The effect of Majorana fermions on the Andreev ... · spectroscopy applied to topological multiband...
Transcript of The effect of Majorana fermions on the Andreev ... · spectroscopy applied to topological multiband...
The effect of Majorana fermions on the Andreevspectroscopy applied to topological multiband
superconductors
Ana Cristina Oliveira Silva
Thesis to obtain the Master of Science Degree in
Engineering Physics
Supervisors: Professor Pedro Domingos Santos do SacramentoProfessor Miguel António da Nova Araújo
Examination Committee
Chairperson: Professor Ana Maria Vergueiro Monteiro Cidade MourãoSupervisor: Professor Miguel António da Nova AraújoMember of the Committee: Professor Eduardo Filipe Vieira de Castro
October 2014
Acknowledgments
I would like to express my sincere gratitude towards my supervisors, P.D.Sacramento and M.A.N. Araujo,
for all they have taught me and for their endless patience.
Thank you.
iii
Resumo
E estudada a condutancia diferencial na interface entre um metal normal e um supercondutor topologico
com duas orbitais por sıtio da rede. As condicoes de fronteira que asseguram uma funcao de onda
contınua e injectiva foram obtidas no contexto da teoria de ”Quantum waveguide”. Na interface entre
um metal normal e um supercondutor, esta presente uma forma adicional de reflexao, a reflexao de
Andreev: um electrao incidente forma um par de Cooper na interface e e reflectida uma lacuna para a
regiao do metal normal. Um supercondutor topologico e caracterizado pela presenca de estados con-
dutores de energia nula localizados ao longo da superfıcie, “edge-states”, para os quais os operadores
de criacao coincidem com os proprios operadores de destruicao, podendo portanto ser identificados
como fermioes Majorana. O Hamiltoniano escolhido para representar o sistema permite o desacopla-
mento da matriz de Bogoliubov-De Gennes em duas matrizes 4× 4. Considerando somente um destes
sub-espacos, o sub-sistema resultante apresenta duas fases distintas - uma fase trivial e uma fase
topologica. A fase topologica caracteriza-se pela existencia de um numero de Chern nao nulo. Este
por sua vez esta interligado com o numero de “edge-states” e, portanto, com o numero de fermioes
Majorana localizados na interface. A condutancia diferencial, obtida a partir da teoria de ”Quantum
waveguide”, mostra ser consistente com o ındice topologico do sistema em contraste com a aplicacao
do modelo de Blonder-Tinkham-Klapwijck , revelando igualmente a existencia de efeitos de interferencia
destrutiva entre as bandas, nao contabilizados se estas forem consideradas independentemente.
Palavras-chave: Teoria de quantum waveguide, fermiao Majorana , supercondutor topologico,
estados ligados de Andreev, condutancia diferencial.
v
Abstract
The differential conductance between a normal metal and a two-band topological superconductor is
studied. To address the wavefunction’s matching conditions at the interface, the extension of quantum
waveguide theory to the Andreev scattering problem is applied. For a normal metal-superconductor
junction, the scattering mechanism known as the Andreev reflection will occur for small energy bias.
It can be briefly defined as the process in which an incident electron will be reflected back as hole,
with the transmission of a Cooper pair into the superconductor’s side. A topological superconductor
is characterized by the existence of zero-energy conducting states at the interface which can be iden-
tified as Majorana fermions. To represent this system we considered a Hamiltonian model in which
the Bogoliubov-De Gennes matrix will split into two 4 × 4 matrices. Considering only one of these two
subspaces the subsystem can be tuned either to a trivial or topological phase by altering one of the
band structure parameters and keeping the remaining parameters fixed. The topological phase will be
characterized by a non-zero Chern number. As this quantity can be related to the number of edge-
states by the bulk-boundary correspondence, its value gives the number of localized Majorana fermions.
The calculated differential conductance within the framework of quantum waveguide theory is shown to
be consistent with the topological index, in contrast with the historically established Blonder-Tinkham-
Klapwijck model, and it also reveals the existence of destructive interference effects, which cannot be
attained if the conduction paths are treated independently.
Keywords: Quantum waveguide theory, Majorana fermion, topological superconductor, An-
dreev bound state, differential conductance
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Contents
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Resumo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1 Fundamental concepts 1
1.1 An introduction to Topological band theory . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Symmetry transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 Time-reversal symmetry and the Z2 invariant . . . . . . . . . . . . . . . . . . . . . 5
1.2.2 Particle-hole symmetry in Bogolubov-De Gennes Hamiltonians and the existence
of Majorana fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Kitaev’s model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Andreev spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5 Overview of the BTK model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.6 Differential conductance and ABS in a d-wave superconductor . . . . . . . . . . . . . . . 19
1.7 Pseudo-spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2 Andreev spectroscopy in a multiband topological superconductor 24
2.1 The model for the Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2 Quantum waveguide theory of Andreev spectroscopy . . . . . . . . . . . . . . . . . . . . . 29
2.3 Application of the waveguide theory to the N-S junction . . . . . . . . . . . . . . . . . . . 33
2.4 Differential conductance of the N-S boundary . . . . . . . . . . . . . . . . . . . . . . . . . 37
3 Conclusions and future work 45
Bibliography 49
A Derivation of the Bogolubov equations for a non-uniform superconductor 51
B Fukui-Hatsugai-Suzuki method for calculating the Chern number on a discretized Brillouin
zone. 54
ix
List of Figures
1.1 Representation of the Brillouin zone in 2D as a torus. . . . . . . . . . . . . . . . . . . . . . 2
1.2 Schematic representation of the Hamiltonian in eq(1.38), in a trivial (a) and non-trivial (b)
limits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 Schematic illustration of an Andreev point contact experiment between a normal metal
(yellow) and a superconductor (blue). The Andreev reflection process is represented: an
incoming electron (red dot) enters the superconductor by forming a Cooper pair (two red
dots) and emitting a hole back (blue dot). Spin orientation is denoted by the white arrows.
Image from reference [18]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4 Schematic illustration of the tunneling transport between a normal metal and a supercon-
ductor. µN represents the chemical potential at the normal state region. In both sides the
density of states is represented: the blue color represents occupied states. Only when
µN > ∆ the current, represented by ΓN→S , will flow through the interface. Image from
reference [22]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.5 Normalized differential conductance as a function of the incident electron’s energy from
reference [20]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.6 Schematic illustration of two consecutive Andreev reflections at the interface between a
normal metal and d-wave superconductor, when the interface is oriented perpendicular to
a nodal direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.7 Normalized differential conductance for various barrier strengths and in two distinct cases:
a) the pairing potential is isotropic and one observes essentially the same behavior as in
fig(1.5) ; b) Normalized differential conductance corresponding to the situation described
in fig(1.6). A zero bias conductance peak appears, insensitive to the dimensionless barrier
strength Z. Image from reference [21]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.8 Schematic representation of a normal metal- topological superconductor junction. The
red stars indicate the localized Majorana fermions. Image from reference [26]. . . . . . . 13
1.9 Quasiparticle excitations with positive group velocity in the superconductor. Image adapted
from reference [33]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
xi
1.10 Quasiparticle excitations at both sides of the normal metal (N) - superconductor (S) in-
terface. The digits indicate the momentum of the excitations. The relevant ones indicate
as follows: 0 is the incoming electron, 2 is the transmitted excitation crossing through the
Fermi surface, 4 is the transmitted quasiparticle excitation with no Fermi surface crossing,
5 represents the normal reflection at the interface and 6 the Andreev reflection. Image
from reference [33]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.11 Superconductor’s excitation spectrum. The dashed lines are the normal metal’s excitation
spectrum. Image adapted from reference [19] . . . . . . . . . . . . . . . . . . . . . . . . 18
1.12 Illustration of the scattering processes in a normal-anisotropic superconductor junction.
Image from reference [21]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.1 Fermi surface and ξ+(k) band in the trivial phase. . . . . . . . . . . . . . . . . . . . . . . 28
2.2 Fermi surface and ξ+(k) band in the topological phase. . . . . . . . . . . . . . . . . . . . 28
2.3 Energy spectrum of the Hamiltonian model eq(2.7) for an infinite ribbon in the yy (or xx)
direction. The Majorana fermion can be seen for the longitudinal momentum π. This figure
was obtained by M.A.N. Araujo. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.4 Node occurring at the interface between a normal metal and a two-band superconductor.
Image from reference [28]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.5 Tight-binding showing the splitting of the incoming electron into two channels of the su-
perconductor, similar to a waveguide. Image from reference [28]. . . . . . . . . . . . . . 30
2.6 Tight-binding showing the splitting of the incoming electron into two pseudo-spin channels
of the superconductor, similar to a waveguide. . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.7 Conductance gs and gn as function of the incident electron energy for t2 = +0.08 in a
microconstriction, Z=0, for a fixed transverse momentum py= 0. Calculation performed
with waveguide theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.8 Conductance gs and gn as function of the incident electron energy for t2 = +0.08 and Z=1,
for a fixed transverse momentum py=0. Calculation performed with waveguide theory. . . 38
2.9 Conductance gs and gn as function of the incident electron energy for t2 = +0.08 and Z=2,
for a fixed transverse momentum py= 0. Calculation performed with waveguide theory. . 38
2.10 Conductance gs as function of the incident electron energy for t2 = +0.08 in a microcon-
striction, Z=0, computed with BTK generalized model, for a fixed transverse momentum
py=0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.11 Conductance gs and gn as function of the incident electron energy for t2 = +0.08 and Z=0,
for a fixed transverse momentum py=π. Calculation performed with waveguide theory. . . 39
2.12 Conductance gs and gn as function of the incident electron energy for t2 = +0.08 and Z=1,
for a fixed transverse momentum py=π. Calculation performed with waveguide theory. . . 40
2.13 Conductance gs and gn as function of the incident electron energy for t2 = +0.08 and Z=2,
for a fixed transverse momentum py=π. Calculation performed with waveguide theory. . . 40
xii
2.14 Conductance gs and gn as function of the incident electron energy for t2 = −0.08 and Z=0
and a fixed transverse momentum py=0. Calculation performed with waveguide theory. . 41
2.15 Conductance gs and gn as function of the incident electron energy for t2 = −0.08 and Z=1
and a fixed transverse momentum py=0. Calculation performed with waveguide theory. . 41
2.16 Conductance gs and gn as function of the incident electron energy for t2 = −0.08 and Z=2
and a fixed transverse momentum py=0. Calculation performed with waveguide theory. . 42
2.17 Conductance gs as function of the incident electron energy for t2 = −0.08 with various
strength barriers and a fixed transverse momentum py=π. Calculation performed using
the generalized BTK model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.18 Conductance gs as function of the incident electron energy for t2 = −0.08 with various
strength barriers and a fixed transverse momentum py=π. Calculation performed using
waveguide theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
xiii
Chapter 1
Fundamental concepts
1.1 An introduction to Topological band theory
The building blocks of matter can interact in various forms, which subjected to certain external condi-
tions, can lead to different states of matter. In the theory of phase transitions, one usually defines a local
order parameter which acquires a distinct expectation value as the different phases are established. By
comparison with this framework, topological states of matter emerge as a new paradigm [1]: not only is
it impossible to define a local order parameter, but also the different topological states are differentiated
upon the inability to adiabatically deform the energy bands into one another, without closing the bulk
gap. Thus, a material in a topological phase, in contact with another in a trivial phase, has to undergo
a phase transition at the interface, accomplished by the vanishing of the bulk gap. At the points where
the bulk gap collapses, gapless conducting states arise. These states are, therefore, able to conduct
current along the interface and are known as edge-states [2, 3]. This scenario can occur, for example,
at the interface between a topological insulator and the vacuum. In this case, one can see more easily
that the appearance of such edge-states is highly interconnected to the non-trivial topology of the bulk,
a phenomena known as bulk-edge correspondence [2].
The gapless modes exhibit more robustness in face of local disorder, which is another peculiarity of the
topological phase, and are related with the existence of a non-zero topological invariant [4].
In this thesis, the dominant focus falls to one topological invariant, the Chern number (present in systems
without time-reversal symmetry), although some remarks will be given to a second invariant, the Z2 (for
time-invariant systems).
The mathematical construction of the Chern number is rooted in the Berry phase (γn) : as the Hamilto-
nian of the system is adiabatically transformed by slowly varying its parameters, the eigenstates gain a
phase, γ, which contains a new term, besides the usual dinamical phase [5] ,
γn = i
∮〈ψn(R)| ∂
∂R|ψn(R)〉 · δR . (1.1)
The function integrated in the equation above is called the Berry connection or Berry vector potential.
