# Unconventional Fermi surface spin textures in the Bi_ {x} Pb_ {1− x}/Ag (111) surface alloy

**ABSTRACT** The Fermi and Rashba energies of surface states in the BixPb1−x/Ag(111) alloy can be tuned simultaneously by changing the composition parameter x. We report on unconventional Fermi surface spin textures observed by spin and angle-resolved photoemission spectroscopy that are correlated with a topological transition of the Fermi surface occurring at x=0.5. We show that the surface states remain fully spin polarized upon alloying and that the spin-polarization vectors are approximately tangential to the constant energy contours. We discuss the implications of the topological transition for the transport of spin.

**0**Bookmarks

**·**

**99**Views

- [Show abstract] [Hide abstract]

**ABSTRACT:**Using spin and angle-resolved photoemission spectroscopy we investigate a momentum region in Pb quantum well states on Si(111) where hybridization between Rashba-split bands alters the band structure significantly. Starting from the Rashba regime where the dispersion of the quasi-free two-dimensional electron gas is well described by two spin-polarized parabolas, we find a breakdown of the Rashba behavior which manifests itself (i) in a spin splitting that is no longer proportional to the in-plane momentum and (ii) in a reversal of the sign of the momentum splitting. Our experimental findings are well explained by including interband spin-orbit coupling that mixes Rashba-split states with anti-parallel rather than parallel spins. Similar results for Pb/Cu(111) reveal that the proposed hybridization scenario is independent on the supporting substrate.New Journal of Physics 06/2013; · 4.06 Impact Factor

Page 1

Unconventional Fermi surface spin textures in the BixPb1−x/Ag(111) surface alloy

Fabian Meier1,2, Vladimir Petrov3, Sebastian Guerrero4, Christopher

Mudry4, Luc Patthey2, J¨ urg Osterwalder1, and J. Hugo Dil1,2

1Physik-Institut, Universit¨ at Z¨ urich, Winterthurerstrasse 190, CH-8057 Z¨ urich, Switzerland

2Swiss Light Source, Paul Scherrer Institut, CH-5232 Villigen, Switzerland

3St. Petersburg Polytechnical University, 29 Polytechnicheskaya St, 195251 St Petersburg, Russia

4Condensed matter theory group, Paul Scherrer Institut, CH-5232 Villigen, Switzerland

(Dated: March 3, 2009)

The Fermi and Rashba energies of surface states in the BixPb1−x/Ag(111) alloy can be tuned si-

multaneously by changing the composition parameter x. We report on unconventional Fermi surface

spin textures observed by spin and angle-resolved photoemission spectroscopy that are correlated

with a topological transition of the Fermi surface occurring at x = 0.5. We show that the surface

states remain fully spin polarized upon alloying and that the spin polarization vectors are approx-

imately tangential to the constant energy contours. We discuss the implications of the topological

transition for the transport of spin.

PACS numbers: 73.20.At, 71.70.Ej, 79.60.-i

Controlling the spin degree of freedom of the electron

lies at the heart of spintronics [1]. One possibility to

manipulate the electron spin without the need of any

external magnetic field is found in the Rashba-Bychkov

(RB) effect [2]. It appears in (quasi) two-dimensional

electron or hole systems with a lack of inversion sym-

metry and plays a prominent role for a proposed spin

field-effect transistor [3]. For most systems, the RB ef-

fect is small. Therefore, many of the related intriguing

effects, such as a renormalization of the Fermi liquid pa-

rameters [4], changes in the electron-phonon coupling [5],

enhanced superconductivity transition temperatures [6],

and real space spin accumulation [7, 8, 9] remain for the

most part experimentally unobservable.

Recently, it has been shown that the RB effect is dra-

matically enhanced in the Bi/Ag(111)(√3×√3)R30◦and

Pb/Ag(111)(√3×√3)R30◦surface alloys due to an addi-

tional in-plane inversion asymmetry [10, 11, 12, 13, 14].

