A novel platform for two-dimensional chiral topological superconductivity

5.1 MB·English

by  Jian Li

A novel platform for two-dimensional chiral topological superconductivity
Jian Li,1 Titus Neupert,2 Zhi Jun Wang,1 A. H. MacDonald,3 A. Yazdani,1 and B. Andrei Bernevig1
1Department of Physics, Princeton University, Princeton, NJ 08544, USA
2Princeton Center for Theoretical Science, Princeton University, Princeton, NJ 08544, USA
3Department of Physics, University of Texas at Austin, Austin, TX 78712, USA
(Dated: January 7, 2015)
We show that the surface of an s-wave superconductor decorated with a two-dimensional lattice
of magnetic impurities can exhibit chiral topological superconductivity. If impurities order ferro-
magneticallyandthesuperconductingsurfacesupportsasufficientlystrongRashba-typespin-orbit
coupling,Shibasub-gapstatesatimpuritylocationscanhybridizeintoBogoliubovbandswithnon-
5
vanishing, sometimes large, Chern number C. This topological superconductor supports C chiral
1
Majorana edge modes. We construct phase diagrams for model two-dimensional superconductors,
0
accessing the dilute and dense magnetic impurity limits analytically and the intermediate regime
2
numerically. To address potential experimental systems, we identify stable configurations of fer-

n romagnetic iron atoms on the Pb (111) surface and conclude that ferromagnetic adatoms on Pb
a surfaces can provide a versatile platform for two-dimensional topological superconductivity.
J

5
The chiral p-wave superconductor in two dimensions

] (2D) and the closely related fractional quantum Hall
l
l Pfaffian state at ν = 5/2 are the archetypal examples
a ferromagnetic
h of topologically ordered states of matter that support island
- non-Abelian anyonic excitations. [1–3] The theoretical
s
e explorationofthesestateshasshapedourunderstanding
m of topological order and is foundational for the distant s-wave superconductor (Pb)
. goal of building a topological quantum computer. [4–
t
a 6] In contrast to the ν = 5/2 fractional quantum Hall
FIG. 1. Concept of building a topological p-wave supercon-
m
state, however, to date chiral p-wave superconductiv- ductor from magnetic impurity bound states on the surface
- ity has not been confirmed in any experimental system. of an s-wave superconductor like Pb with strong spin-orbit
d
The most prominent candidate system, superconducting coupling. A very thin layer of magnetic adatoms, such as Fe
n
o Sr2RuO4, [7] has been the subject of an ongoing debate or Co, is deposited on the surface of Pb, and forms islands
around which chiral Majorana edge modes are present (yel-
c about its actual paring state. [8–11]
low). Dependingonthedensityofthemagneticatoms,oneor
[
Chiral superconductors break time-reversal symmetry manyMajoranamodesmayappear. Aone-dimensionalchain

1 (TRS).ThishinderstheformationofCooperpairs,since ofmagneticatomsmaysupportlocalizedMajoranamodesat
v orbital(andpossiblyparamagnetic)pair-breakingeffects its end. These modes can be detected by tunneling tech-
9 cancomeintoplay. Depairingisalsothemainhurdlefor niques.
9
realizingalineofproposalsinwhichlayeredheterostruc-
9
0 turesinvolvingferromagnetsands-wavesuperconductors
0 are used to build an artificial 2D p-wave superconduc-
1. tor. [12–14] The guiding principle for these proposals is As the phase-space for the existence of a topological su-
0 to design a band structure with a single normal-state perconductoriscontrolledbytheZeemangap[23], these
5 Fermi surface with no spin-degeneracy. If Rashba spin- systems require a delicate balance of the parameters in-
1 orbit coupling is present so that the states on this Fermi volved (spin-orbit coupling, magnetic field and chemical
:
v surface are not fully spin-polarized, and if s-wave super- potential)inordertocreatethetopologicalsuperconduc-
i conductivity is proximity-induced in such a system, the tor.