From this quantity we define the Berry curvature:
1
Ωn = ∇×An (1.2)
Integrating the Berry curvature over all the Brillouin zone defines the topological invariant Chern
number, an integer which is non-zero for the topological phases.
Cn =1
2π
∫TBZ
Ωn dS . (1.3)
It is important to note that this integral is directly related to the Berry phase: the integrals in eq(1.3)
and in eq(1.1) are equivalent as a result of the application of Stokes’ theorem.Therefore one can also
define the Chern number as the accumulated Berry phase around the Brillouin zone divided by 2π, which
will be invariant under a smooth gauge transformation of the eigenstates [1]. For a two-band model, a
geometric interpretation of the Chern number is also possible. In these systems, the Hamiltonian can
be written as H = h(k).σ and it can be showed that eq(1.3) counts the number of times the unit vector
h/|h| covers the entire unit sphere as a function of the momentum k [2].
To determine the Chern number, it is necessary to solve eq(1.3). One strategy is to apply Stokes’ the-
orem. However, as the Brillouin zone is a torus (fig. 1.1) it has no boundary, so if the Berry curvature
is a well behaved function one would always obtain a zero Chern number, which is precisely what hap-
pens for non-topological phases [6]. Nevertheless, the procedure breaks down if the Berry connection
presents singularities. The origin of such problematic points lies on the impossibility of picking a global
gauge that guarantees a smooth and single-valued wavefunction [6, 3]. Instead, it is necessary to define
distinct gauges in different regions of the Brillouin zone and separately apply the Stokes’ theorem to the
different patches.
Figure 1.1: Brillouin zone in 2D. Image adapted from reference [6].
In order to clarify how this can lead to an integer number, it is useful to present the example given by
the authors B. Andrei Bernevig and Taylor L.Hughes [3]: one first considers the existence of two regions
in the Brillouin zone, R and TBZ −R. Inside each patch, it is possible to choose a local gauge such
that the wavefunction is well-behaved and vanishes when the region boundaries are crossed. Identifying
eif(k)ψ1 with the wavefunctin at TBZ−R and eig(k)ψ2 with the region R , where eif(k) and eig(k) are the
respective gauge transformations, allows to state the boundary condition as the following:
2
eif(k)ψ1 = eig(k)ψ2 (1.4)
⇔ ψ2 = e−iθ(k)ψ1, with θ(k) = g(k)− f(k) (1.5)
Accordingly, the two Berry connections are related:
A2(k) = A1(k) +∇θ(k) (1.6)
The next step is to explicitly evaluate the integral for the Chern number:
Cn =1
2π
(∫R∇×A1(k) +
∫TBZ−R
∇×A2(k))
(1.7)
Remembering that the torus has no boundary, the application of Stokes’ theorem to the last equa-
tion enables the line integral of the two connections to be calculated over the same boundary ∂R with
opposite orientations: ∂(TBZ −R) = −∂R . Then eq( 1.7 ) can then be written as:
Cn =1
2π
∫∂R
dk∇θ(k) (1.8)
Thus, the promised result has been achieved. If ∂R is a close loop, the wavefunction has to be the
same in the starting and end points which forces the previous integration to be proportional to 2π . This
is the same reasoning as to why the winding number in a vortex is quantized.
The definition of the Chern number, eq.(1.3), can yet be presented differently: one starts from explicitly
computing the Berry curvature for a given occupied band n,
Ωnkx,ky = ∇×An =∂Any∂kx
− ∂Anx∂ky
(1.9)
which by virtue of eq(1.1) becomes:
Ωnkx,ky = i[〈∂ψn∂kx|∂ψn∂ky〉 − 〈∂ψn
∂ky|∂ψn∂kx〉] (1.10)
Taking the inner product of the state 〈ψ′n| in both sides of the eigenvalue equation, H|ψ〉 = E|ψ〉 , and
differentiating the resulting equation, produces an equality similar to the Hellmann-Feynman theorem:
〈ψn′ |∂ψn∂kx〉 =
1
En − En′〈ψn′ |
∂H
∂kx|ψn〉 (1.11)
After which is possible to rewrite the state |∂ψn
∂kx〉 as a linear expansion in the eigenstates |ψn′〉 , with
coefficients for n 6= n′ given by the last equation. Hence, using this linear expansion representation of
the states |∂ψn
∂kx〉, the Berry curvature gains the new form:
Ωnkx,ky = i∑n 6=n
〈ψn| ∂H∂kx |ψn′〉〈ψn′ |∂H∂ky|ψn〉 − 〈ψn| ∂H∂ky |ψn′〉〈ψn′ |
∂H∂kx|ψn〉
(En − En′)2(1.12)
This alternative expression can be shown to be related to the Kubo formula for the Hall conductance
3
[7]:
σH =e2
h
1
2π
∫Ωnkx,kydkxdky, (1.13)
where one can recognize the integral above as the Chern number defined in eq.(1.3), thus allowing to
relate the Chern number with an observable quantity, the Hall conductance. Additionally, eq(1.9) reveals
that the sum of the Chern number of all bands must be zero, owing to the anticomutative property of the
cross product. For example, in a two-band system, taking |−〉 to be the eigenstate of the lower energy
band and |+〉 the eigenstate of the upper band, the Berry potential in each band is given by:
Ω− = i〈−|∇H(k)|+〉 × 〈+|∇H(k)|−〉
(E− − E+)2(1.14)
Ω+ = i〈+|∇H(k)|−〉 × 〈−|∇H(k)|+〉
(E+ − E−)2(1.15)
⇒ C+ = −C− , (1.16)
explicitly illustrating how the sum in all bands of the Chern number will be zero.
The Chern number can also be calculated through the bulk-boundary correspondence [2], which math-
ematically is stated in the following way: if there are NR edge states with positive group velocity and NL
with negative, the change in the change in the Chern number across the interface will be quantified by
∆C = NR −NL . (1.17)
Then if the interface demarks the separation between a trivial and a topological system, the Chern
number can be determined directly by the above equation.
1.2 Symmetry transformations
Provided that two physical systems share the same topological invariant, they are topologically equiva-
lent, which suggests that there must be some underlying symmetries enabling the division of systems
into topological classes [8].
Two important symmetries are time-reversal symmetry (TRS) and particle-hole symmetry (PHS), both
implemented by antiunitary operators on the Hilbert space. Any antiunitary operator will be defined as
the general operator A satisfying:
〈Aφ|Aψ〉 = 〈φ∗|ψ∗〉 (1.18)
Such operators will be in general represented by the complex conjugation operator (K) and an unitary
operator (U ) in the form KU , which can either be squared +1 or −1 [8, 3].
4
1.2.1 Time-reversal symmetry and the Z2 invariant
The single-particle Hamiltonian possesses time-reversal symmetry if it verifies [3]:
H = T HT −1 ,with: T 2 = ±1 , (1.19)
where T is the time-reversal operator. The ± sign regards two distinct cases: the + sign corresponds
to the TRS operator for spinless (or integer spin) particles, while the − sign is applicable for half-integer
spinful particles. In the spinless case, the TRS operator is simply given by the complex conjugation
operator, T = K, whereas for half-integer spin particles T = iσyK.
If the Hamiltonian of the system is said to possess TRS, the single-particle Hamiltonian in momentum
space will have to verify the conditions [8]:
H(k) = H∗(−k) Spinless particles , (1.20)
H(k) = iσyH∗(−k)(−iσy) Spinful particles . (1.21)
In the case of the superconductor’s Hamiltonian, written in the general form:
H =1
2
∑k
a†k. HBdG .ak , HBdG =
Ξ ∆
∆† −Ξ∗
, (1.22)
where ak = (ck↑ ck↓ c†−k↑ c†−k↓)T . The condition for TRS is similar to eq(1.21):
HBdG(k) = iσyH∗BdG(−k)(−iσy) . (1.23)
For spinful particles, the fact that the TRS operator squares to −1 implies that such systems obey
the Kramers’ theorem: for an odd number of particles with half-integer spin, there are at least two
degenerate eigenstates, Ψ and T Ψ [9, 3]. This result is established by proving that Ψ and T Ψ are
orthogonal eigenstates. To prove it, one first must notice that:
〈T ψ|T 2ψ〉 = 〈ψ|T ψ〉∗ , because T is anti-unitary . (1.24)
Nevertheless, for spinful particles it is also true that:
〈T ψ|T 2ψ〉 = −〈T ψ|ψ〉 = −〈ψ|T ψ〉∗ , (1.25)
and thus
〈ψ|T ψ〉∗ = −〈ψ|T ψ〉∗ ⇒ 〈ψ|T ψ〉 = 0 . (1.26)
A relevant consequence of this theorem applies to the elements of the scattering matrix 〈T Ψ|H|Ψ〉,
particularly if the scattering process concerns an odd number of excitations, then the probability of
5
scattering between the two states is zero. This can be seen by noticing that:
〈ψ|H|T ψ〉 = 〈T ψ|T HT ψ〉∗ = 〈T ψ|T HT −1T 2ψ〉∗ = 〈T ψ|HT 2ψ〉∗ = −〈T ψ|H|ψ〉∗ , (1.27)
where it has been used the time-reversal invariance of the Hamiltonian of the system (H = T HT −1).
However, because H is Hermitian, then:
〈T ψ|H|ψ〉∗ = 〈ψ|H|T ψ〉 , (1.28)
which when substituting in eq( 1.27 ), leads to:
〈ψ|H|T ψ〉 = −〈ψ|H|T ψ〉 ⇒ 〈ψ|H|T ψ〉 = 0 . (1.29)
One way of viewing the underlying implications of this result is by imagining a 2D-geometry where a
fermion moves in the x-direction. It will never be backscattered, unless TRS has been broken.
Under TRS the Berry curvature changes sign. Therefore a TRS system has C = −C = 0 [3].
In time-reversal invariant systems, a non-trivial phase will be characterized by a different topological
invariant, the Z2, which can take upon two values: ν = 0 (trivial phase) and ν = 1 (topological phase)
[2]. This topological invariant was first introduced by Kane and Mele [10] when studying the effect of
spin-orbit interaction in the Hamiltonian of a single plane of graphene. To exemplify how a topological
phase can emerge in the presence of TRS, it is sufficient to address the simplest form of the spin-orbit
term considered by these authors, in which the Sz spin component is conserved [11]. In this case, the
Hamiltonian will decouple into two independent contributions: the Hamiltonian for spin-up electrons and
the correspondent term for spin-down electrons [11]. The system altogether respects TRS, but each
spin contribution taken separately does not and hence spin-up and spin-down carry opposite Chern
numbers [11]. As a consequence, at the boundaries of the system counter-propagating edge-modes
will both appear at time-reversal invariant points, but because TRS is present, spin-up and spin-down
edge-states will not mix ( single-particle backscattering is prohibited) [3]. At other momentum points,
the spin-orbit interaction will lift the degeneracy imposed by Kramers’ theorem. The connection between
states at different time invariant points, for example k = 0 and k = π, can occur in two distinct forms,
depending on the topological properties of the bulk system. If the energy dispersion relation of each
pair crosses the Fermi energy an even number of times, there will be an even number of Kramers’ pairs
and a gap will open due to backscattering. The system will be topologically equivalent to a trivial phase.
However, if the energy dispersion relation of each pair intersects the Fermi energy an odd number of
times, backscattering cannot occur: they are protected from opening a gap by TRS and the edge-states
will propagate freely even under small perturbations, leading to the propagation of ”spin-filtered” current
[10, 3, 2]. In this case, the system is in a non-trivial phase which by analogy to the quantum hall effect
is called the quantum spin hall phase [11]. The value of the Z2 invariant can therefore be calculated by
the bulk-boundary correspondence, relating the number of Kramers’ pairs edge-states ( NK ) with the
6
change of the topological invariant at the boundary of the system (∆ν) [2]:
NK = ∆ν mod 2 (1.30)
It should be noted that the eq(1.30) is not the only way of calculating the Z2 invariant [10, 2, 3]. Other
methods include, for example, interpreting the Z2 as the obstruction to Stokes’ theorem in half of the
Brillouin zone [3].
1.2.2 Particle-hole symmetry in Bogolubov-De Gennes Hamiltonians and the
existence of Majorana fermions
The Bogolubov-De Gennes Hamiltonian (HBdG) written as in eq(1.22) has PHS, defined as [8]:
HBdG(k) = −τxH∗BdG(−k)τx ,where the Pauli matrix τx operates in particle-hole space . (1.31)
This is an intrinsic symmetry of the HBdG matrix in eq(1.22) due to the Hermiticity of Ξ and because
∆(k) = −∆T (k), by imposition of the Fermi statistics [12, 8], with the pairing potential ∆ either a singlet
or a triplet pairing [12]:
Singlet pairing: ∆(k) = ψ(k)iσy =
0 ψ(k)
−ψ(k) 0
, ψ(k) even function of k (1.32)
Triplet pairing: ∆(k) = i(d(k).σ)σy =
−dx(k) + idy(k) dz(k)
dz(k) dx(k) + idy(k)
, d(k) odd function of k .