Furthermore, the band structure can be continuously

tuned between these two systems by substituting Bi

with Pb [15], as schematically illustrated in Fig. 1(a).

The large RB effect combined with the tunability of the

Fermi and the RB energies make BixPb1−x/Ag(111)(√3×

√3)R30◦, henceforth BixPb1−x/Ag(111), an ideal model

RB system to study the geometrical and the topolog-

ical changes in the Fermi surface of its surface states

[5, 16]. It is clear that the large conductivity of the Ag

substrate short-circuits possible spin currents at the sur-

face of BixPb1−x/Ag(111), but RB semiconductors [17] or

thin metallic films [18] might be found that are equally

tunable and suited for technological applications.

We present in this work spin and angle resolved photoe-

mission spectroscopy (SARPES) data on surface states of

BixPb1−x/Ag(111) to resolve the changes in their Fermi

surface spin textures (FSST) as a function of composi-

tion x. We will argue that the spin transport is strongly

affected by a topological transition of the Fermi surface

taking place at the critical value xc= 0.5.

The RB effect occurs at interfaces or surfaces whenever

kx

E

ky

Γ

E0

k0

S

EΓ

E

kx

(b)(a)

x=1

x=0.6

x=0

K1

K2

(c)

EΓ

DOS (arb. units)

E (arb. units)

(d)

EΓ > EF

E0 > EF > EΓ

o

i

oioi

i

o

FIG. 1: (color online) (a) Qualitative plot of the surface state

band structure of Bi/Ag(111) (x = 1) along the direction¯Γ¯ K

in momentum space (adapted from Ref. [13]) showing the two

Kramer’s pairs K1 and K2.

level (dashed lines) lowers continuously and the spin splitting

becomes smaller (not shown). (b) Schematic picture of the

Rashba effect for a two-dimensional hole gas around¯Γ and

illustration of the relevant parameters. The yellow (light gray)

arrows are the spin expectation values of the eigenspinors.

(c) Density of states for the outer (o) and inner (i) constant

energy contours. (d) Hole Fermi seas (gray regions) and Fermi

surfaces (thick lines) when E¯Γ> EFand E0> EF> E¯Γ.

As x is decreased, the Fermi

the absence of the space inversion symmetry lifts the spin

degeneracy due to the spin-orbit coupling. The simplest

example of a RB Hamiltonian is given by [19],

H = H0+ HRB,

(1a)

where the kinetic energy of the two-dimensional hole gas

with negative effective mass m∗is

?

H0= σ0

E¯Γ−

?2

2m∗∇2

?

,

(1b)

arXiv:0903.0233v1 [cond-mat.mtrl-sci] 2 Mar 2009

Page 2

2

while the RB term is

HRB= −αRB

?

iσy

∂

∂x− iσx

∂

∂y

?

.

(1c)

The positive coupling constant αRBreflects the RB cou-

pling. The unit 2 × 2 matrix is denoted by σ0, while σx

and σyare the standard Pauli matrices in the basis in

which the quantization axis is along the z direction. The

eigenenergies of H yield the upper (+) and lower (−) RB

branches

E±(k) = E¯Γ+?2|k|2

2m∗± αRB|k|.

(2a)

with the corresponding eigenspinors

?k,±| =?ei(ϕ±π/2), 1?/√2,

where ϕ = arctan(ky/kx) and the two-dimensional mo-

mentum k is measured relative to the¯Γ point. Although

HRBbreaks the spin-rotation symmetry of H0, it pre-

serves time-reversal symmetry. The mechanism of the

enhanced spin splitting in the BixPb1−x/Ag(111) surface

alloy goes beyond this simple model. Nevertheless, many

of the fundamental properties of this system are captured

by the simple nearly free electron RB (NFERB) model

described above .