X
effective pairing near the single Fermi surface is equiva-
Recently, a 1D topological superconductor was real-
r
a lent to that of a chiral p-wave superconductor. ized in a system that is quite distinct but employs sim-
In one dimension (1D), based on the principle of com- ilar microscopic ingredients – spin-orbit coupling, ferro-
bining spin-orbit coupling and externally applied mag- magnetism, and s-wave superconductivity. [24] A chain
netic fields, various groups have proposed engineering ofmagneticFeatomsisdepositedonthesurfaceofans-
artificial realizations of p-wave superconductors. [15–21]. wavesuperconductor withstrong spin-orbitinteractions.
An experimental realization of these proposals employ- The Fe chain is ferromagnetically ordered [24] with a
ing semiconductor nanowires with strong spin-orbit cou- largemagneticmoment,whichtakestheroleofthemag-
pling [22] has reported Majorana fermion signatures. In netic field in the nanowire experiments. Unlike previous
this setup, the externally applied magnetic field has to proposals,this“magneticfield”ismostlylocalizedonthe
berathersmall(toavoidsuppressingsuperconductivity). Fe chain, with small leakage outside. Superconductivity
2
isnotdestroyedalongthechain. Inthissetup,theenergy consider the mean-field Hamiltonian
scaleoftheexchangecouplingoftheFeatomsistypically H =(cid:88)(cid:104)t(cid:0)c†c +c†c (cid:1)− µc†c +∆c† c†
much larger than that of the Rashba spin-orbit coupling r r+eˆ1 r r+eˆ2 2 r r r,↑ r,↓
and the superconducting pairing. The ferromagnetically r∈Λ
orderedFeatomsinducelocalizedShibastateswithinthe +iα(cid:0)c†rσ2cr+eˆ1 −c†rσ1cr+eˆ2(cid:1)+h.c.(cid:3)
gap of the superconductor. [25–27] The hybridization of +J (cid:88) c†σ c .
r 3 r
these states forms the band structure of a 1D p-wave
r∈Λ∗
superconductor that supports Majorana end states. [28] (1)
Because the Fe bands are fully spin split, no additional
control over the chemical potential is necessary. A sim- Here,c† =(c† ,c† )isaspinorofthecreationoperators
r r,↑ r,↓
ilar scenario applies when the Fe orbitals are magnetic for electrons at site r with spin ↑,↓, and σ , i = 1,2,3,
i
but itinerant. [23] are the three Pauli matrices. We denote by t the near-
est neighbor hopping integral in the superconductor, µ
In this Letter we point out that this strategy can also thechemicalpotential,andαthestrengthoftheRashba
be successful in 2D. Magnetic adatoms on the surface of spin-orbit coupling. Classical magnetic moments ferro-
a superconductor with strong spin-orbit coupling, when magnetically aligned normal to the plane in the σ3 di-
arranged in a 2D lattice, [29] can yield a 2D topologi- rection are positioned on a sublattice Λ∗ of Λ and are
cal chiral p-wave superconductor whose chiral Majorana exchange-coupled to electrons via the term proportional
edge modes can be observed in Scanning Tunneling Mi- to J. Drawing experience from the 1D situation and the
croscope (STM) measurements (see Fig. 1). To shed ab-initiocalculationspresentedbelow,thephysicallyrel-
light on the rich range of possibilities, we analyze the evant hierarchy of energy scales that we consider here is
topological properties of a system with dense local mo- given by
mentsthatareexchangecoupledtoamodel2Dsupercon-
t(cid:29)J (cid:29)α∼∆. (2)
ductor, demonstrating that topological superconductors
with higher Chern numbers, and consequently multiple Dense impurity limit —Asawarmup, itisinstructive
chiral Majorana edge channels, can easily occur. We are to consider the simplest situation in which each lattice
also able to analyze the model’s dilute magnetic impu- site is coupled to a magnetic moment, i.e., Λ=Λ∗. This
rity limit analytically and obtain numerical topological limit is representative of system with self-assembled is-
phasediagramsforintermediateimpurityconcentrations. landsofmagneticadatoms. Itsconsiderationallowsusto
Based on density-functional-theory (DFT) calculations, highlight the difference between the regime Eq. (2), and
wefurtherproposerealizing2Dtopologicalchiralp-wave that of small J that has been studied previously. [12–14]
