevise someofthe properties ofDira
ele
trons inthe
ontext ofgraphene,
given the a

umulated amount of eviden
e for Dira
fermions in graphene.
How
an the low-energy des
ription of a solid be given by an e(cid:27)e
tive Dira
-like
equation? In one-dimensional physi
s, this takes pla
e due to the linearization of the
S
attering in one-dimensional heterostru
tures des
ribed by the Dira
equation 2
spe
trum
lose to the Fermi momentum, whi
h introdu
es right- and left-movers with
a linear energy-momentum relation. Another possibility, that we dis
uss in the bulk of
the paper, is by having two atoms per unit
ell with spe
ial type of hopping. In two-
dimensions, this possibility has been realized by graphene. The ele
trons in graphene
π
are
on(cid:28)ned to move within the −orbitals of system, formed from the overlap of the
2p
z
atomi
orbitals of a single
arbon atom. The transport and opti
al properties of
π
graphene are determined essentially by the behavior of the −ele
trons. The ele
troni
π
density is su
h that graphene has one −ele
tron per
arbon atom, and is therefore a
half (cid:28)lled system. It turns out that the band stru
ture of graphene around the Fermi
surfa
eishighlynon-standard, sin
etherelationbetween themomentumandtheenergy
v 1 106
F
is linear, with a proportionality
oe(cid:30)
ient given by ≃ × m/s and termed Fermi
velo
ity. In fa
t, the Fermi surfa
e is redu
ed to two points (the so
alled Dira
points
K K′
and ) in the Brillouinzone, with a zero density of states at the Fermi-points. This
fa
t gives graphene its non-
onventional properties.
k p Γ
Starting from a · approa
h [5℄, adapted to degenerate bands o(cid:27) the −point
K
[6℄, we obtain an e(cid:27)e
tive Hamiltonian in real spa
e, valid around the point, having
the form
H = v σ p ,
F
· (1)
σ = (σ ,σ ) p = i~(∂ ,∂ )
x y x y
where is a ve
tor using the Pauli matri
es and − . The
Hamiltonian (1) is nothing but the Dira
Hamiltonian for massless parti
les in 2+1
v
F
dimensions, with an e(cid:27)e
tive light-velo
ity . Clearly, one expe
ts that the physi
al
properties of su
h a system will be di(cid:27)erent from those where the S
hrödinger equation
is valid. Note that this situation is an example of
omplex emergent behavior [12℄, sin
e
the original problem was that of independent ele
trons (and therefore S
hrödinger like)
in a periodi
potential.
K K′
Those ele
trons in graphene lying
lose to the Fermi points ( and ), will have
a di(cid:27)erent response to external potentials than those in other materials des
ribed by
the S
hrödinger equation. In fa
t, for S
hrödinger ele
trons, if we
onsider that the
V
0
potential varies at a given point in spa
e from zero to a (cid:28)nite value (the so
alled
step potential) there is always a (cid:28)nite fra
tion of impinging ele
trons that are re(cid:29)e
ted
V
0
ba
k. More over, if is greater than the energy of the impinging parti
les, there will
be an exponential attenuated wave fun
tion in the region where the potential is (cid:28)nite.
For Dira
ele
trons the situation is di(cid:27)erent. It was shown by Klein [16℄ that Dira
V
0
parti
les
an not be
on(cid:28)ned by an arbitrary large potential . In fa
t, they pass
through strong repulsive potential without the exponential de
ay that
hara
terizes
S
hrödinger parti
les. This is
alled Klein tunneling. If the parti
les are massless
and moving in one-dimension the situation is even more dramati
, sin
e there is no
ba
ks
attered probability (cid:29)ux, no matter how large the potential is [13, 14℄.
In addition to external potentials, a parti
le propagating in solid state systems
an
fa
e a situation where its e(cid:27)e
tive mass
hanges in spa
e. This possibility takes pla
e in
m(r) r
heterostru
tures, where the mass of the parti
le is written as , being the position
S
attering in one-dimensional heterostru
tures des
ribed by the Dira
equation 3
of the parti
le. In this situation, the S
h(cid:4)rodinger equation has to be modi(cid:28)ed, in other
to
omply with hermiti
ity and (cid:29)ux
onservation.
If the parti
les are des
ribed by the Dira
equation, the informationon the material
propertiesisen
odedbothintheFermivelo
ity[6℄andinthemass. We
anthenimagine
v (r)
F
the possibility of produ
ing a herestru
ture where the Fermi velo
ity, , and the
mass
hange withthepositionoftheparti
le. Aswe showbellow,even inthis
ase, there
will be not ba
ks
attering if the parti
le is massless and the velo
ity is allowed to vary
from point to point. In the
ontext of graphene physi
s, a position-dependent Fermi
velo
ity was already
onsidered in the
ase of
urved graphene [7℄. Another possibility
of produ
ing both a Fermi velo
ity and a mass term that are position dependent is to
subje
t the system to strain. It has already been shown using ab-inito
al
ulations, in
the
ontext of graphene physi
s, that strain leads to gap formation [8, 9℄. It is to be
expe
ted that for general strain, di(cid:27)erent parts of the material presents lo
al values
of Fermi velo
ity and energy gap. Also in the
ase of graphene, it was shown that
depositing the material on top of Boron Nitride leads to gap-formation [10℄.
In this paper we dis
uss several possibilities for the s
attering of Dira
parti
les
througharegionwhere boththevelo
ityandthemassarepositiondependent. Although
our motivationis to study graphene strips [15℄ in a latter publi
ation, we start by giving
two exa
t solutions for one-dimensional systems. The generalization to a quasi-one-
dimensional system, su
h as a narrow nano-wire, is easy to obtained and will be given
in a follow up publi
ation. In order to see the di(cid:27)eren
es between the S
hrödinger and
the Dira
problems, we start by revising the
ase of one-dimensional heteros
tru
tures
des
ribed by the S
hrödinger equations and latter move the study of the Dira