We plot in Fig. 1(b) the dispersion of the NFERB

model. The dispersion along any cut passing through

the¯Γ point can be assigned two distinct colors that dis-

tinguish the anti-parallel alignments of the spin expecta-

tion values for the eigenspinors. This gives two spin-split

bands colored in blue and red in Fig. 1(b). They are offset

by two opposite wave vectors of magnitude k0when mea-

sured from¯Γ. The Rashba energy ER = ?2k2

characterizes the strength of the RB effect. The spin

polarization vectors S±(k), defined as the spin expec-

tation values of the eigenspinors |k,±?, are parallel to

the basal plane of Fig. 1 and are orthogonal to k, as

depicted by the yellow arrows in Fig. 1 (b). Below E¯Γ,

the spin polarization rotates counterclockwise along the

outer constant energy contour and clockwise for the inner

contour. Above E¯Γ, the spin polarization rotates coun-

terclockwise along both contours. The experimentally

determined spin polarization vectors will be denoted by

P = (Px,Py,Pz) and will be shown to obey this simple

rule.

The density of states (DOS) ν(EF) of the NFERB

Hamiltonian is also sensitive to the change in the ge-

ometry and topology of the Fermi surface upon tuning of

the Fermi energy EF. The DOS νo,i(EF) of the outer (o)

and the inner (i) Fermi contour shown in Fig. 1(c) are

given by

(2b)

0/(2|m∗|)

νo,i(EF) = Θ(E0− EF)ν2D

?????1 ±

?

E0− E¯Γ

E0− EF

?????,

(3)

whereby Θ is the Heaviside function and ν2D

|m∗|/(2π?2). The + refers to the outer Fermi contour,

=

the − to the inner one. The sum νo(EF) + νi(EF) re-

duces to the constant DOS 2ν2Dof a spin degenerate

two-dimensional hole gas with parabolic dispersion when

E¯Γ> EF, has a singular derivative when EF= E¯Γ,

while it displays the one-dimensional Van Hove singu-

larity ν(EF) ∼ (E0− EF)−1/2in the limit EF→ E0.

The BixPb1−x/Ag(111) sample preparation was car-

ried out in situ under ultrahigh vacuum conditions with

a base pressure better than 2×10−10mbar. The Ag(111)

crystal was cleaned by multiple cycles of Ar+sputtering

and annealing. Bi and Pb were simultaneously deposited

from a calibrated evaporator at a pressure below 4×10−10

mbar, with the total amount corresponding to 1/3 of a

mono-layer. The sample quality was affirmed by low-

energy electron diffraction, which showed sharp (√3 ×

√3)R30◦spots and no further superstructure, and angle-

resolved photoemission spectroscopy (ARPES), which

showed a continuous tuning of the band structure and

no superposition of the Bi/Ag(111) and the Pb/Ag(111)

band structures. These are both strong indications that,

although the surface is well ordered, Bi and Pb are ran-

domly substituted.

The experiments were performed at room temperature

at the Surface and Interface Spectroscopy beamline at the

Swiss Light Source of the Paul Scherrer Institute using

the COPHEE spectrometer [20]. The data were obtained

using horizontally polarized light with a photon energy

of 24 eV. Because of the inherently low efficiency of Mott

detectors the energy and angular resolution were sacri-

ficed in the spin-resolved measurements up to 80 meV

and 1.5 degree, respectively. The coordinate system for

the measurements is such that a momentum distribution

curve (MDC) is taken along the kx-axis. This means that

a spin polarization vector P is expected to point in the

±y-direction if the NFERB model holds qualitatively.

A detailed description of the band structure of

Bi/Ag(111) and Pb/Ag(111) can be found in Refs. [10,

11, 12, 13]. There are two Kramer’s pairs K1 and K2

of bands that are qualitatively drawn in Fig. 1(a). The

inner one (K1) is mostly of spzsymmetry. The outer

one (K2) is mostly of px,ysymmetry. For Pb/Ag(111)

(x = 0), both K1 and K2 are only partially occupied. For

Bi/Ag(111) (x = 1), K1 is fully occupied, while K2 is only

partially occupied. Irrespective of x = 0 or x = 1, the

spin polarization vectors for K1 are nearly parallel to the

surface plane and are approximately perpendicular to the

momenta, in agreement with the NFERB model. In con-

trast, the spin polarization vectors for K2 feature signif-

icant out-of-plane components depending on the crystal-

lographic direction. This is a consequence of the stronger

coupling to in-plane potential gradients [10, 12]. We will

only consider K1 from now on.