superconductors experimentally by depositing transition In particular, it is possible to access a phase with Chern
metaladatomsonsuperconductingPb. Thetypeofmag- number 2 in the large J limit. For the case Λ = Λ∗ and
netic ion can be varied to access different strengths of α = 0, the Hamiltonian (1) has gapless lines in momen-
the magnetic moment. In the case of Fe adatoms on a tum space defined by
Pb (111) surface, we show that strong magnetic order in (cid:112)
ε =± J2−∆2, (3)
general leads to an odd number of 2D Fermi surface seg- k
ments. As a consequence the proximity-induced super- whereε =2t(cos k +cos k )−µ. Addingspin-orbitcou-
k 1 2
conducting phases can have nonzero Chern numbers and plingα(cid:54)=0willgenericallyliftthesedegeneracylinesand
chiral Majorana edge modes. Below we first present our gapthespectrumexceptatthefourinversion-symmetric
modelsystemresultswhichbringoutanumberofgeneric momenta [k =(1−i,1−j)π/2 with i,j =±] at which
ij
features of superconductor surfaces with ferromagneti- thespin-orbitcouplingvanishesandatwhichphasetran-
callyorderedmagneticadatoms,andthenwesummarize sitionsbetweendifferentChernnumberphasesoccur. At
our DFT results by focusing on the specific case of Fe each of these momenta, the condition (3) is met for two
adatoms on a Pb (111) surface. values of the chemical potential
(cid:112)
To render the analytical calculations tractable, we µi,j,λ =−2(i+j)t+λ J2−∆2, i,j,λ=±. (4)
consider a Hamiltonian that models only the surface
Around each of these gapless points in the three-
layer of a bulk 3D s-wave superconductor on which
dimensional µ-k parameter space, we can reduce the
the s-wave superconducting order parameter ∆ is in-
Hamiltonian to an effective two-band model and expand
duced from the bulk. On this superconducting layer, we
it to linear order in the deviations δk from k and the
ij
model the magnetic impurities as classical spins whose
deviations δµ from µ yielding
i,j,λ
only interaction with the electrons in the superconduc-
(cid:114)
tor is through Zeeman-like couplings. [25–27] Employing ∆α ∆2
Heff =− (jδk σ −iδk σ )+λ 1− δµσ .
a tight-binding description on a 2D square lattice Λ that i,j,λ J 2 1 1 2 J2 3
is spanned by the primitive lattice vectors eˆ and eˆ , we (5)
1 2
3
potentials,morethantwoFermisurfacescanexistinthe
reduced Brillouin zone defined by Λ∗ as the Fermi wave-
lengths are significantly smaller than the lattice spacing
ofΛ∗, asaconsequencehigherChernnumberscanoccur
[e.g. 8 in Fig. 2(a) and 15 in Fig. 2(b)] but the trade-
off is an overall smaller induced gap; iii) the phases with
different Chern numbers are generally separated by lines
(in the 2D parameter space) defined by conditions simi-
lar to Eq. (4), and across each specific line the change of
Chern numbers is a constant determined essentially the
same way as in our preceding analysis.
Dilute impurity limit — To better understand the di-
lute impurity limit, we complement our results on the
lattice by a calculation in which we treat the underlying
2D superconductor in the continuum limit and consider
sparsely distributed Shiba impurities that are arranged
inasquarelatticeontopofit. Thisallowsustoderivean
FIG. 2. Phase diagrams for Hamiltonian (1), in terms of
Chern numbers [(a) and (b)] and superconducting gaps [(c) effectivetwo-bandmodelforthehybridizingShibastates.
and (d)] as a function of the chemical potential µ and the This effective Hamiltonian represents a chiral p-wave su-
strength of the magnetic moments J. The panels (a) and (c) perconductor in the appropriate parameter regime.
[(b) and (d)] correspond to the case of one magnetic adatom The strategy of our derivation is inspired by the cal-
every 2×2 (3×3) lattice sites. In these plots, ∆ = 0.06t and
culation for 1D Shiba chains of Ref. [28] (for details,
α = 0.1t; µ has been shifted such that µ = 0 implies the
see [30]). We start from a 4×4 Bogoliubov-de-Gennes
chemical potential lying in the center between the two spin-
(BdG) Hamiltonian
split band bottoms.