ase.
2. Warming up: the S
hrödinger ele
trons in 1D
In order to
ompare how the s
attering of S
hrödinger ele
trons di(cid:27)ers from that of
Dira
parti
les, we give a brief a

ount of the s
attering of ele
trons by the interfa
e of
a heterostru
ture.
Let us
onsider the
ase of 1D S
hrödinger ele
trons with position dependent
m(x) m(x) = m− x < 0 m(x) = m+ x > 0
e(cid:27)e
tive mass , su
h that , for and , for .
The boundary
onditions in this
ase are (see (cid:28)nal paragraph of this se
tion; for more
general boundary
onditions see Ref. [11℄ )
ψ−(0) = ψ+(0),
(2)
1 d 1 d
ψ−(0) = ψ+(0).
m−dx m+dx (3)
We
onsider that the parti
les are moving from the left to the right. Therefore the wave
fun
tion is
ψ−(x) = eikx +re−ikx,
(4)
ψ+(x) = teipx,
(5)
S
attering in one-dimensional heterostru
tures des
ribed by the Dira
equation 4
k p
where the values of and are (cid:28)xed by energy
onservation:
~2k2 ~2p2
E = = .
2m− 2m+ (6)
r 2
The re(cid:29)e
tan
e | | is determined from the boundary
onditions as
(m+k m−p)2
2
r = − ,
| | (m+k +m−p)2 (7)
r 2 1 m+ t 2
whi
h has the limiting value | | → for → ∞. The transmittan
e
oe(cid:30)
ient | | is
4(m+k)2
2
t = ,
| | (m+k +m−p)2 (8)
We
an
he
k now that the boundary
onditions used for the wave fun
tion are the

orre
t ones, sin
e they satisfy the (cid:29)ux
urrent
onservation. Using absolute values, we
have indeed
jinc = jreft +jtrans,
(9)
jinc = ~k/m+ jreft = r 2~k/m+ jtrans = t 2~p/m−
with , | | , and | | .
The problem just des
ribed represents an heterostru
ture, where two materials are
glued together (due to similar latti
e
onstants), having di(cid:27)erent e(cid:27)e
tive masses in the
two materials. The used boundary
onditions are obtained writing the Hamiltonian as
~2 d 1 d
H = .
−2mdxm(x)dx (10)
(cid:1) dx(Hψ)∗ψ = (cid:1) dxψ∗Hψ
This
hoi
e de(cid:28)nes an Hermitian problem ‡ sin
e . The