(1.33)
The Hamiltonian in eq( 1.22 ) can be diagonalized by performing a Bogolubov transformation, allow-
ing to express the Hamiltonian in terms of the non-interacting quasiparticle operators [13]. In real space
the quasiparticle destruction operator will be:
γ =
∫dr u∗↑(r)ψ↑ + u∗↓(r)ψ↓ + v∗↑(r)ψ†↑ + v∗↓(r)ψ†↓ . (1.34)
The eigenstates of the HBdG are given by: φ(r) = (u↑(r) u↓(r) v↑(r) v↓(r))T . Eq( 1.31 ) implies
that if the eigenstate φ has energy E, the eigenstate τxφ∗ will have −E energy. Thus, if E = 0 and this
is a non-degenerate state, PHS will lead to:
φ(r) = τxφ∗(r)⇔ γ = γ† ⇒ Majorana fermion . (1.35)
In analogy with particle physics, this equality can be viewed as the equivalence between particle and
anti-particle, which is exactly the definition of a Majorana particle, or in this case, a Majorana fermion.
Superconductors are, therefore, ideal candidates for creating Majorana fermions from the quasiparticle
7
excitations [14, 15, 16].
1.3 Kitaev’s model
Conversely, it is possible to describe an usual fermion as superposition of Majorana modes [14, 15, 3].
This can be clarified by constructing an algebraic representation where the usual fermionic operators
are written in terms of Majorana operators (eq(1.35)), which satisfy [14, 15, 16]:
γi, γj = 2δij . (1.36)
The fermionic operators can then be presented as [17, 3, 14]:
cj = 12 (γ2j−1 + iγ2j)
c†j = 12 (γ2j−1 − iγ2j)
(1.37)
explicitly showing how a conventional fermion can be thought as a combination of Majoranas in a su-
perconductor. A question that can then arise is whether it will be possible to have single Majoranas in
a superconductor, without them pairing up to form an usual fermion. This indeed is accomplishable in
a topological superconductor, where they will appear as edge states bounded to an interface delimiting
a topological phase transition [3, 14, 15, 16]. A simple model that illustrates qualitatively this feature
is Kitaev’s model [17], devised for a p-wave superconductor. In Kitaev’s paper, the superconductucting
gap is written in the polar form ∆ = |∆|eiθ , but for simplicity, it shall be use the same assumption as in
reference [3], where it is assumed that a global phase rotation was applied to the Hamiltonian, in order to
consider ∆ real. Then, the Hamiltonian for the one-dimensional p-wave superconductor can be written
as:
H =∑j
[− t(c†jcj+1 + c†j+1cj)− µc
†jcj + |∆|(c†j+1c
†j + cjcj+1)
], (1.38)
where, the parameters µ, t, ∆ are, respectively, the chemical potential, the hopping parameter and
the superconductor pairing potential and the geometry considered is a wire with L sites. By inserting eq(
1.37 ) in eq( 1.38 ), the Hamiltonian will be written in terms of the Majorana operators:
H =i
2
∑i
(−µγ2j−1γ2j + (t+ |∆|)γ2jγ2j+1 + (−t+ |∆|)γ2j−1γ2j+2 (1.39)
This representation will allow one to understand in a simpler form the existence of two very distinct
cases, attained with the following choice of parameters [17, 3, 14] :
a)µ < 0, |∆| = t = 0
b)µ = 0, |∆| = t > 0
To gain better insight on what are the key properties exclusive to a particular choice of parameters,
it is necessary to probe the Hamiltonian in the different limits:
8
a)H → −µ i2∑j(γ2j−1γ2j)
b)H → it∑j(γ2jγ2j+1)
In a) one can check that i2γ2j−1γ2j = c†jcj − 1
2 , thus enabling the identification of the ground-state
with the vacuum. This means a) must be a trivial phase (see fig.(1.2) (a)). However, in b) the coupling
between Majorana fermions neglects the sites j=1 and j=L, leading to the existence of two widely sepa-
rated and localized Majorana fermions. The rest of the Majorana operators pair up to form conventional
fermions, in the same way as in a), but the existence of these two zero energy modes at the end of the
chain reveals the existence of a non-trivial topological phase (see fig.(1.2) (b)) [17].
Figure 1.2: Schematic representation of the two distinct phases in Kitaev’s model, where the oval struc-tures are the conventional fermionic operators in the lattice sites and the interior circles are the twoMajorana operators. a) represents the trivial phase, where all Majorana fermions are paired to formusual fermions; b) exemplifies the topological phase, where unpaired Majorana fermions exist at theedges of the chain and are identified by the digits 1 and 2. Image adapted from reference [3].
The given example does not correspond to any particular material and was only intended as a toy-
model, which may then impose the question: is there any known material who possess this state of
matter?
A candidate for this topological phase is the electron-doped insulator CuxBi2Se3, which becomes su-
perconductor below 3K [18].
1.4 Andreev spectroscopy
When a normal metal and a superconductor are in close vicinity, new features of electronic transport can
be observed. The proximity between the two structures is mediated by an interface, which may or may
not include a tunnel barrier. In the absence of such barrier, the interface is called a microconstriction - an
inter-metalic point contact [19]. Different electronic transport mechanisms are privileged whether the in-
terface is a ’pure’ point contact or a tunnel barrier. In the first case, the predominant process of scattering
is the Andreev reflection, which can be understood in the following way: an incoming electron from the
normal metal, with energy below the superconducting gap ∆ , tries to propagate through the interface
to the superconductor region. The only way the superconductor can accommodate a freely propagating
quasiparticle excitation is in the energy states above the superconducting gap, which means the incom-
ing excitation from the normal metal can only be transmitted if it exponentially decays to the ground state
[19, 20]. This is accomplished by a process of condensation of the first electron with another electron
9
of the normal metal to form a Cooper pair [20]. The second electron can be viewed as reflecting a hole
back to the normal metal, and so the Andreev reflection can be briefly defined as a process of reflection
of an electron into a hole, with the transmission of a Cooper pair into the superconductor’s side (fig.(1.3))
[19].
Figure 1.3: Schematic illustration of an Andreev point contact experiment between a normal metal (yel-low) and a superconductor (blue). The Andreev reflection process is represented: an incoming electron(red dot) enters the superconductor by forming a Cooper pair (two red dots) and emitting a hole back(blue dot). Spin orientation is denoted by the white arrows. Image from reference [18].
The current that flows through a clean interface into the superconductor is thus twice as the observed
in the normal metal [20, 19].
The presence of disorder (tunneling barrier) at the interface reduces the transmission (fig.(1.4)), except
if the conduction occurs through an Andreev Bound State (ABS) [21].
Figure 1.4: Schematic illustration of the tunneling transport between a normal metal and a superconduc-tor. µN represents the chemical potential at the normal state region. In both sides the density of statesis represented: the blue color represents occupied states. Only when µN > ∆ the current, representedby ΓN→S , will flow through the interface. Image from reference [22].
10
In 1982, Blonder, Tinkham and Klapwijk (BTK) constructed a theory in which the differential conduc-
tance of a normal-superconductor junction could be determined for various kinds of interfaces, ranging
from a point contact microconstriction to a tunneling junction [20]. The variation from both regimes was
successfully accomplished by introducing a localized repulsive potential at the interface, with a strength
controlled by a dimensionless parameter, Z. Later on, a revision of the microscopic construction of this
theory will be presented, explicitly showing how this parameter emerges from the boundary conditions
at the interface.
Figure 1.5: Normalized differential conductance attained in the BTK paper [20] as a function of appliedbias energy, for various barrier strengths. In the microconstriction case (Z=0), the Andreev scatteringprocess dominates for bias voltages less then ∆. As Z increases and particularly when Z = 5, anenhanced peak appears when there is an energy ∆ available.
Fig(1.5) is one of the calculations included in the BTK paper; it is visible the phenomena of excess
current at zero bias due to the dominant Andreev reflection and also that an enhanced peak at V = ∆
bias exists when the strength of the repulsive potential increases, illustrating how point contact spec-
troscopy can be used to bring into knowledge important properties of the superconducting state, such
as the value of the superconductor’s pairing potential ∆ [23]. It is important to point out that the BTK
theory was devised for s-wave superconductors. In fact, different behaviours can arise as soon as the
pairing symmetry is modified and d-wave superconductivity is a good example of such, with the appear-
ance of zero-energy localized sates at the interface known as Andreev bound states (ABS) [24, 21].
For d-wave superconductors the pairing potential is anisotropic and has nodes where it changes sign
[24]. The relationship between the existence of Andreev bound states and the symmetry of the pairing
potential can be understood imagining a normal metal-superconductor junction, where the normal metal
side can be thought as a slab with a finite size thickness [24, 25]. The specified geometry is just a tool
for a better mental picture of the process, because in fact what follows is as easily valid if the thickness
11
of the normal metal slab is taken to be zero [21, 24, 25]. Then, if an incoming electron from the metal
strikes the interface at zero bias voltage, it will be submitted to the Andreev reflection, resulting in a
hole being propagated to the normal metal side, which will be specularly reflected at the back of the
slab. The hole will propagate back to the interface and again will be Andreev reflected, but now under
the influence of a pair potential which is the sign-reversed version of the one who acted upon the first
Andreev reflection. The Andreev bound state can be thought as this electron-hole pair, confined to the
normal slab region, undergoing repeated Andreev reflections (fig.(1.6)) [24, 25]. If the thickness of the
slab is taken to be infinitely small, then the bound state will be localized at the interface, where a node
of the pair potential occurs [21].
Figure 1.6: Schematic representation of the ABS for a normal metal - d-wave superconductor junction,when the interface is oriented prependicular to the nodal direction. Image adapted from reference [25].
In 1996, Kashiwaya,Tanaka and Kajimura developed a generalization of the BTK framework for
anisotropic superconductors [21]. The authors found that, for d-wave superconductivity, the presence of
ABS have an impact on the differential conductivity, as it is showed in fig(1.7, b) ): now the enhanced
peak for finite strength barriers is located at zero bias, rather then at a V = ∆ (fig(1.5)), a phenomena
known as zero-bias anomaly or zero-bias conductance peak (ZBCP).
Figure 1.7: Normalized differential conductance for various barrier strengths and in two distinct cases:a) the pairing potential is isotropic and one observes essentially the same behavior as in fig(1.5) ; b)Normalized differential conductance corresponding to the situation described in fig(1.6). A zero biasconductance peak appears, insensitive to the dimensionless barrier strength Z. Image from reference[21].
12
As Z increases, the tunneling transport through the interface becomes increasingly dominant, and so
the occurrence of ZBCP may seem at first glimpse a contradiction with what have been said previously
when discussing strong tunneling barriers. However, remembering the existence of zero-energy ABS at
the interface is the key-point to dilute the apparent contradiction [24, 21]: in the presence of an ABS, the
superconductor’s differential conductivity (gs) will be independent of the dimensionless barrier strength
Z at zero-bias energy, but the normal state conductance (gn) will diminish as Z increases, thus implying
that the normalized conductivity ( gsgn ) will increase at an energy bias which matches the ABS energy.
This will become more clear in section(1.6).
It is crucial to address two last questions:
The first one, concerns the issue of how ABS and Majorana fermions are related and can be answered
with topology - at the interface between a normal metal and a topological superconductor, a phase
transition will occur, leading to the emergence of gapless edge-states which will be Majorana fermions
due to PHS (fig.(1.8)) [16]; on the other hand, if the pairing-potential has a node plan where it changes
sign, a zero-energy ABS will be present. Thus, in this situation, one expects the ABS to be a Majorana
fermion.
Figure 1.8: Schematic representation of a normal metal- topological superconductor junction. The redstars indicate the localized Majorana fermions. Image from reference [26].
The second regards the issue of multiband superconductivity, already prominent for the candidate
CuxBi2Se3, which is perceived in the framework of a two-band model [27]. This concrete example
serves as motivation to study a superconductor with two orbitals per lattice site. However, the above
presented theories for studying the differential conductance in a normal metal-superconductor junction
have been developed in the assumption of one band superconductors. As the incoming normal metal
electron now approaches the superconductor it will encounter two possible electron pockets enabling
transmission, meaning that there are two channels for the transmission into the superconductor [28].