(cid:18) (cid:19)
ξ ∆
HBdG = k (7a)
Here,thePaulimatricesactonasubspacedefinedbythe k ∆∗ −ξ
k
two bands that satisfy condition (4). In the µ-k space,
Hamiltonian (5) represents a source of Berry flux that that acts on 4-spinor valued wave functions Ψ(r) =
corresponds to a unit topological charge i×j×λ=±1. (ψ ,ψ ,ψ†,−ψ†)T(r), r ∈R2, with ξ =k2/(2m)−µ+
↑ ↓ ↓ ↑ k
As the chemical potential ramps through the gapless α(k σ −k σ ). In addition, the magnetic impurities are
1 2 2 1
pointatµi,j,λ,th√eChernnumberchangesbyi×j×λ. Let represented by the Hamiltonian
us assume 2t > J2−∆2, so that the critical chemical
(cid:88)
potentials are in the order µ−,−,− <µ−,−,+ <µ−,+,− = Himp =−J δ(r−r∗)S3, (7b)
µ < µ = µ < µ < µ . Triv-
+,−,− −,+,+ +,−,+ +,+,− +,+,+ r∗∈Λ∗
ially, for µ < µ the Chern number is zero. By
−,−,−
increasing µ, the system thus exhibits phases with the whereS3 =σ3⊗τ0,withτ3 theidentitymatrixactingon
Chern numbers particle-holespace. Ifwerestrictthewave-functionΨ(r)
to the locations r∗ ∈Λ∗ of the impurities, we obtain the
C = −1, 0, +2, 0, −1, (6)
self-consistency equation
wherethephasetransitionsoccurattherespectiveµ .
Hence, with homogeneous magnetization, the superic,oj,nλ- (cid:0)E−HqBdG(cid:1)−1 (−JS3)Ψ(cid:101)(q)=Ψ(cid:101)(q) (8)
ductor may already exhibit Chern number equal to 2.
In cases where magnetic impurities are spaced more for the Fourier transforms Ψ(cid:101)(q)=(cid:80) e−iq·r∗Ψ(r∗),
r∗∈Λ∗
sparsely,i.e.,ifΛ∗ isasublatticeofΛ,evenhigherChern wherethemomentumq ∈[0,2π)2 nowbelongstotheΛ∗
numbers can be obtained. Brillouin zone and we have set the impurity spacing to
We solved Hamiltonian (1) numerically for the case of unity.
onemagneticimpurityevery2×2and3×3plaquettesof We need two more steps to reduce Eq. (8) to an ef-
thesquarelattice,andshowthephasediagramsinFig.2. fective two-band model for the Shiba states, assuming
From the phase diagrams, we can read off three general they are deep in the superconducting gap and dilute
features: i)atsmallchemicalpotentialsaroundtheband compared with the Fermi wavelength. First, the left-
bottom, where the Fermi wavelengths are larger than or hand side is expanded to linear order in the energy E
comparable to the lattice spacing of Λ∗, the sequence of to cast the equation in the form of the time-independent
Chernnumbers(−1,0,+2)alwaysoccurswhenJ/tisnot Schroedinger equation. Second, we project the effective
toolarge–thisuniversalfeaturecorrespondstothedense HamiltonianintotheeigenstatesofanisolatedShibaim-
√
limitthatwehavediscussedabove; ii)atlargerchemical purity on every site, given by Ψ = (1,0,1,0)T/ 2 and
+
4
mod(C,2)=0 mod(C,2)=1 thevalidityofHamiltonian(9)requiresaself-consistency
thatpermitsthelow-energyexpansionofEq.(8). There-
fore, instead of computing the Chern numbers in an ex-
tended parameter space, we focus on information that
can be obtained at special points of the Brillouin zone
atinfinitesimalenergy. Tothatend,observethatHamil-
tonian (9) has C rotational symmetry. Thus, any gap
4
closing at points other than the C -symmetric momenta
4
k = (0,0) and k = (π,π) changes the Chern number by
an even integer due to the symmetry-imposed multiplic-
ity of the nodal points. By expanding the Hamiltonian
into Dirac form around the C -symmetric momenta, we
4
obtain the expression [30]
(−1)C =sgn(cid:2)d (cid:3)sgn(cid:2)d (cid:3) (η =1) (10)
0,(0,0) 0,(π,π)
FIG. 3. Parity of the Chern number for the BdG band
for the parity of the Chern number. The numerical eval-
structure of an s-wave superconductor decorated with dilute
Shiba impurities, following Eq. (10). Here, aΛ∗ is the lattice uation of this equation is shown in Fig. (3) in the form
spacing of the Shiba impurities, λ is the Fermi wavelength of a phase diagram.