onstru
tion of the (10) guaranties that the total probability (cid:29)ux is
onserved. The
boundary
ondition (3), follows immediately from the writting the Hamiltonian in the
Sturm-Liouville form.
3. Dira
ele
trons in 1D
Let us now study the
ase of the Dira
Hamiltonian. We are interested in
ases where
both the velo
ity and the mass of the parti
les depend on their position in spa
e. We
study (cid:28)rst the
ase of massless Dira
parti
les, like those present on graphene. Latter
we add up the mass term. The Hamiltonian for a massless quasiparti
le des
ribed by
v
F
an e(cid:27)e
tive Dira
equation, with an e(cid:27)e
tive velo
ity of light , is given by
~ d
H = v σ ,
F x
i dx (11)
‡ The Eq. (10) is a parti
ular
ase of a Sturm-Liouville operator, whi
h has the general form:
d d
p(x) +q(x)y =λw(x)y,
−dx dx
(cid:18) (cid:19)
λ p(x) q(x) w(x)
where istobedeterminedfromtheboundary
onditions,and , ,and aregivenfun
tions.
Sturm-Liouville problems are Hermitian.
S
attering in one-dimensional heterostru
tures des
ribed by the Dira
equation 5
σ x
x
where is the Pauli matrix. Let us now
onsider that we make an heterostru
ture,
made of two di(cid:27)erent materials, both of them des
ribed by an e(cid:27)e
tive Dira
equation,
su
h as (11). An example
ould be a strip of the material where part of it is subje
ted
to strain and other part is strain free; this leads to di(cid:27)erent velo
ities in the two
v
F
parts of the material. This situation
alls for a model where is position dependent:
v = v (x)
F F
. Although this situation does not make sense in high-energy physi
s, it is
v
F
quite
on
eivable in
ondensed matterphysi
s, sin
e thevalue of isdeterminedby the
material under
onsideration (cid:21) di(cid:27)erent materials
an have di(cid:27)erent Fermi velo
ities.
v v (x)
F F
The trivial repla
ement → renders the problem non-Hermitian, as
an be
seen by applying the de(cid:28)nition given above, when we dis
ussed the S
hrödinger
ase.
There is however a way out. It is a simple matter to show that the operator
~ d
H = v (x)σ v (x),
F x F
i dx (12)
p p
v (x) = v
F F
is Hermitian and redu
es to Eq. (11) in the parti
ular
ase . Note that the
v (x)ψ(x)
F
derivative will a
t on the produ
t . We stress that we are not studying the
s
attering of relativisti
parti
les ipn
ondensed matter systems, but instead des
ribing
the s
attering of parti
les represented by an e(cid:27)e
tive low energy model that is formally
equivalent to the Dira
equation (a situation that takes pla
e in graphene). It would
be interesting to derive Eq. (12) from a mi
ros
opi
Hamiltonian. In Appendix A we
give a tight-binding model whose e(cid:27)e
tive low-energy theory is given by Eq. (12). The
ψ† = (ψ∗,ψ∗)
1 2
problem (12) has spinorial wave fun
tion of the form and the probability
S = v (x)ψ†σ ψ
x F x
(cid:29)ux is
omputed as , as
an be shown using the traditional derivation
of
omputing the time
hange of the probability density [17℄.
Hψ = Eψ H
Theeigenproblem , with givenbyEq. (12),
orrespondstotwo
oupled
(cid:28)rst-order di(cid:27)erential equations of the form
~d[ v(x)ψ (x)]
2
v(x) = Eψ (x),
1
i dx (13)
p
p ~d[ v(x)ψ (x)]
1
v(x) = Eψ (x).
2
i dx (14)
p
p
It is straightforward to show that the two (cid:28)rst-order di(cid:27)erential equations given above