A simple way to address the problem is to apply the anisotropic version of the BTK theory separately
to the two bands and then add them together, therefore treating the multiband superconductor as a
parallel conductor [29]. This, however neglects any possible quantum mechanical interference effects
that may occur at the interface, urging for a new approach. This issue has close resemblance with a
quantum waveguide problem [30], which lead the authors M.A.N. Araujo and P.D.Sacramento to adapt
the model of a quantum waveguide to the interface between a normal-metal and a multiband supercon-
ductor [31, 28].
As an end note to this section, it is never enough to point out the importance of all the above con-
cepts that, assembled together, form the background in which the question of how and why studying the
13
differential conductance in a metal-topological superconductor is pertinent and can be approached, thus
providing the substructure sustaining the present thesis.
1.5 Overview of the BTK model
The starting point for the construction of the BTK formalism [20] are the Bogolubov equations for a
one-dimensional superconductor:
[−(
~2
2m∇2)− µ(x) + V (x)
]u(x, t) + ∆(x)v(x, t) = i~∂u(x,t)
∂t
−[−(
~2
2m∇2)− µ(x) + V (x)
]v(x, t) + ∆(x)u(x, t) = i~∂v(x,t)
∂t
(1.40)
where µ(x) is the chemical potential, ∆(x) the pairing potential, m the quasiparticle mass and V (x)
represents a general potential that may include , for example, the Hartree-Fock averaged Coulomb
interaction or the ion cores potential [13, 32].
This system of coupled equations can be viewed as Schrodinger-like equations for the coherence factors
u(x, t) and v(x, t) , respectively the electron and hole amplitudes which define the mixed character of
the quasiparticle excitation operators. It is then possible to perceive the Bogolubov equations as the set
equations that govern the evolution in time of the quasiparticle wavefunction:
ψ =
u(x, t)
v(x, t)
(1.41)
As a means of simplification, BTK took µ(x) ,V (x) and ∆ to be constant at both sides of the interface,
which allowed them to take as trial solutions the plane-waves:
ψ =
ueikx−iEt/~
veikx−iEt/~
(1.42)
The geometry considered was the one-dimensional case. The interface was localized at x = 0, the
normal-metal region at x < 0 and the superconductor side occupied the region x > 0. This clear partition
in two regions was taken under the assumption that the pairing potential reaches its asymptotic value
over length scale smaller than the coherence length ξ. Also, the effect of the interface on the scattering
process was captured by letting V (x) be a localized repulsive potential of the form:
V (x) = Hδ(x) (1.43)
where the parameter H is taken to be a constant which quantifies the strength of the barrier potential
and simulates the localized disorder at the interface [20].
Thus, at each junction side V (x) vanishes, granting a simplified form to the coupled equations(1.40) at
the superconductor region:
14
~2k2
2m − µ ∆
∆ −~2k2
2m + µ
uv
= E
uv
(1.44)
To attain the non-trivial solution it is necessary to solve the eigenvalue equation, yielding:
E2 =(~2k2
2m− µ
)2
+ ∆2 (1.45)
The above equation provides the means for determining both the magnitude of the momentum of the
excitations, as well as the amplitude of the coherence factors - u and v -, although the accomplishment
of the last task requires the condition for normalization, u2 + v2 = 1 , to be taken alongside:
~2(k±)2
2m− µ = ±
√E2 −∆2 ↔ ~k± =
√2m[µ± (E2 −∆2)1/2]1/2 (1.46)
1− v2± = u2
± =1
2
(1± (E2 −∆2)1/2
E
), (1.47)
where the indices ± for the functions u and v denote the two possible solutions allowed by the
term ±(E2 − ∆2)1/2 . It is important to notice that eq(1.46) enables to ascribe the plus solution (E2 −
∆2)1/2 to the momentum ±k+ and the minus solution −(E2 −∆2)1/2 with the momentum ±k− , where
the consideration of both signs for each momentum’s magnitude is imposed by the BCS pairing of
momentum k and −k. Also, from the same eq(1.46) one can see that excitations with energies above
the energy gap ∆ occurring at ±k+ are predominantly electron-like and that excitations with ±k− are
predominantly hole-like, a statement which becomes exact in the limit E >> ∆. Thus, for a given energy
E, the two possible solutions for the wavefunction ψ are:
ψ±k+ = ei±k+x
u+
v+
and: ψ±k− = ei±k−x
u−v−
. (1.48)
From eq(1.47), BTK labeled as u0 and v0 the quantities:
1− v20 = u2
0 =1
2
(1 +
(E2 −∆2)1/2
E
), with E > ∆ , (1.49)
allowing to rewrite the two solutions for the excitations in the superconductor as:
ψ±k+ = ei±k+x
u+
v+
= ei±k+x
u0
v0
(1.50)
ψ±k− = ei±k−x
u−v−
= ei±k+x
v0
u0
. (1.51)
Another crucial information attained from eq(1.47) comes from considering energies below the en-
ergy gap: u and v will be complex functions and the momentum k± will have a small imaginary compo-
nent, which will dictate excitations to exponential decay over the superconductor region, in accordance
15
with the expected dominant Andreev reflection process in such energy range. BTK reveals this is ex-
actly the case by showing that the quasiparticle current is converted into a supercurrent carried by the
condensate, over a distance of the order of ξ(T ), where T denotes the temperature.
With these considerations in mind, it is then possible to construct the quasiparticles wavefunctions as:
ψ±k+
S = ei±k+x
u0
v0
→ electron-like quasiparticle (1.52)
ψ±k−
S = ei±k−x
v0
u0
→ hole-like quasiparticle (1.53)
The wavefunctions from metal side are obtained when the same eq(1.52) and eq(1.53) are probed
in the limit ∆ = 0 . In this case, the allowed momentum takes the form:
~p± =√
2m√µ± E (1.54)
Now one can see more easily that ±p+ are essentially electron excitations and ±p− hole excitations,
aided also from the drastically simplification of the coherence factors u0 and v0 that become restrained
to take only two values: zero or one. Hence, the metal wavefuctions are represented by the vectors:
ψ±p+
N = ei±p+x
1
0
→ electron excitation (1.55)
ψ±p−
N = ei±p−x
0
1
→ hole excitation (1.56)
With the excitation wavefunctions at both side of the junction specified, BTK addressed the scattering
problem at the interface, by considering the following boundary conditions:
1. The continuity of the wavefunction must be guaranteed at the interface, meaning:
ψN (0) = ψS(0) (1.57)
2. The matching condition for the derivative of the wavefunction in the presence of a delta-function
potential:~
2m
(dψSdx− dψN
dx
)= HψN (0) (1.58)
3. The wave-vectors of transmitted and reflected excitations must be chosen such that the correct group
velocity, dEd(~k) , is assured. More precisely, an incoming electron from the metal region must have positive
group velocity, as well as the transmitted excitation in the superconductor side; a reflected electron to
the metal region must have a negative group velocity and a propagating hole in the same side also has
to carry a negative group velocity (fig.(1.9) and fig.(1.11)).
16
Figure 1.9: Quasiparticle excitations with positive group velocity in the superconductor. Image adaptedfrom reference [33].
Thus, the allowed incident, reflected and transmitted wavefunctions are:
ψinc = eip+x
1
0
(1.59)
ψrefl = aeip−x
0
1
+ be−ip+x
1
0
(1.60)
ψtrans = ceik+x
u0
v0
+ de−ik−x
v0
u0
(1.61)
The coefficients a, b, c and d are the probability amplitudes for each corresponding scattering pro-
cess: a is the probability amplitude for the Andreev reflection, b concerns the specular reflection at the
interface and finally, c and d are the transmitted probability amplitudes that account for the two existing
wave-vectors with positive group velocity, for a given energy of interest in the superconductor excitation
spectrum.
Figure 1.10: Quasiparticle excitations at both sides of the normal metal (N) - superconductor (S) in-terface. The digits indicate the momentum of the excitations. The relevant ones indicate as follows:0 is the incoming electron, 2 is the transmitted excitation crossing through the Fermi surface, 4 is thetransmitted quasiparticle excitation with no Fermi surface crossing, 5 represents the normal reflection atthe interface and 6 the Andreev reflection. Image from reference [33].
By applying the boundary conditions 1 , 2 and 3, a set of equations will be derived, which will then
17
allow the determination of the probability amplitudes:
1.→ ψN (0) = ψS(0)⇔ (1.62) b+ 1 = cu0 + dv0
a = cv0 + du0
2.→ ~2m
(dψSdx− dψN
dx
)= HψN (0)⇔ (1.63)
i~2m
(cu0k
+ − dv0k−(1− b)p+
)= H(1 + b)
i~2m
(cv0k
+ − du0k− − ap−
)= Ha
As means of simplifying the end result expressions, BTK replaced the parameter H by the dimen-
sionless barrier strength:
Z =mH
~2kF(1.64)
where kF denotes the Fermi momentum. It should be noted that as one departs from energies around
∆ ,the superconductor excitation spectrum becomes increasingly closer to the spectrum of the normal
metal (fig.(1.11)). Thus, the incident energy should lie in a small range closer to the superconductor’s
pairing potential.
Figure 1.11: Superconductor’s excitation spectrum. The dashed lines are the normal metal’s excitationspectrum. Image adapted from reference [19]
Furthermore, in metals µ >> ∆ [34] which, along with the close range of energies, imposes that
when evaluating equation (1.47), one should look for wave-vectors near kF :
k± ∼ kF
√1± ∆
µ(1.65)
Finally, if one introduces Z and solves the system of equations for p+ = p− = k+ = k− = kF , the
probability amplitudes a, b, c and d will be fully determined.
With the knowledge of these coefficients, it is possible to attain an expression for the differential conduc-
tance, gs:
gs = 1 + |a|2 − |b|2 (1.66)
In the BTK paper [20] a rigorous demonstration for this statement is given by calculating the current
that flows through the interface, where the right side of eq(1.66) appears in the final form of its expres-
18
sion and is nominated as the ”transmission coefficient for electrical current”. Nevertheless, gs can be
understood qualitatively in the following way: in the metal region an incoming electron carries a positive
current. At the interface, the electron can be specularly reflected and as it moves in the negative direc-
tion, it will be accounted as a negative current. However, if the incoming electron is Andreev reflected,
the hole moving in the negative direction will generate a positive current, owing to its sign reversed
charge with respect to the electron [20]. Hence, the probability of Andreev reflection has to enter with a
positive contribution to the current transmission coefficient, which in the case of a N-N juntion would just
be given by the usual expression 1− |b|2 .
1.6 Differential conductance and ABS in a d-wave superconductor
The BTK formalism, being one-dimensional, does not account for the anisotropy of the pairing potential,
such as in the case of d-wave superconductors where ∆(k) = ∆ cos(2θk) , with θk the angle between
the quasiparticle wave-vector and the nodal direction [25].
Yet, Kashiwaya,Tanaka and Kajimura made an extension of the BTK model to two spatial dimensions
which essentially follows similar steps with the key difference that it takes into account the dependence
of the gap function on quasiparticle’s momentum [21].
As before, the coherence coefficients can be related using the Bogolubov equations, from which one
obtains:
u(py, k+) =
∆(k+,py)E−ξ+ v(k+, py)
u(py,−k−) =∆(−k−,py)E−ξ− v(−k−, py) ,
where ξ = ~2k2
2m − µ.
Considering the usual display of the N-S junction: the interface is localized at x = 0, accommodating
the repulsive potential Hδ(x) , the normal metal placed at x < 0 and the superconductor filling the x > 0
region; if an incoming electron strikes the interface with a wave-vector having an arbitrary angle θN with
respect to the interface, all vector components that are parallel to the interface will be conserved in all
possible occurring scattering processes (fig.(1.12)).
Thus, the wavefunctions for the quasiparticle excitations can be written in a very similar way as in the
BTK model:
ψinc(x, y) = eipy.yψN (x) , with: ψN (x) = eip+x
1
0
+ be−ip+x
1
0
+ aeip−x
0
1
(1.67)
ψS(x, y) = cei(k+x+pyy)
u(k+, py)
v(k+, py)
+ de−i(k−x−pyy)
u(−k−, py)
v(−k−, py)
(1.68)
In the special case where the interface is positioned such that its normal vector coincides with the
nodal plane, electron-like and hole-like excitations will be subject to pair potentials that differ from one
another by an opposite sign: ∆(py, k+) = −∆(py,−k−). For simplicity, we shall use the notation:
19
Figure 1.12: Scattering processes in a N-anisotropic SC junction. The momentum’s component parallelto the interface is always conserved. Image from reference [21].
u(k+, py) = u+
v(k+, py) = v+
u(−k−, py) = u−
v(−k−, py) = v−
∆(k+, py) = ∆+
∆(−k−, py) = ∆−
∆ = |∆+| = |∆−|
The next step is to impose the BTK boundary conditions 1 and 2 , from which the coupled equations
are attained:
1.→ ψN (0) = ψS(0)⇔
u+ u−
v+ v−
cd
=
b+ 1
a
(1.69)
2.→ ~2m
(dψSdx− dψN
dx
)= HψN (0)⇔
λ+u+ −λ−u−λ+v+ −λ−v−
cd
=
−2iZ(b+ 1) + (1− b)
(1− 2iZ)a
(1.70)
where λ± = k±kF
arises when substituting H by the dimensionless strength barrier Z .