F
in the limit α → 0 and the Hamiltonian (9) was evaluated We have also performed calculations for magnetic or-
with ξ=30λF. ders other than simple ferromagnetism. In particular,
the case where the magnetic configurations corresponds
√ to 2D helices is related to previous studies on 1D he-
Ψ =(0,1,0,−1)T/ 2[28]. Weobtaintheeffectivetwo-
− lices. [31–40] We obtained criteria for such a system to
band Hamiltonian
be fully gapped by proximity effect, and found that the
1 fully-gapped superconducting phases can be generically
Heff = [(1−η−d )σ −d σ +d σ ],
q η+d 0,q 3 2,q 1 1,q 2 topologically nontrivial. The results and phase diagrams
0,q
(9a) are presented in the supplementary material. [30]
where In closing, we complement our simple model consider-
ations with a specific material proposal to realize a su-
d = J (cid:88) e−iq·r∗f (|r∗|), (9b) perconductor with nonzero Chern numbers. For that,
0,q 2 +
we consider transition metal atoms, in particular Fe, de-
r∗(cid:54)=0
di,q =iJ2 (cid:88) e−iq·r∗|rri∗∗|f−(|r∗|), (9c) ppoerscitoendduonctothr.e T(1h1e1)sasmurefaccoemobfinPabt,ioansotrfomngatteyrpiael-sI,Ibsuut-
r∗(cid:54)=0 a different surface of Pb, was used in the experimental
realization for the 1D p-wave superconductor. [24] The
are defined in terms of the functions
Pb atoms on the (111) surface form a triangular lattice.
f (r)= e−√r/ξ (cid:88)(±1)(1−λ)/2(cid:113)k(λ)cos(cid:16)k(λ)r− π(cid:17), Through ab-initio calculations that assume a ferromag-
± v 2πr F F 4 neticalignmentoftheFemagneticmomentsandinclude
F λ=± spin-orbit coupling, we compared the relaxation energies
(9d)
for various densities and arrangements of Fe adatoms on
In the equation above, ξ is the coherence length of
the Pb (111) surface, and found that a deposition with
the 2D superconductor (without the magnetic impuri-
one Fe atom in each triangular plaquette is particularly
ties), v = (cid:112)α2+2µ/m is its Fermi velocity, k(±) =
F F favorable(fordetails,see[30]). Inthiscase,theFeatoms
(cid:112)
2mµ+α2m2∓mα are the Fermi wave-vectors for its formahoneycomblatticeinwhichtheatomssitatdiffer-
two spin-split bands, and η = J[k(+) +k(−)]/4v is a ent heights in each sublattice [see Fig. 4(a)]. We further
F F F
dimensionless parameter. In the “deep Shiba limit” in performed DFT calculations of the electronic structure.
which the projection in the states Ψ and Ψ is justi- The resulting (not spin-degenerate) Fermi surface and
+ −
fied, we have η ∼1. band structure restricted to the Fe d-orbitals is shown in
The Hamiltonian (9) represents the effective super- Fig. 4(b) and (c). Critically, we find an odd number of
conductor formed by the Shiba bound states within the Fermi surfaces. The Chern number of the corresponding
gap of the underlying s-wave superconductor. Similar to BdGHamiltonianisindeednonzerooveralargerangeof
the case of the effective two-band model in Eq. (5), this the chemical potential [see Fig. 4(d)].