an be put in Sturm-Liouville form:
d d E2
[v (x) y (x)] = y (x),
− dx F dx 1 v (x)~2 1 (15)
F
y (x) = v (x)ψ (x) p(x) = v (x) λ = E2/~2 w(x) = 1/v (x) ψ (x)
1 F 1 F F 2
with , , , and , and
obtained fromp
~ d
ψ (x) = v (x) [ v (x)ψ (x)].
2 F F 1
iE dx (16)
p p
If the velo
ity pro(cid:28)le
hanges
ontinuously, one expe
ts that the
ontinuity of the wave
fun
tion should be valid. If the velo
ity pro(cid:28)le
hanges abruptly at a given point in
x = 0 v(x) = v x < 0 v(x) = v x > 0
− +
spa
e, say at , from , for , to , for , one has to use
S
attering in one-dimensional heterostru
tures des
ribed by the Dira
equation 6
y (x) = v (x)ψ (x)
2 F 2
Eqs. (13) and (14) to derive the boundary
onditions. De(cid:28)ning
Eqs. (13) and (14)
an be written as p
~d[y (x)] y (x)
2 1
= E ,
i dx v(x) (17)
~d[y (x)] y (x)
1 2
= E .
i dx v(x) (18)
x = 0
Integrating Eqs. (17) and (18) using a symmetri
in(cid:28)nitesimal interval around
one obtains the
ondition
y (0−) = y (0+),
i i
(19)
i = 1,2
with . Note that
ondition (19) implies the dis
ontinuity of the wave fun
tion.
Let us now move to the solution of several parti
ular
ases.
3.1. The step-like velo
ity pro(cid:28)le
Let us
onsider the
ase
v (x) = v−θ( x)+v+θ(x).
F F − F (20)
The solution of the Dira
equation reads
1 1
ψ (x) = eiqx +r e−iqx,
−
1 ! 1 ! (21)
−
and
1
ψ (x) = t eipx,
+
1 (22)
!
E = v−q~ E = v+p~
with F and F . The above wave fun
tions were obtained by solving the
x ≷ 0
Dira
equation for , where the velo
ity is
onstant. Using these solutions and the
boundary
ondition (19) one obtains
√v−(1+r) = √v+t,
( √v−(1 r) = √v+t, (23)
−
r = 0
whi
h is satis(cid:28)ed only for . Had we
onsider the general
ase of both a velo
ity
pro(cid:28)le (20) and a potential pro(cid:28)le of the form
V(x) = V θ(x),
0
(24)
E > V r = 0
0
under the
ondition , the boundary
onditions would still give . This result
is
alled Klein tunnelling [16℄. If we
onsider the same velo
ity and potential pro(cid:28)les,
E < V
0
but now take the energy , the boundary
onditions still gives the result (23), but
x > 0
the wave fun
tion of the propagating mode for is now di(cid:27)erent and given by
1
ψ (x) = t e−ipx.
+
1 (25)
!
r = 0
In this
ase the result will also be . The
on
lusion is that it is not possible to
ba
ks
atterer massless Dira
ele
trons with a step-like velo
ity and potential pro(cid:28)les in
1D.
S
attering in one-dimensional heterostru
tures des
ribed by the Dira
equation 7
3.2. The linear velo
ity pro(cid:28)le with massless parti
les
We have seen that an abrupt
hange of thevelo
ityattheinterfa
e produ
es no re(cid:29)e
ted
v− v+
parti
les. Let us now study the
ase of a smooth
hange in the velo
ity from F to F.
(The result
an be guessed from the outset!) To that end, we
hoose the pro(cid:28)le
v (x) = v−θ( δ x)+(v¯+x∆)θ(δ x )+v+θ(x δ),
F F − − −| | F − (26)
v¯ ∆
where we have de(cid:28)ned and as
v− +v+
v¯ = F F ,
2 (27)
v+ v−
∆ = F − F .
2δ (28)
x < δ I x > δ III
For the
ases − (region ) or (region ) the Fermi velo
ity is
onstant and
the solution of the Dira
equation is elementary, as we have seen before. The interesting
x < δ II