Inverting eq(1.69), one can express the probability amplitude coefficients c and d as functions of a and
b:
c = 1Λ
[v−(1 + b)− u−a
]d = 1
Λ
[u+a− v+(1 + b)
],
with Λ = u+v− − u−v+ the determinant of the matrix in eq(1.69). By replacing these expressions in
eq(1.70) and defining the quantities:
20
ζ11 = λ+u+v− − λ−u−v+
ζ12 = −(λ+ + λ−)u+u−
ζ21 = (λ+ + λ−)v+v−
ζ22 = −λ+u−v+ − λ−u+v− ,
it will be possible to present the final form of the a and b in a less cumbersome way:
a =2ζ21
(1 + 4Z2)Λ + ζ11 − ζ22 − 2iZ(ζ22 + ζ11)− ζ11ζ22−ζ12ζ21Λ
(1.71)
b =(1− 2iZ)2Λ2 − (1− 2iZ)(ζ11 + ζ22) + ζ11ζ22−ζ12ζ21
Λ
(1 + 4Z2)Λ + ζ11 − ζ22 − 2iZ(ζ22 + ζ11)− ζ11ζ22−ζ12ζ21Λ
(1.72)
One can inquire what will happen to the probability amplitudes a and b if Λ → 0. To probe this limit
it is useful to use the l’Hopital rule and notice that some of the linear combinations of the quantities
ζij , (i, j) ∈ 1, 2 are proportional to Λ :
ζ11 + ζ22 = (λ+ − λ−)Λ
ζ11ζ22 − ζ12ζ21 = −λ+λ−Λ2 .
Thus:
b→ Λ2[(1− 2iZ)(λ+ − λ−) + λ+λ−]
ζ11 − ζ22→ 0 (1.73)
a→ 2ζ21
ζ11 − ζ226= 0 (1.74)
If Λ = 0 , then u+
v+= u−
v−. Using the Bogolubov equations, one can retrieve that for E = 0, ξ± = ±i∆
which in turn leads to:
u+
v+=u−v−
= i . (1.75)
Thus, not only it verifies a vanishing Λ , as it demands the Andreev reflection probability to reach
its maximum value, which can be verified when substituting the ζij coefficients by their expressions
and attaining: a = u+
v+. In such conditions, gs equals 2. However, in the normal metal-normal metal
junction (N-N), as Z becomes larger it is increasingly more difficult for an electron to tunnel through the
interface. Therefore, the normal conductance - gn - decreases, meaning that at zero energy the relative
conductance, defined as gsgn
, will increase with Z.
Hence, the existence of ABS is connected with a vanishing Λ and it will be under this requirement that
these bound states will be examined in the following chapters.
21
1.7 Pseudo-spin
An electron that moves in a solid can be described by a tight-binding model. Within this approach,
if there are two atoms per lattice site, then the Hamiltonian written in the momentum space, which is
attainable by Fourier transforming the Hamiltonian in real space, can be presented in a matrix form, in
which a 2× 2 matrix will contain essentially all the physical information of the system. As an illustration,
one may consider a 1D chain in which two generic atoms, A and B, occur in alternate sequence along
the chain. The unit cell will be composed by two different sites, corresponding to the A and B atoms.The
Hamiltonian in the k space will be given by:
H = (a†k b†k)
εa V (k)
V (k) εb
akbk
. (1.76)
The term V (k) arises from the Fourier transform of the hopping term in the real space tigh-binding
Hamiltonian. The operators a†k and b†k act on a sub-lattice composed only by atoms of a unique type:
sub-lattice containing only atoms of type A and the sub-lattice formed by type B atoms, respectively.
The above 2 × 2 matrix allows to infer important information about the system and is called the Bloch
Hamiltonian h(k). Such information will be related to the eigenvalues and eigenvectors, which can be
calculated by performing a suitable transformation that diagonalizes h(k). This task is accomplished by
performing a unitary transformation such that:
h(k) = UDU−1 , (1.77)
where D is the diagonal matrix formed by the eigenvalues of h and U is the matrix containing as
columns its eigenvectors, which when applied to the operators will carry the transformation:
d1(k)
d1(k)
= U−1
akbk
, (1.78)
enabling the Hamiltonian to be written with the diagonal form
H = (d†1(k) d†2k)
ε1 0
0 ε2
d1(k)
d2(k)
⇔ H =∑k
ε1d†1(k)d1(k) +
∑k
ε2d†2(k)d2(k) , (1.79)
where ε1 and ε2 are the eigenvalues of the Bloch Hamiltonian. Thus, the described tight-binding
model has two energy bands per unit cell, which means that the Bloch Hamiltonian represents a two
band model.
Alternatively, with the foreknowledge that two energy bands exist, one can perform a Bogolubov trans-
formation in hope to diagonalize the Hamiltonian with new operators written as linear combinations of
the operators ak and bk . The new operators would be d1 and d2, expressed as:
22
d1(k) = α1ak + β1bk
d2(k) = α2ak + β2bk, (1.80)
Then, by requiring the commutation relations to be preserved, the Hamiltonian written in terms of the
new operators will have the same form as in eq(1.79) if the following coupled equations, for each energy
band, are verified:
εaα+ βV = αε
V α+ εbβ = εβ⇔ h(k)
αβ
= ε
αβ
, (1.81)
which corresponds to the eigenvalue equation for the Bloch Hamiltonian and where α and β repre-
sent the probability amplitudes for the two pseudo-spin components (the two available orbitals).
Therefore, in the case of the tight-binding model, one can identify the ak and bk as the operators for
the plane waves over the A and B atoms and d1(k) and d2(k) as the operators for Bloch waves in the
energy bands 1 and 2, respectively.
It is then possible to specify the kinetic energy of a two band model by the Bloch Hamiltonian h(k) and
because it is a 2×2 matrix ,it can be written as a linear combination of the Pauli matrices plus the identity
in the pseudo-spin space (always possible as the identity and the Pauli matrices span the whole space
of a 2× 2 matrix):
h(k) = h0(k)τ0 + hx(k)τx + hy(k)τy + hz(k)τz (1.82)
⇔ h = h0(k)τ0 + h(k).τ , (1.83)
where τ0 corresponds to the identity matrix 1. To bring more clarity to the notation, it shall be as-
sumed for the remaining of this thesis that the set of Pauli matrices τµ, with µ = 0, x, y, z act on the
pseudo-spin indices, while the σµ are reserved for the physical spin indices.
23
Chapter 2
Andreev spectroscopy in a multiband
topological superconductor
This chapter concerns the theoretical formalism (sections 2.1 and 2.2) providing the means for deter-
mining the differential conductance at the normal metal-superconductor junction, which will illustrate the
main propose of this work (section 2.4):
a) The ABS predicted by quantum waveguide theory correctly reconciles the Andreev problem with the
topological invariant.
b) Observe the interference effects on the differential conductance, as the incident electrons go from a
single band metal to a multiband superconductor, as described by quantum waveguide theory.
2.1 The model for the Hamiltonian
One shall consider a topological superconductor with two orbitals per lattice site described by the
particle-hole symmetric Hamiltonian:
H =1
2
∑k
(c†k↑ c†k↓ c−k↑ c−k↓)
Ξ(k) ∆(k)
∆†(k) −Ξ∗(−k)
ck↑
ck↓
c†−k↑
c†−k↓
, (2.1)
where the operators c and c† are itself two component vectors due to the two orbitals available in
each lattice site. Ξ and ∆ are , respectively, the kinetic energy and the pairing potential, both being 4× 4
matrices, defined as:
Ξ(k) =
h(k).τ + h0(k).τ0 0
0 h∗(−k).τ∗ + h∗0(−k).τ0
(2.2)
24
with:
hx = sin(ky)
hy = −sin(kx)
hz = 2t1(cos(kx) + cos(ky)) + 4t2cos(kx)cos(ky)
h0 = −µ− t1(cos(kx) + cos(ky))
(2.3)
and
∆(k) = dz(k)σx ⊗ τ0 ,with: dz(k) = ∆(sin(kx)− isin(ky)) (2.4)
which is equivalent to
∆(k) = dz(k)
0 τ0
τ0 0
, (2.5)
and represents a p-wave superconducting pairing. The above choice for the pairing function breaks
time-reversal symmetry. The system is therefore in the symmetry class D and has, for two spatial
dimensions, a Z topological index, the Chern number [8].
The model describes a system in which the electrons with spin-up are paired with spin-down elec-
trons, both having kinetic energies related by a time-reversal operation: Ξ↓ = Ξ∗↑(−k) .
The matrix in eq(2.1) is the HBdG Hamiltonian which has to satisfy the Bogolubov equations:
Ξ(k) ∆(k)
∆†(k) −Ξ∗(−k)
u↑
u↓
v↑
v↓
= E
u↑
u↓
v↑
v↓
, (2.6)
in which u and v are both two component vectors accounting for the two orbitals available in each
site.
Writting explicity the equations resulting from eq(2.6) , allows to see that the HBdG matrix only couples
u↑ with v↓ and u↓ with v↑ , showing that HBdG can be decoupled into two separate systems: a 4 × 4
matrix for the (u↑ v↓) subspace and another 4× 4 matrix which acts on the subspace (u↓ v↑) .
In what follows, we shall only consider the sub-system (u↑ v↓) :
h(k).τ + h0(k).τ0 dz(k).τ0
d∗z(k).τ0 −h(k).τ − h0(k).τ0
u↑v↓
. (2.7)
This Hamiltonian, eq(2.7), is in symmetry class A (unitary symmetry class) and has a Z topological
index in two dimensions. It describes a spinful chiral p+ip superconductor [8].
(u↑ v↓) are the eigenvectors of the system and can be written in the form:
25
u↑v↓
=
uk
αβ
vk
αβ
(2.8)
where α and β compose the orbital (pseudo-spin) eigenstates of the Bloch Hamiltonian h, obeying
the Schrodinger equation:
hz + h0 hx − ihyhx + ihy −hz + h0
αβ
= ξ(k)
αβ
, (2.9)
enabling to rewrite eq(2.7) as eq(2.11):
eq.(2.7) =
u[(hz + h0)α+ (hx − ihy)β]︸ ︷︷ ︸ξα
+dzvα = Euα
u[(−hz + h0)β + (hx + ihy)α]︸ ︷︷ ︸ξβ
+dzvβ = Euβ
−v[(hz + h0)α+ (hx − ihy)β]︸ ︷︷ ︸ξα
+d∗zuα = Evα
−v[(−hz + h0)β + (hx + ihy)α]︸ ︷︷ ︸ξβ
+d∗zuβ = Evβ
(2.10)
⇒
ξ dz
d∗z −ξ
uv
= E
uv
⇔ h′.r
uv
= E(k) , (2.11)
where ξ± = h0 ±√h2x + h2
y + h2z. The h′ vector is defined such that h′x − ih′y = dz(k) and h′z = ξ(k).
The r matrices are the Pauli matrices acting on the (u v) space.
From the last equation, one can calculate the superconductor’s energy bands more easily and because
for a given momentum k , the kinetic energy ξ can either take the values ξ+ or ξ− , the two supercon-
ductor’s negative energy bands will be given by E± = −√ξ2± + |dz|2 . The subsystem (u↑ v↓) does not
verify the time-reversal symmetry, because the hx component disrupts the condition in eq(1.20) . In fact,
this kinetic energy is already topologically non-trivial.
By restricting our analysis solely to the subsytem (u↑ v↓), we are not addresing a complete description
of the problem. This is due to PHS transformation (eq(1.31)), which in general, when applied to the
matrix in eq(2.9), will project (u↑ v↓) into the (u↓ v↑) subsytem and therefore will require the Majorana
fermion to be constructed as a sobreposition of the two subspaces. However, in the special cases where
the transverse momentum is either py = 0 or py = π, the two values considered in the calculations for
the differential conductance, the matrix in eq(2.9) will preserve PHS, allowing us to illustrate the two
main goals stated in the begining of this chapter.
The topological properties of the superconductor do not stem from the topological properties of the
kinetic energy, however. This can be seen as follows: the Berry connection can be written as a sum of
26
the two contributions arising from the eigenstates (u v) and (α β) :
A = i(u†↑ v†↓)∂
∂k
u↑v↓
= i(u† v†)∂
∂k
uv
+ i(α† β†)∂
∂k
αβ
= A′ + A′′ , (2.12)
where A′ and A′′ represent the Berry connections of the eigenvectors (u v) and (α β), respectively.