Hamiltoniancanhavenodalpointsink-µ-spaceatwhich In conclusion, we have proposed a versatile platform
the Chern number changes. However, unlike in Eq. (5), for realizing chiral superconductors in 2D. We have ob-
the nodes can occur at any point in the Brillouin zone, tainedanalyticallythetopologicalphasediagram(Chern
making an analytic treatment intractable. In addition, numberandgapsofthesuperconductor)ofthediluteand
5
DARPA under SPAWAR Grant No.: N66001-11-1-4110,
the Packard Foundation, and a Keck grant. JL acknowl-
edges support from Swiss National Science Foundation.
AHM acknowledges support from the Welch Foundation
under grant TBF1473. AY acknowledges support from
NSFMRESEC-DMR-0819860 and NSF-DMR-1104612.
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0952428, ONR-N00014-11-1-0635, MURI-130-6082,
6
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7
Supplemental Material for “A novel platform for two-dimensional chiral topological
superconductivity”
CONTENTS
Acknowledgments 5
References 5
Derivation of the effective Hamiltonian in the dilute Shiba limit 7
General strategy 7
Shiba lattice in the 2D Rashba superconductor 8
Two-dimensional Shiba lattice with helical magnetic order 14
Commuting helix rotations 16
Non-commuting helix rotations 17
Details of density functional theory analyses 17
Derivation of the effective Hamiltonian in the dilute Shiba limit
General strategy
We want a general formalism to find the low energy excitations and effective Hamiltonian of the following Hamil-
tonian
(cid:88)
H =H +H , H = V δ(r−r ), (11)
0 1 1 n n
n∈c
where H is the original Hamiltonian which is gapped (≈∆) around zero energy, V are matrices associated with the
0 n
inner degrees of freedom (spin, particle-hole ect.) which can induce in-gap states, and r is implicitly d-dimensional
(also n, m, k, q, R, n below).
b
We start with the Schr¨odinger equation
(H +H )Ψ=EΨ, (E <∆). (12)
0 1
It follows that
G (E)H Ψ=Ψ, (13)
0 1
where G (E)=(E−H )−1 is the Green function.
0 0
Because H is composed of delta-functions for a small set c, G H is nonzero only in the columns corresponding to
1 0 1
c, thus Eq. 13 is equivalent to
(cid:88)
G (E;r,r )V ψ(r )=ψ(r). (14)
0 n n n
n∈c
In the simplest case, if c contains only one single point, labeled by 0, then Eq. 14 implies
G (E;r ,r )V ψ(r )=ψ(r ). (15)
0 0 0 0 0 0
The energy of the excitations can be obtained by solving
Det[1−G (E;r ,r )V ]=0, (16)
0 0 0 0
(cid:90)
G (E;r ,r )=G (E;0)= dk G (E;k). (17)
0 0 0 0 0
8
In more complicated cases, the following equation, again implied by Eq. 14, can serve as the starting point to extract
an effective Hamiltonian
(cid:88)
∀m∈c: G (E;r ,r )V ψ(r )=ψ(r ), (18)
0 m n n n m
n∈c
(cid:90)
G (E;r ,r )=G (E;r −r )= dk G (E;k)eik(rm−rn). (19)
0 m n 0 m n 0
In addition if ∀n: V =V and r =nR (n∈Zd with d the dimension), Eq. (18) can be transformed to k-space:
n 0 n
(cid:88)
ψ(q)= e−iqrmψ(r ) (q ∈[−π/R,π/R)) (20)
m
m
(cid:88)
= e−iqrmG (E;r −r )V ψ(r ) (21)
0 m n 0 n
m,n
(cid:90)
(cid:88)
= dk G (E;k)ei(k−q)(rm−rn)V e−iqrnψ(r ) (22)
0 0 n
m,n
(cid:34) (cid:35) (cid:34) (cid:35)
(cid:90)
= dk G (E;k) (cid:88)ei(k−q)m(cid:48)R V (cid:88)e−iqrnψ(r ) (23)
0 0 n
m(cid:48) n
(cid:34) (cid:35)
(cid:90)
(cid:88)
= dk G (E;k) δ(k−q−2n π/R) V ψ(q) (24)
0 b 0
nb
=G (E;q)V ψ(q), (25)
0 0
where
(cid:88)
G (E;q):= G (E;q+2n π/R). (26)
0 0 b
nb
n can be interpreted as the band index when the k-space is folded into the Brillouin zone defined by [−π/R,π/R).