ase is therefore the region | | (region ). In this
ase we have to use the Dira
Hψ = Eψ H
equation in the form (12). Expli
itly we have to solve the problem with
given by
~ d
H = √v¯+x∆σ √v¯+x∆.
x
i dx (29)
ψ
1
Writting the di(cid:27)erential equations satis(cid:28)ed by the spinors, we obtain for the spinor
y = √v¯+x∆ψ
1
(
onsidering the substitution ) the equation
d2y dy
2 2
v (x) ∆v (x) = ǫ y,
− F dx2 − F dx (30)
ǫ2 = E2/~2
with . Making the repla
ement
v¯+x∆
z = θ(x),
v¯ ≡ (31)
we obtain the result
d2y dy
2 2
z +z +ν y = 0,
dz2 dz (32)
ν2 = ǫ2/∆2 ω = lnz
with . Making the additional repla
ement , Eq. (32) is redu
ed to
that of the harmoni
os
illator §
d2y
2
+ν y = 0.
dω2 (33)
y(x)
The general solution of Eq. (33) is elementary and from it one obtains given by
y(x) = Asin[νLθ(x)]+Bcos[νLθ(x)],
(34)
with
v¯+x∆
Lθ(x) = ln .
v¯ (35)
m
y = z
§ One shoumld2 =noteν2that the soluyti(ozn)=ofAEzqi.ν +(32B)z
−aiνn also be obtained by noti
ing that is a
solution if − , leading to .
S
attering in one-dimensional heterostru
tures des
ribed by the Dira
equation 8
ψ ψ (x) = y(x)/ v (x) ψ
1 1 F 2
The
omponent of the spinor is obtained from , and is
obtained using Eq. (16) and reads p
~ dy(x)
ψ (x) = v (x) .
2 F
iE dx (36)
p ψI( δ) = ψII( δ) ψII(δ) = ψIII(δ)
The boundary
onditions are − − and , and
an be
written as k
eikδ S / v− C / v− 0 r e−ikδ
− − F − F
eikδ C λ− S λ+ 0 A e−ikδ
 −p − −p   =  
0 S / v+ C / v+ eiqδ B 0 (37)
 + F + F −    
 0 C+pλ− −S+pλ+ −eiqδ  t   0 
    
with
S = sin[νLθ( δ)],
±
± (38)
C = sin[νLθ( δ)],
±
± (39)
~ v±∆ ν
λ = F .
±
iE v¯ θ( δ) (40)
p ±
r 2 1 r 2 =
The fra
tion of re(cid:29)e
ted (cid:29)ux is given by | | and the transmitted (cid:29)ux is − | |
v+ t 2/v−
F| | F. The expli
it evaluation (whi
h is some what lengthy) of the
oe(cid:30)
ients gives
2
r = 0,
| | (41)
v−
t 2 = F ,
+
| | v (42)
F
and therefore the ele
trons are totally transmitted a
ross the heterojun
tion. Of
ourse
that this result
ould have been anti
ipated from the
on
lusions of Se
. 3.1, sin
e it is
always possibletorepresent a well-behaved fun
tionby a sum ofin(cid:28)nitesimalre
tangles.
(It is however elegant to have an exa
t solution to a given problem.)
3.3. The linear velo
ity pro(cid:28)le with massive parti
les
We
an now use what we have just learned to a deal with a more general situation
where the ele
troni
spe
trum
hanges due to a
hange of the material. The simplest
x = δ

ase is that where for the mass of the quasi-parti
les jumps from zero to a (cid:28)nite
value. The
onne
tion between the two materials is represented by the linear pro(cid:28)le of
the velo
ity dis
ussed in Se
. 3.2. The mass pro(cid:28)le is
m(x) = mθ(x δ),
− (43)
that is, the
hange is velo
ity takes pla
e outside the region of (cid:28)nite mass. Sin
e the
E = v2q2 +m2v4
+ +
parti
le has a (cid:28)nite mass the spe
trum
hanges to ± , and the wave
fun
tion in this region
hanges to p
E2 m2v4
ψIII(x) = t − + e−iqx.
E mv2 (44)
p − + !
A B L/T λ± T/L
k Note that the dimensions of and are , and has dimensions of .
p p
S
attering in one-dimensional heterostru
tures des
ribed by the Dira
equation 9
eiq E2 m2v4eiq
+
The overall modi(cid:28)
ation is the repla
ement of by − in the third row,
eiq (E mv2)eiq r 2
and of by − + in the fourth row. Workipng out the
al
ulation of | | we
obtain
r 2 = [1+2ǫ( ǫ+√ǫ2 1)],
| | − − − (45)
ǫ = E/(mv2) δ v±
with + . As expe
ted, the result does not depend on and on F, due to
the reasons dis
ussed in Se
. 3.2. A plot of Eq. (45) is given in Fig. 1. The situation
is same if the mass pro(cid:28)le had been
hosen as
m(x) = mθ(x+δ),
(46)
this is, the linear pro(cid:28)le rises up inside the massive region.
1
0.8
0.6
2 |r|
0.4
0.2
0
1 1.2 1.4 1.6 1.8 2
ε
ǫ
Figure 1. Left: plot of Eq. (45) as fun
tion of . Right: plot of Eq. (45) as fun
tion
ǫ
of
3.4. The linear mass pro(cid:28)le
We have seen above that the linear velo
ity pro(cid:28)le does not
ontribute to the
ba
ks
attering of Dira
-fermions, even when itis
ombined witha regionwhere fermions
are massive. In fa
t, any velo
ity pro(cid:28)le produ
es no ba
ks
attering in one-dimension.
All the ba
ks
attering
omes from the
hange in the dispersion, due to the presen
e of
the mass term. This fa
t motivates the question of what is form of the wave fun
tion
r 2
and of the
oe(cid:30)
ient | | if the mass does not
hange abruptly but in a smooth way? A