From eq( 1.1 ), the total Chern number will be calculated by the circulation around the Brillouin zone,
summed over the two negative bands of eq( 2.11 ):
C =∑∮
A.δk =∑∮
A′.δk +∑±
∮A′′.δk (2.13)
or in terms of the Berry’s connections:
C =∑∫
Ω′−dS′ +
∑±
∫Ω′′dS′′ , (2.14)
where the last summation is over the two h.τ energy bands and therefore vannishes. Thus, the
topology arising from h.τ will not contribute to the total Chern number C, as it will cancel upon the
summation over all its energy bands.
Only A′ will contribute to the topology of the system (total Chern number). From eq( 2.11 ), we choose
dz in the form of p+ip pairing in order to have Dirac points contributing to a Chern number, in the first
term of eq( 2.14 ).
Distinct choices of the Hamiltonian parameters can lead to distinct topological phases and thus different
Chern numbers. Fixing µ = 0.7 and ∆ = 0.1 , a topological state will have to be probed by varying
the hopping parameters t1 and t2. Choosing t1 = 0.07 with t2 either +0.08 or −0.08 and calculating the
Chern number for each choice, one obtains:
t2 = −0.08⇒ C = 1⇒ Topological phase
t2 = 0.08⇒ C = 0⇒ Trivial phase. (2.15)
To compute the Chern number, the method developed by Fukui, Hatsugai and Suzuki for multiband
superconductors ( see appendix B) [35, 36] was implemented.
If we had chosen the chemical potential µ lying within the energy gap, for example µ = −0.14, for the
same values of the hopping parameters, the kinetic energy would be topologically non-trivial with a
Chern number C = −2, but it would not contribute to the topological properties of the superconductor,
as we have proved. Furthermore, ξ+ would always be positive (h′ would only visit the north pole) and
ξ− always negative (h′ would only visit the south pole), thus leading to a null Chern number for the
superconductor (see appendix C).
In eq(2.15) the Fermi surfaces are not the same in each phase. When topology is present, the Fermi
surface contains 3 electron pockets (fig(2.2)) centered at momentum coordinates ( kx = 0 , ky = 0 ) , (
kx = 0 , ky = π ) , ( kx = π , ky = 0 ). In the trivial phase, the Fermi surface will contain another pocket
located at ( kx = π , ky = π ), fig(2.1). The surfaces are obtained by determining the Fermi momentum
in the upper band: ξ+(k) = 0
27
(a) Fermi surface for t2 = 0.08 with 4 electron pock-ets.
(b) ξ+(k) band structure for t2 = 0.08.
Figure 2.1: Fermi surface and ξ+(k) band in the trivial phase.
(a) Fermi surface for t2 = −0.08 with 3 electronpockets.
(b) ξ+(k) band structure for t2 = −0.08.
Figure 2.2: Fermi surface and ξ+(k) band in the topological phase.
The number of edge states is related to the non-trivial topology of the system. This is the bulk-
boundary correspondence ( eq.(1.17) ). Thus, if C = 1 there can only precisely exist a single Majorana
fermion (fig.(2.3)) and one must expect to infer its existence by solely indentifying one ABS in the differ-
ential conductance, as discussed in chapter 1.
Figure 2.3: Energy spectrum of the Hamiltonian model eq(2.7) for an infinite ribbon in the yy (or xx)direction. The Majorana fermion can be seen for the longitudinal momentum π. This figure was obtainedby M.A.N. Araujo.
28
2.2 Quantum waveguide theory of Andreev spectroscopy
To study the scattering processes occurring at an interface between a normal metal and a multiband su-
perconductor a new theory must be introduced, such that the quantum interference between transmitted
waves into the superconductor is taken into account. The incoming electron from the single band metal
has to split into two conduction channels in the two-band superconductor. This situation is analogous to
a waveguide [30], as represented in figures (2.4) and (2.5), leading the authors M.A.N. Araujo and P.D.
Sacramento to adapt the quantum waveguide problem to treat the matching conditions at the interface
between a normal metal and a multiband superconductor [28].
This chapter is dedicated to presenting the detailed formalism of the theory [31, 28], which will latter be
applied, to the interface between a normal metal and a topological superconductor.
Figure 2.4: Analogy between the NS interface and a quantum waveguide system. Image from reference[28].
If one is interested in studying how an incoming electron should be transmited when encountering
a biffurcation along the path, then the first thing to address are the boundary conditions at the node
[30, 28, 31]:
1. It should be guaranteed that the wavefunction is continuous and single-valued at the node in fig(2.4):
ψ1(O) = ψ2(O) = ψ3(O) . (2.16)
2. The probability and charge current, j(r), should be conserved at the node.
One can see how to formally apply these conditions by starting with a simpler problem, given in ref-
erence [28] , where instead of the node O lining off the separation between the normal metal and the
superconductor regions, the right side is occupied by a metal with two orbitals per site. The example will
also be usefull to provide a means for calculating the normal conductance gn. The model is represented
by a tight-binding chain, where the two branches are hybridized by an operator V .
The Hamiltonian matrix for the coupled chain is given by H1+2 ( see section 1.7) ,
H1+2 =
ε1(k) V (k)
V (k) ε2(k)
(2.17)
with the correspondent eigenstates determined by the equation:
29
Figure 2.5: Schematic illustration of the interface between a one band model (branch 3) and a two bandtight-binding model, where the electron will hopp along branch 2 ( with hopping parameter t’), as well asalong branch 1 ( with hopping parameter t), but it also will be able to change between the two branchs(with hopping parameter V). The integer n represents the unit cells. Image from reference [28].
ε1(k) V (k)
V (k) ε2(k)
αkβk
= ε
αkβk
, (2.18)
allowing the wavefunction of the electron in the coupled chain to be the Block state:
ψ(n) = eikn
αkβk
, (2.19)
where n represents a unit cell containing two atoms. One can then construct the incident and trans-
mited wavefunctions as:
ψtrans(n) = Ceikn
αkβk
+Deik′n
α′kβ′k
(2.20)
ψinc(n) = eipn + be−ipn (2.21)
The momenta k and k′ are determined by the eigenvalue equation resulting from eq(2.18), subjected
to the energy conservation condition:
Enormal metal = ε−(k) = ε+(k′)
ε± = 12 (ε1 + ε2)± 1
2
√4V 2 + (ε1 − ε2)2
The boundary condition 1 can be applied directly, keeping in mind that the lower index in the wave-
function ψ indicates whether the wavefunction belongs to the branch 1, 2 or 3 from fig(2.5):
ψ3(n→ 0) = ψ2(n→ 0) = ψ1(n→ 0)↔ 1 + b = Cαk +Dα′k = Cβk +Dβ′k . (2.22)
However condition 2 depends upon specifying the expression for the probability current. It shall be
used in this example and in the further sections the definition given in reference [28]:
30
j(r) = Reψ†(r)∂H
∂kψ(r) ,with: k = −i∇ . (2.23)
With this definition and owing to the fact that for the 1D tight-binding model the operator k reduces
simply to −i ∂∂n , the probability current for each side of the node can be expressed as:
jinc(r) = Re−i ψ†inc ∂E∂p ψinc = Re−im ψ†inc∂∂nψinc =
n→0Re pm (1 + b∗)(1− b)
jtrans(r) = Reψ†trans(r)∂H1+2
∂kψtrans(r) .
Then, condition 2 may be imposed as:
limn→0−
(∂Enomal metal∂p
ψinc
)= (1 1). limn→0+
(∂H1+2
∂kψtrans
), (2.24)
leading to the equation:
p
m(1− b) = C
(1 1).∂H1+2
∂k.
αkβk
+D(1 1).∂H1+2
∂k.
α′kβ′k
. (2.25)
To justify this statement, one can verify that by inserting the equation above into the definition of
j(r)inc, while taking into account eq(2.22), the end result will be j(r)trans, showing that indeed eq(2.24)
provides the conservation of the probability current at the node:
(1 + b∗) using eq(2.22) =1
2(C∗α∗k +D∗α∗k′ + C∗β∗k +D∗β∗k′) =
1
2C∗(α∗k β
∗k
)1
1
+1
2D∗(α∗k′ β
∗k′)1
1
(2.26)
⇒ Re( pm
(1 + b∗)(1− b))
= j(r)trans . (2.27)
With eq(2.22) and eq(2.24) the waveguide boundary conditions are established. Furthermore, they
provide the system of equations which uniquely determine the probability amplitudes b, C and D.
A next consideration should be how to incorporate the repulsive potential, localized at the interface of a
N-N junction or a N-S junction, whithin the waveguide theory boundary conditions. The task is fulfilled by
slightly shifiting the barrier potential such that it occupies the normal metal region [31, 28], which means
the barrier potential at site l takes the form:
Hδ(n− l) . (2.28)
In this way the electronic transport through the barrier can be worked out as a 1-band problem and the
wavefunction in the normal metal region can be written as
ψN (n ≤ l) = eipn + be−ipn (2.29)
and
31
ψN (l < n < 0) = αeipn + βe−ipn . (2.30)
The boundary conditions at n = l are now analogous to the BTK boundary conditions [20],
ψN (l−) = ψN (l+)
∂ψN (l+)∂n − ∂ψN (l−)
∂n = 2m~2 HψN (l) .
(2.31)
Letting l→ 0 enables to express the amplitudes α and β as functions of the reflexion coefficient b :
1 + b = α+ β (2.32)
α− β = (1− b)− 2iZ(1 + b)kFp
+ (1− b) , (2.33)
where H has already been substituted by its dimensionless version, Z. It is important to notice that
when carring out the limit l→ 0 , the repulsive potential is shifted to the node and so ψN (l < n < 0) has
also to obey the waveguide theory boundary conditions:
α+ β = Cαk +Dα′k = Cβk +Dβ′k (2.34)
p
m(α− β) = C
(1 1).∂H1+2
∂k.
αkβk
+D(1 1).∂H1+2
∂k.
α′kβ′k
. (2.35)
Thus replacing eq(2.33) in eq(2.35) reveals how the reflexion coefficient b is affected by the presence
of the repulsive potential:
p
m
[(1− b)− 2iZ(1 + b)
kFp
]= C
(1 1).∂H1+2
∂k.
αkβk
+D(1 1).∂H1+2
∂k.
α′kβ′k
, (2.36)
by comparing this equation with eq(2.25) it is then clear that
1− b→ (1− b)− 2iZ(1 + b)kFp. (2.37)
The normal conductance in the N-N junction will be given by gn = 1− |b|2 and can be determined by
solving eq(2.25) coupled with eq(2.22), from which the amplitudes C and D can be eliminated:
C = 1A [β′k(1 + b)− α′k(1 + b)] , with: A = 1
αkβ′k−βkα′k
D = 1A [α′k(1 + b)− β′k(1 + b)] .
Inserting these expressions into eq(2.25) along with the effect of the barrier potential H, eq(2.37),
and introducing the auxiliar variables:
32
Σ(k) = m
p
(1 1).∂H1+2
∂k .
αkβk
F = Σ(k)
A (β′k − α′k) + Σ(k′)A (αk − βk) ,
enables to express the final result for the amplitude b as, in the approximation p ≈ kF :
b =(1− 2iZ)− F(1 + 2iZ) + F
, (2.38)
from which gn is attained.
2.3 Application of the waveguide theory to the N-S junction
With the outline of the essential steps of waveguide theory in the last section, one can now proceed
with the task of applying it to study the transport phenomena occuring at an interface between a normal
metal and the topological superconductor described by the Hamiltonian of section(2.1).
As in the previous case, the interface will act as a node where the incoming electron wavefunction will
be splitted between the two conduction paths: the two pseudo-spin channels at the superconductor
(fig.(2.6)). Assuming the edge of the superconductor to be along the y direction, an incident electron
wave-vector with an arbitrary angle with respect to the interface normal vector, will have its vector com-
ponents parallel to the interface conserved in the scattering processes, just as discussed in section(1.6),
allowing to impose the boundary conditions in the same limit as in the 1D problem. This means that in
the xy plane, the boundary conditions at the interface will be probed in the limit x→ 0 .
Figure 2.6: Schematic representation of the interface between a one band normal-metal and a topolog-ical superconductor with two pseudo-spin channels available for the electron to be transmitted.