b
Shiba lattice in the 2D Rashba superconductor
We start with the BdG Hamiltonian
(cid:18) (cid:19)
ξ ∆
H (k)= k k , (27)
BdG ∆† −ξ
k k
(cid:126)2
ξ = k2−µ+α(k σ −k σ ), (28)
k 2m∗ x y y x
∆ =∆ σ +δ (k σ −k σ ). (29)
k s 0 p x y y x
First we make the Hamiltonian dimensionless by defining
k =(cid:112)2m∗µ/(cid:126)2, k˜ =k/k , (30)
µ µ
l =(cid:126)2/(2m∗α), α˜ =1/(l k ), (31)
α α µ
l =(cid:126)2/(2m∗δ ), ∆ =1/(l k ), (32)
p p p p µ
H˜ =H/µ, ∆˜ =∆ /µ, (33)
s s
and then omit the tildes, the bulk Hamiltonian becomes
H (k)=τ ⊗[(k2−1)σ +α(k σ −k σ )]
BdG z 0 x y y x
+τ ⊗[∆ σ +∆ (k σ −k σ )]. (34)
x s 0 p x y y x
In this dimensionless Hamiltonian, energies are measured in units of µ (assumed to be large compared with other
energy scales), lengths (wave vectors) are measured in units of the Fermi wavelength (wave vector).
9
Next we go to the basis where the normal state Hamiltonian is diagonalized
H˜ (k)=U(ϕ )†H (k)U(ϕ ) (35)
BdG k BdG k
=τ ⊗[(k2−1)σ +αkσ ]+τ ⊗[∆ σ +∆ kσ ] (36)
z 0 z x s 0 p z
=[E (k)τ +∆ (k)τ ]⊕[E (k)τ +∆ (k)τ ], (37)
+ z + x − z − x
where
(cid:18) (cid:19)
1 1 −1
U(ϕ )=τ ⊗ √ , (38)
k 0 2 ieiϕk ieiϕk
E (k)=k2−1±αk, (39)
±
∆ (k)=∆ ±∆ k. (40)
± s p
According to the general formalism (Sec. ), we need to evaluate
(cid:90)
G (E,r)= dkG (E,k)eik·r (41)
0 0
(cid:90) ∞ kdk (cid:90) 2π dϕ
= 2π 2πk U(ϕk)[E−H˜BdG(k)]−1U(ϕk)†eikrcos(ϕk−ϕr) (42)
0 0
(cid:20)(cid:90) 2π dϕ(cid:48) (cid:21)
=R(ϕ ) k U(ϕ(cid:48))Γ(E,rcosϕ(cid:48))U(ϕ(cid:48))† R(ϕ )†, (43)
r 2π k k k r
0
where
(cid:90) ∞ kdk
Γ(E,x)= eikx[E−H˜ (k)]−1, (44)
2π BdG
0
(cid:18) (cid:19)
1 0
R(ϕ )=τ ⊗ . (45)
r 0 0 eiϕr
To evaluate Γ(E,x), we first perform the integral
(cid:90) ∞ kdk
Γ (E,x)= eikx[E−E (k)τ −∆ (k)τ ]−1 (46)
b=± 2π b z b x
0
(cid:39)ρb(cid:90) dEbei[kF(b)+Eb/vF]x EE+2E−bEτz2+−∆∆b2τx, (47)
b b
where v = 2k (k ≡ (cid:112)1+α2/4) is the Fermi velocity (the same for ± bands), k(±) = k ∓α/2 is the Fermi wave
F 0 0 F 0
vector for +/−-band, ρ = k(b)/2πv is the density of states for b-band at the Fermi energy, and ∆ = ∆ (k(b)) is
b F F b b F
the pairing gap at the Fermi energy. Following Pientka et al. [42], we have
(cid:90) 1 π
dEbeiEbx/vF E2+a2 = ae−a|x|/vF, (48)
b
(cid:90) E ω2
dEbeiEbx/vF E2+ba2E2+Dω2
b b D
iπω2
= D sgn(x)(e−a|x|/vF −e−ωD|x|/vF), (49)
ω2 −a2
D
(cid:112)
where a= ∆2−E2 is a short-hand notation, and ω is the Debye frequency which will be sent to +∞ in the end.