hoi
e for the
hange of the mass pro(cid:28)le, leading to an exa
t solution, is that des
ribed
by:
x
m(x) = mθ(x δ)+m θ(x)θ(δ x).
− δ − (47)
We approa
h the solution of the s
attering problem by solving (cid:28)rst the Dira
m(x) = mx
equation subje
ted a mass pro(cid:28)le δ. The method of solution is inspired
in that used for the 3+1 Dira
equation [18, 19℄. In this
ase the Fermi velo
ity is
σ mv2 xψ

onstant, and therefore we have to solve Eq. (11) with the additional term z F δ :
~dψ x
2
v σ +σ mv ψ = Eψ.
F xi dx z F δ (48)
S
attering in one-dimensional heterostru
tures des
ribed by the Dira
equation 10
The solution of this problem pro
eeds in several steps. The (cid:28)rst is to operate on the
left of Eq. (48) with the operator
d
σ σ .
y z
dx (49)
After some algebra we obtain the following result
d2ψ
~2v2 = [(E2 m2v4x2/δ2)+σ ~mv3/δ]ψ.
F dx2 − − F y F (50)
The two
omponents of the spinor are still
oupled in Eq. (50). In order to de
oupled
them we use the unitary transformation
φ = U†ψ,
(51)
1 0
U = .
0 i (52)
!
σ σ˜ = U†σ U
y y y
The above transformation
hanges to , whi
h, still mixing the spinors in
the Eq. (53) below, does it with the same sign (this is a
ru
ial step). After applying
the unitary transformation we obtain
d2φ
~2v2 = [(E2 m2v4x2/δ2)+σ˜ ~mv3/δ]φ.
F dx2 − − F y F (53)
F = φ φ φ i = 1,2
± 1 2 i
We now introdu
e two new fun
tions de(cid:28)ned by ± , where (with )
φ F
±
are the
omponents of the spinor . In terms of the fun
tions the eigenproblem takes
the form:
d2F
~2v2 ± +[m2v4x2/δ2 ǫ2]F = 0,
− F dx2 F − ± ± (54)
ǫ2 = E2 ~mv3/δ
where ± ± F . The general solutionof Eq. (54) isgiven interms of paraboli
D (x)
ν

ylinder fun
tions (see Appendix B)
F±(x) = A±Dν1±(√2(b/a)1/4x)+B±Dν2±(i√2(b/a)1/4x), (55)
with
1 ǫ2
ν± = + ± = ν (1 1)/2,
1
− 2 2√ab − ∓ (56)
1 ǫ2
ν± = ± = ν± 1
2 1
− 2 − 2√ab − − (57)
E2δ
ν = ,
2~v3m (58)
F
a = ~2v2 ,
F (59)
4 2 2
b = v m /δ .
F (60)
x
If we were looking for the solution of the problem valid to all values of , the fun
tions
D (x)
ν
with imaginary argument, sin
e are not real and are not normalizable in the
x
in(cid:28)nite volume, had to be ex
luded. Therefore, in the limit → ±∞ the normalizable