Recalling that we are only considering the subsystem (u↑ v↓) , we follow eqs(2.7) and (2.8) for the
wavefunctions in the superconductor and write:
h(k).τ + h0(k).τ0 dz(k).τ0
dz(k)∗τ0 −h(k).τ − h0(k).τ0
ukαk
ukβk
vkαk
vkβk
= E(k)
ukαk
ukβk
vkαk
vkβk ,
(2.39)
33
As the y component of the wavevector is a conserved quantity, one can construct the transmited
wavefunction in the superconductor’s region as eipyyΨS , where
ΨS(x ≥ 0) = Cψk+eik+x +Dψk−e
−ik−x + Eψq+eiq+x + Fψq−e
−iq−x . (2.40)
Each ψk denotes a four-dimensional column eigenvector of the BdG matrix from eq(2.39). The
transmited excitations are superpositions of the wave-vectors k± and q± from the two Fermi-surface
pockets. Their x-component is chosen such that the group velocity is positive for an energy E above
the gap (as in fig(1.9)). For the normal-metal, which is a single-band metal, the wavefunctions are those
introduced in eq.(1.67), eipyyΨN with:
ΨN (x ≤ 0) = eip+x
1
0
+ be−ip+x
1
0
+ aeip−x
0
1
(2.41)
In the approximation where k± and q± both take the value of their respective Fermi momentum, their
determination can be achieved by finding the zeros of the electron-like energy band, which are closest
to the center of correspondent Fermi-pocket.
With both wavefunctions in each region specified, one can now proceed by applying the waveguide
boundary conditions 1 and 2 . The boundary condition which imposes the wavefunction to be single
valued at the node follows immediatly if one evaluates the wavefunctions, in each region, in the limit
x→ 0 :
uk+αk+ uk−αk− uq+αk+ uq−αq−
uk+βk+ uk−βk− uq+βk+ uq−βq−
vk+αk+ vk−αk− vq+αk+ vq−αq−
vk+βk+ vk−βk− vq+βk− vq−βq−
C
D
E
F
=
1 + b
1 + b
a
a
(2.42)
To evaluate condition 2 it is necessary to probe each member of eq(2.24):
∂ε
∂p=
pm 0
0 − pm
,with: p = −i ∂∂x⇒ ∂ε
∂pΨN =
x→0−
p+
m
1
0
− bp+
m
1
0
− ap−m
0
1
. (2.43)
∂H
∂k.ψk =
x→0+
∂(h(k).τ+h0(k).τ0)
∂ku↑ + ∂dz
∂kv↓
∂d∗z∂ku↑ − ∂(h(k).τ+h0(k).τ0)
∂kv↓
=
∂(h(k).τ+h0(k).τ0)
∂kuk
αβ
+ ∂dz∂kvk
αβ
∂d∗z∂kuk
αβ
− ∂(h(k).τ+h0(k).τ0)
∂kvk
αβ
(2.44)
Thus, the current conservation at the node can be expressed as:
1− b = CΘ(k+) +DΘ(k−) + EΘ(q+) + FΘ(q−)
a = CΦ(k+) +DΦ(k−) + EΦ(q+) + FΦ(q−) ,(2.45)
34
where Θ(k) and Φ(k) are the auxiliar functions defined as follow:
Θ(k) =m
p+(1 1).
(uk∂h(k).τ + h0(k).τ0
∂kx+ vk
∂dz(k)
∂kx
)αk
βk
. (2.46)
Φ(k) =m
p−(1 1).
(vk∂h(k).τ + h0(k).τ0
∂kx− uk
∂d∗z(k)
∂kx
)αk
βk
. (2.47)
From eq(2.42) it is possible to invert the system and attain the amplitudes C, D, E and F in terms of
a and b:
C = (1+b)Γ1+aΓ2
Λ
D = (1+b)Γ3+aΓ4
Λ
E = (1+b)Γ5+aΓ6
Λ
F = (1+b)Γ7+aΓ8
Λ
(2.48)
with: Λ =
∣∣∣∣∣∣∣∣∣∣∣∣
uk+αk+ uk−αk− uq+αk+ uq−αq−
uk+βk+ uk−βk− uq+βk+ uq−βq−
vk+αk+ vk−αk− vq+αk+ vq−αq−
vk+βk+ vk−βk− vq+βk− vq−βq−
∣∣∣∣∣∣∣∣∣∣∣∣(2.49)
The coefficients Γi are given by linear combinations of the cofactor elements Clj that form the inverse
matrix of system (2.42). For example, Γ1 = C11 + C12.
It is now explicit that the reflection and the Andreev amplitudes will become dependent on the determi-
nant of the matrix Λ. Also, one will expect, as argued in section(1.6), that when Λ→ 0 ,these amplitudes
will become independent of the barrier strength, with the transmission occuring via the localized ABS.
It is still necessary to introduce the effect of the repulsive potential at the interface. As in the last section,
the problem can be more easily addressed if the repulsive potential is slightly shifted from the node
point, thus allowing it to be treated at the normal single-band region. The only difference will be due to
the presence of the superconductor, which will require for the metal’s scattering wavefunction, both at
the right and left side of the barrier potential, to have electron and hole non-null components:
ψN (x ≤ ε) = eip+x
1
0
+ be−ip+x
1
0
+ aeip−x
0
1
(2.50)
ψN (ε < x < 0) = αeip+x
1
0
+ βe−ip+x
1
0
+ γeip−x
0
1
+ ηe−ip−x
0
1
(2.51)
Taking the same boundary conditions specified in eq(2.31), the end result in the limit ε → 0 will be
equivalent with only two extra equations, owing to the presence of hole-like excitations:
γ + η = a
γ − η = ( 2mUi~2p− + 1)a .
35
Thus, the reflection coefficient b will be affected in the same manner, with its dependence upon
the dimensionless barrier parameter Z given by eq(2.37). The Andreev reflection amplitude will also
be modified, which can be seen when imposing the waveguide matching conditions to the last set of
equations, particulary, the requirement 2 will lead to the equality:
a(1− 2ikFZ
p−) = CΦ(k+) +DΦ(k−) + EΦ(q+) + FΦ(q−) (2.52)
,
which requires, by comparison with the second equation in system(2.45), for a to be modified in this
equation as:
a→ a(1− 2ikFZ
p−) (2.53)
.
With these replacements plus eq(2.48), it is possible to rewrite the equations coming from the bound-
ary condition 2. as:
(1− b)− 2iZ kFp+ (1 + b) = 1
Λ (ζ11(1 + b) + aζ12)
a(1− 2iZ kFp− ) = 1
Λ (ζ21(1 + b) + aζ22) ,(2.54)
where ζij , (i, j) = 1, 2 are the quantities arising from the replacement of C, D, E and F in eq(2.52)
and are defined as:
ζ11 = Γ1Θ(k+) + Γ3Θ(k−) + Γ5Θ(q+) + Γ7Θ(q−)
ζ12 = Γ2Θ(k+) + Γ4Θ(k−) + Γ6Θ(q+) + Γ8Θ(q−)
ζ21 = Γ1Φ(k+) + Γ3Φ(k−) + Γ5Φ(q+) + Γ7Φ(q−)
ζ22 = Γ2Φ(k+) + Γ4Φ(k−) + Γ6Φ(q+) + Γ8Φ(q−) .
(2.55)
Solving the system of eqs(2.54), one obtains the final form for the amplitudes of normal and Andreev
reflection:
a =2ζ21Λ
(1 + 2iZ + ζ11Λ )(1− 2iZ − ζ22
Λ ) + ζ12ζ21Λ2
(2.56)
b =(1− 2iZ − ζ11
Λ )(1− 2iZ − ζ22Λ )− ζ12ζ21
Λ2
(1 + 2iZ + ζ11Λ )(1− 2iZ − ζ22
Λ ) + ζ12ζ21Λ2
(2.57)
It is then possible to calculate the differential conductance of the N-S junction, gs = 1 + |a|2 −
|b|2, although a concrete value can only be attained with the knowledge of the amplitudes uk and vk
. As already argued, such task can be accomplished using the Bogolubov equations together with the
normalization condition. Since py will be a fixed quantity and particularly because it will only take upon
the two values - 0 or π - the pairing potential will be a real function of the momentum in the x direction,
if kx (or qx ) is real. Thus, a possible solution for E ≤ ∆ is :
36
v(l+, py) = 1√2
u(l+, py) =dz(l+,py)
E−i√|dz(l+,py)|2−E2
v(l+, py)
u(l−, py) = 1√2
v(l−, py) =dz(l−,py)
E−i√|dz(l−,py)|2−E2
v(l−, py)
, (2.58)
where l = k, q . For E > ∆ :
u(l+, py) =
√1 + 1
2
√E2−|dz(l+,py)|2
E
v(l+, py) =dz(l+,py)
E+√E2−|dz(l+,py)|2
u(l+, py)
v(l−, py) =
√1 + 1
2
√E2−|dz(l−,py)|2
E
u(l−, py) =dz(l−,py)
E+√E2−|dz(l−,py)|2
v(l−, py)
. (2.59)
A final remark should be given to the case where py = π and t2 = −0.08 .For such transverse momen-
tum only one real electron pocket will be intersected, the (0, π) pocket. However, quantum waveguide
theory can still be applied if one allows the momentum of the ”missing” pocket to be complex.
2.4 Differential conductance of the N-S boundary
All the above construction can be applied to determine the behavior of the differential conductance gs as
a function of the bias voltage. Particularly, gs will be computed in two distinct choices for the Hamiltonian
parameter t2 ( t2 = +0.08 and t2 = −0.08 ) which, as mentioned before, lead the system into two distinct
phases: a trivial and a topological phase, respectively. For the transverse momentum fixed at normal
incidence, py = 0 , and considering the non-topological case, waveguide theory provides the results in
fig(2.7)-(2.9) , which account for different magnitudes of the strength barrier, Z.
Figure 2.7: Conductance gs and gn as function of the incident electron energy for t2 = +0.08 in amicroconstriction, Z=0, for a fixed transverse momentum py= 0. Calculation performed with waveguidetheory.
37
As one may see, the electron transport at zero energy-bias E → 0 is supressed, even though ex-
citations k+ and q+ are submitted to a sign-reversed pairing potential version of their correspondent
counterpart k− and q− , which might lead to expect a similiar situation as in d-wave superconductivity,
fig(1.7 b) ). Also, the observed vanishment of gs for E → 0 becomes more pronounced with the increase
of the disorder parameter Z.
Figure 2.8: Conductance gs and gn as function of the incident electron energy for t2 = +0.08 and Z=1,for a fixed transverse momentum py=0. Calculation performed with waveguide theory.
Figure 2.9: Conductance gs and gn as function of the incident electron energy for t2 = +0.08 and Z=2,for a fixed transverse momentum py= 0. Calculation performed with waveguide theory.
If one applies the generalized form of BTK theory for an anisotropic pair potential, the resulting
differential conductance will behave similar to fig(1.7 b) ), predicting an ABS at zero energy in each
38
Fermi-pocket , identifiable by the condition discussed in section(1.6) : Λ→ 0 . This condition was never
satisfied for the differential conductances attained with waveguide theory for t2 = +0.08 .
Figure 2.10: Conductance gs as function of the incident electron energy for t2 = +0.08 in a microcon-striction, Z=0, computed with BTK generalized model, for a fixed transverse momentum py=0.
With the same choice of the parameter t2 , but altering the transversed momentum to py = π ,
waveguide theory predicts the behaviour of the differential conductances in fig(2.11)-(2.13) . Similarly to
py = 0 there are no bound states and the conductivity at zero bias energy is supressed.
Figure 2.11: Conductance gs and gn as function of the incident electron energy for t2 = +0.08 and Z=0,for a fixed transverse momentum py=π. Calculation performed with waveguide theory.
One may notice the presence of peaks and valleys which occur at bias energies coinciding with the
value of the pair potential at the momentum kF and qF .
39
Figure 2.12: Conductance gs and gn as function of the incident electron energy for t2 = +0.08 and Z=1,for a fixed transverse momentum py=π. Calculation performed with waveguide theory.
Furthermore, the assymptotic behaviour of the differential conductance gs as the bias energy in-
creases is similar in all the present figures: it becomes increasingly closer to the normal conductance gn
, illustrating that the special features of the N-S junction are circumscribed to a range of energies near
the gap energy.
Figure 2.13: Conductance gs and gn as function of the incident electron energy for t2 = +0.08 and Z=2,for a fixed transverse momentum py=π. Calculation performed with waveguide theory.
For the topological phase, considering a normal incidence, py = 0 , and applying quantum waveguide
theory provides a similar qualitative picture as for t2 = +0.08 , with also zero conductance at E → 0 .
40
Figure 2.14: Conductance gs and gn as function of the incident electron energy for t2 = −0.08 and Z=0and a fixed transverse momentum py=0. Calculation performed with waveguide theory.