b D
Then
Γ =γ +γ τ +γ τ , (50)
b 0b 1b x 3b z
 
γ +γ 0 γ 0
0+ 3+ 1+
Γ=Γ+⊕Γ− = γ10+ γ0−+0 γ3− γ0+−0 γ3+ γ10− , (51)
0 γ 0 γ −γ
1− 0− 3−
10
where
√
γ0b =−πρb (cid:112) E eikF(b)x− ∆2b−E2|x|/vF, (52)
∆2−E2
b
√
γ1b =−πρb (cid:112) ∆b eikF(b)x− ∆2b−E2|x|/vF, (53)
∆2−E2
b
γ3b =−πρb ω2 +iEωD22−∆2 sgn(x)eikF(b)x(e−√∆2b−E2|x|/vF −e−ωD|x|/vF). (54)
D b
Next we find
 A+ −ie−iϕkA− C+ −ie−iϕkC−
U(ϕk)ΓU(ϕk)† = 21 ieiCϕk+A− −ie−Ai+ϕkC− ieiBϕk+C− −ie−Ciϕ+kB−, (55)
ieiϕkC− C+ ieiϕkB− B+
where
A =(γ +γ )±(γ +γ ), (56)
± 0+ 3+ 0− 3−
B =(γ −γ )±(γ −γ ), (57)
± 0+ 3+ 0− 3−
C =γ ±γ . (58)
± 1+ 1−
The integral Eq. (43) breaks down to the following basic one
(cid:90) π/2
I (z)=il dϕei(lϕ+zcosϕ) (l=0,±1) (59)
l
−π/2
(cid:90) π/2 (cid:34) +∞ (cid:35)
(cid:88)
=il dϕeilϕ inJ (z)einϕ (Jacobi-Anger expansion) (60)
n
−π/2 n=−∞
+∞ (cid:90) π/2
(cid:88)
= il+nJ (z) dϕei(l+n)ϕ (61)
n
n=−∞ −π/2
+∞
=πJ (z)+2i (cid:88) J2m+1−l(z), (62)
−l 2m+1
m=−∞
where J (z) is the the n-th order Bessel function of the first kind. Expressed in terms of I (z), we have
n l
(cid:90) 2π d2ϕπk ileilϕkγ0b(cosϕk)=−ρ2b (cid:112)∆2E−E2 (cid:2)Il(zb)+(−1)lIl(−zb∗)(cid:3), (63)
0 b
(cid:90) 2π d2ϕπk ileilϕkγ1b(cosϕk)=−ρ2b (cid:112)∆∆2−b E2 (cid:2)Il(zb)+(−1)lIl(−zb∗)(cid:3), (64)
0 b
(cid:90) 2π dϕ
2πk ileilϕkγ3b(cosϕk) (65)
0
=−ρb iωD2 (cid:2)I (z )−(−1)lI (−z∗)−I (z(cid:48))+(−1)lI (−z(cid:48)∗)(cid:3), (66)
2 ω2 +E2−∆2 l b l b l b l b
D b
(cid:113)
z ≡k(b)r+i ∆2−E2r/v , z(cid:48) ≡k(b)r+iω r/v . (67)
b F b F b F D F
First we look at the limit r →0, such that z ,z(cid:48) →0. In this case
b b
I (z =0)=π, I (z =0)=±2i. (68)
0 ±1
We find
G (E,r →0)=R(ϕ )(A˜τ ⊗σ +C˜τ ⊗σ )R(ϕ )† (69)
0 r 0 0 x 0 r