Figure 2.15: Conductance gs and gn as function of the incident electron energy for t2 = −0.08 and Z=1and a fixed transverse momentum py=0. Calculation performed with waveguide theory.
41
Figure 2.16: Conductance gs and gn as function of the incident electron energy for t2 = −0.08 and Z=2and a fixed transverse momentum py=0. Calculation performed with waveguide theory.
Likewise, applying the extended version of the BTK theory in this case results in the opposite predic-
tion with an enhanced flow of current occuring at zero bias and detecting the presence of an ABS which
would be a Majorana fermion, due to topology. The model also grants the same conclusion if rather
than a normal incidence, the transverse momentum is fixed at py = π. Thus BTK theory predicts the
existence of a Majorana fermion in each Fermi pocket, a contradiction with the topological properties of
the system which only allow one Majorana fermion.
Figure 2.17: Conductance gs as function of the incident electron energy for t2 = −0.08 with variousstrength barriers and a fixed transverse momentum py=π. Calculation performed using the generalizedBTK model.
42
The differential conductance attained by waveguide theory with fixed transverse momentum py = π
and t2 = −0.08, however, breaks the pattern of suppressed zero bias conductivity and meets for the first
time the criterium for ABS.
Figure 2.18: Conductance gs as function of the incident electron energy for t2 = −0.08 with variousstrength barriers and a fixed transverse momentum py=π. Calculation performed using waveguide the-ory.
One can observe that the enhanced peak located at zero energy is conserved even when the barrier
parameter increases to Z = 5 , therefore being independent of the local interface disorder, which is
exactly what one would expect for an ABS. Because the parameter choice imposes a topological phase,
the localized ABS can be identified as a Majorana fermion.
With all these considerations, it is then possible to conclude that in order to attain consistency between
the topological properties of the system and the number of ABS in each Fermi-pocket, it is necessary
that an interference effect occurs between the two traversed Fermi pockets, fixed by the momentum py.
These interference effects are correctly accounted for by waveguide theory, emphasizing the importance
of taking into account from the start the various conduction channels at a multiband superconductor,
rather than treating them independently and constructing the final differential conductance as a sum of
all separate contributions. Furthermore, when applied to a single pocket, waveguide theory proved to
attain the correct behaviour expected in the presence of ABS.
43
Chapter 3
Conclusions and future work
For the choice of the Hamiltonian’s parameter t2 = −0.08, the Fermi surface is composed by an odd
number of electron pockets and quantum waveguide theory allows a differential conductivity with a zero-
energy ABS located at the single pocket, which can be identified as a Majorana fermion because with
such value of t2 the system is in a topological phase with C = 1. This ABS is also present when the
differential conductance of the single-pocket is attained in the BTK framework, although the theory also
predicts an ABS in each of the other electron-pockets even when the system is non-topological. This
suggests that for the number of ABS to be reconciled with the topological properties of the system, the
two ABS states predicted by BTK theory for the pockets (0, 0) and (π, 0) should interfere destructively
when the incident electron has transverse momentum py = 0. For quantum waveguide theory this is
exactly what is observed. Thus, quantum waveguide correctly integrates the number of calculated ABS
with the topological features of the model, revealing the presence of important quantum interference
effects suppressing conductivity at zero energy bias, when two real electron pockets are transversed, a
phenomena which will not be observed if the bands are treated as separate conduction channels and
added together to form the differential conductivity at the interface.
As future work it would be interesting to study: the effect of spin-orbit interaction, by introducing a
Rabash term in the Hamiltonian. The Rabash spin-orbit term can lead to the recent discovered phe-
nomena of selective equal spin Andreev reflection [37], which consists of incident electrons, with certain
spin orientations, being reflected as holes with the same spin, at the interface between a normal-metal
and a topological superconductor; or calculate the differential conductance gs for the crossed Andreev
reflection mechanism.
45
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49
Appendix A
Derivation of the Bogolubov equations
for a non-uniform superconductor
The Bogolubov method is a common procedure which allows the diagonalization of a Hamiltonian such
as the BCS Hamiltonian of an inhomogeneous superconductor, where spatial variations are due to a
position dependent pair potential ∆(r) or because electrons also experience an arbitrary local potential
U(r) , which may include, for example, the Hartree-Fock averaged Coulomb potential [13, 32]:
H =
∫d3r
∑α
ψ†α(r)(− 1
2m∇2 + U(r)− µ(r)
)ψα(r)− V
∫d3rψ†↑(r)ψ†↓(r)ψ↓(r)ψ↑(r) ,with: α = ↑, ↓
(A.1)
V arises from the BCS attractive two-body potential Vkk′ mediating the interaction between the elec-
trons forming a Cooper pair and is taken in the Cooper approximation:
Vkk′ =
−V for |ξk| < ~wD0 otherwise
, (A.2)
where ~wD is the Debye energy and ξk is the one-electron energy measured from the Fermi level.
The operators ψ and ψ† are the annihilation/creation operators for fermions in real space and can be
attained from the usual momentum operators by a Fourier transform operation [13]:
ψα(r) =∑k
eik.rcαk (A.3)
As in the BCS theory for homogeneous superconductors the term ψ†↑(r)ψ†↓(r)ψ↓(r)ψ↑(r) can be trans-
formed into a quadratic form by performing the mean field approximation:
ψ†↑(r)ψ†↓(r) = A
ψ↓(r)ψ↑(r) = B(A.4)
51
⇒ A.B = (〈A〉+ δA).(〈B〉+ δB) ≈ 〈A〉B + 〈B〉A− 〈A〉〈B〉 . (A.5)
Then, by defining:
∆(r) = −V 〈ψ↑(r)ψ↓(r)〉 , (A.6)
The effective Hamiltonian from eq. (A.1) can be written as:
Heff =
∫d3r
∑α
ψ†α(r)(− 1
2m∇2 + U(r)− µ(r)
)ψα(r) + ∆∗(r)ψ↓(r)ψ↑(r) + ∆(r)ψ†↑(r)ψ†↓(r) (A.7)
The Hamiltonian is now quadratic in the operators ψ and ψ† but not diagonal. However, eq(A.7) can
be diagonalized by a suitable unitary transformation in which one looks for new operators such that:
Heff = EGroud state +∑n,α
εnγ†nαγnα , (A.8)
where εn is the excitation energy of the n state. This transformation is accomplished if :
ψ↑(r) =∑n(γn↑un(r)− γ†↓v∗n(r))
ψ↓(r) =∑n(γn↓un(r) + γ†↑v
∗n(r))
. (A.9)
Taking the commutator of Heff with the new operators γ and γ† and with the spatial dependent
operators ψ and ψ† , one obtains :
[Heff , γnα] = −εmγnα (A.10)
[Heff , γ†nα] = εmγ
†nα (A.11)
[ψ↑(r), Heff ] = H0ψ↑(r) + ∆(r)ψ†↓ (A.12)
[ψ↓(r), Heff ] = H0ψ↓(r)−∆∗(r)ψ†↑ (A.13)
with H0 the one electron Hamiltonian given by H0 = − 12m∇
2 + U(r) − µ(r). Then inserting eq(A.9)
(and its adjunct form) in the above equations, and using eq(A.10) plus eq(A.11), one will attain the
coupled system of equations: εun(r) = H0un(r) + ∆(r)vn(r)
εvn(r) = −H∗0vn(r) + ∆∗(r)un(r)(A.14)
These are the Bogoliuvbov equations. Particularly, this spatially dependent form is the one used in the
BTK paper and in the limit where ∆ , µ and U are constants, the equations will admit as eigenfunctions
the momentum plane-waves, allowing to express eq(A.14) in the same form as in the original BCS
52
Appendix B
Fukui-Hatsugai-Suzuki method for
calculating the Chern number on a
discretized Brillouin zone.
The calculations for the Chern number followed the numerical implementation of the method which is
here described [35, 36]. It will be first addressed the case where the eigenstates are non-degenerate,
as it will be useful to introduce the key steps of the method and also provide a clearer connection to the
original formula in eq(1.3). Then, the adaptation of the model for a non-Abelian Chern number will be
presented, which was the method applied to attain the results in eq(2.15 ).
The Brillouin zone will be given by the discrete set of points kµ = 2πjN , with j = 1, ..., N and µ = x, y,
connected by a link δkµ which will be a vector in the µ direction with magnitude 2πN . Defining the link
variable:
Uµ(k) =〈n(k)|n(k + δkµ)〉|〈n(k)|n(k + δkµ)〉|
, (B.1)
the Berry curvature can be constructed as:
Fxy(k) = ln(Ux(k)Uy(k + δkx)Ux(k + δky)−1Uy(k)−1) , (B.2)
which is restricted to the interval −π < −iFxy(k) ≤ π. The nth Chern number will thus be given by:
Cn =1
2iπ
∑k
Fxy(k) . (B.3)
One should note that the link variable Uµ(k) is the result of discretizing the derivative operator into
the infinitesimal difference operator:
Aµ = 〈n(k)|∂µ|n(k)〉 → 〈n(k)|n(k + δkµ)〉 − 〈n(k)|n(k)〉 . (B.4)
54
Because 〈n(k)|∂µ|n(k)〉 is a pure imaginary number (a property that follows form the normalization
condition of the Bloch eigenstates) and noticing that:
〈n(k)|∂µ|n(k)〉 → 〈n(k)|n(k + δkµ)〉 − 〈n(k)|n(k)〉 = 1− iα− 1, with α << 1 , (B.5)
⇒ 〈n(k)|n(k + δkµ)〉 = 1− iα ≈ e−iα =〈n(k)|n(k + δkµ)〉|〈n(k)|n(k + δkµ)〉|
⇒ −iα ≈ ln[ 〈n(k)|n(k + δkµ)〉|〈n(k)|n(k + δkµ)〉|
], (B.6)
Thus, the Berry connection in the discretized Brillouin zone, eq(B.4), can be expressed by the loga-
rithm of the link variable, which will provide the Berry curvature given by eq(B.2) if one also applies the
discretized version of the derivative operator to the link variable.
In the case where there exists multiple energy bands bellow the Fermi energy, the generalization
of the method to accommodate the mixing between bands follows in close resemblance to the non-
degenerated case with the only change falling into the link variable:
Uµ =det(φµ)
|det(φµ)|. (B.7)
Here φµ is a matrix formed by the elements (φµ)ij = 〈ni(k)|nj(k + δkµ)〉 , with 0 ≤ i, j ≤ n and where
n is the number of bands bellow the Fermi level. The Berry connection and the Chern number can then
be obtained following the same expressions as in the non-degenerate case: eq( B.2) and eq( B.3 ).
55
Appendix C
The problem of finding a consistent
Gauge in a two-band model
The Bloch Hamiltonian for a two-band model is given by h(k).σ, for which it is possible to express the
eigenstates in spherical coordinates as:
|ψ−〉 =
sin( θ2 )
−eiφcos( θ2 )
(C.1)
|ψ+〉 =
cos( θ2 )
eiφsin( θ2 )
. (C.2)
Restricting our attention in the |ψ−〉 state, if we compute it for θ = 0
|ψ−〉 →
0
−eiφ
(C.3)
we see that the eigenstate depends on the azimuthal angle φ, which is ill-defined at θ = 0 because
if φ is either 0 or π, the correspondent point in space will be exactly the same, but for |ψ−〉 this implies
jumping from (0,-1) to (0,1). Thus, the eigenstate is not single-valued over the full S2 sphere. We can
try to find another gauge in which this ”anomaly” point is no longer present. For example, we could try:
|ψ−〉 → e−iφ|ψ−〉 = |ψ′−〉 . (C.4)
Now the eigenstate is well defined for θ = 0, but it will have the same problem for θ = π and hence, there
is no gauge for which |ψ−〉 will be well defined for both θ = 0 and θ = π [3].
In the vector h space, the sphere can be constructed by the orthogonal unit vectors:
hz =hz|h|
= cos(θ), hx =hx|h|
= sin(θ)cos(φ), hy =hy|h|
= sin(θ)sin(φ) , (C.5)
from which we can see that if hz(k) is positive at all points of the Brillouin zone, cos(θ) > 0 ⇒
56
θ ∈ [0, π2 ] and hence the second gauge choice, |ψ′−〉, will guarantee a global well-defined eigenvector,
because θ will never visit the south pole (θ = π). Likewise, if hz(k) is always negative, θ never reaches
the north pole and the first gauge will provide a well define eigenstate over the entire Brillouin zone.
Nevertheless, if we can define a global gauge, the Chern number will be zero (as discussed in the
first chapter). Hence, it is when hz(k) changes sign and visits both the north and south pole of the S2
sphere, which will be when it is impossible to define a global, continuous and smooth gauge over the
entire Brillouin zone, that a topologically nontrivial phase will exist [3].
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