Fig. 2 shows the experimental band structure of the

BixPb1−x/Ag(111) surface alloys along¯Γ¯K for x =

(0.5),(0.6) and (1) measured with (spin integrated)

ARPES. Second derivative data are also shown to en-

hance the contrast. The band K1 is fully occupied for

x = 1 and, as x is decreased, the Fermi level shifts down

Page 3

3

0.0

0.2

0.4

0.6

0.8

1.0

-0.4 -0.2 0.0

kx (Å-1)

0.2

Binding Energy (eV)

-0.4-0.2 0.0

kx (Å-1)

0.2

-0.4-0.2 0.0

kx (Å-1)

0.2

0.0

0.2

0.4

0.6

0.8

1.0

x=1x=0.6x=0.5

FIG. 2: (color online) Upper graphs: Experimental band

structure of BixPb1−x/Ag(111) for x = (0.5),(0.6) and (1)

(from left to right) along the¯Γ¯ K direction, where dark cor-

responds to a higher photoemission intensity. Lower graphs:

Second derivative data to enhance the contrast.

with respect to the bands so that K1 gets depopulated,

and the spin-splitting decreases. For x = 0.6, the Fermi

level EFlies between the band apex and the crossing

point (E0> EF> E¯Γ). Unconventional FSST are then

expected according to the NFERB model. At x = 0.5,

the Fermi level lies approximately at the crossing point of

K1, where the DOS of the inner Fermi contour vanishes

and a topological transition of the Fermi surface occurs

according to the NFERB model. Note that our calibra-

tion of x is slightly different from that given in Ref. 15.

However, this does not affect the conclusions of this work.

Polarization

-0.3-0.2 -0.1

kx(Å-1)

0.0 0.1 0.2

0.6

0.4

0.2

0.0

-0.2

-0.4

-0.6

Px

Py

Pz

Itot

Iy,up

Iy,dn

Intensity (arb. units)

(b)

(d)

kx

ky

ky

kx

(a)

(c)

-0.2-0.1 0.00.10.2

kx(Å-1)

Px

Py

Pz

0.4

0.2

0.0

-0.2

-0.4

Polarization

Intensity (arb. units)

Itot

Iy,up

Iy,dn

x=0.5x=0.6

Γ

Γ

FIG. 3:

BixPb1−x/Ag(111) for x = 0.5 (left) and x = 0.6 (right).

(a) and (b) Total spin integrated intensity (circles) and spin-

resolved intensity curves projected on the y-axis of a MDC

along¯Γ¯ K. (c) and (d) are the corresponding measured (sym-

bols) and fitted (solid lines) spin polarization data. (Insets)

Schematically drawn FSST. For x = 0.6, both bands of K1

crossing EFbetween¯Γ and the SBZ boundary have parallel

spin polarization vectors, while for x = 0.5, the spin polariza-

tion vectors are anti-parallel.

(color online) Spin resolved ARPES data of

We show in Fig. 3 the experimental spin-resolved

MDCs for x = 0.5 (left column) and x = 0.6 (right

column) providing us with the FSST for E¯Γ> EFand

E0> EF> E¯Γ, respectively. The extraction of the spin

polarization vectors P was done by applying a two-step

fitting routine [12] on the data of Fig. 3. For both com-

positions, we find that the surface states K1 remain fully

spin polarized with spin polarization vectors similar to

those of the surface states of Bi/Ag(111) or Pb/Ag(111)

found in [12]. The spin polarization vectors lie mainly in

the surface plane perpendicular to k and both the out-of-

plane and radial spin polarization components are com-

paratively small. This finding is corroborated by several

similar measurements in different crystallographic direc-

tions and at different binding energies. We thus conclude

that the spin polarization of the surface states K1 is ro-

bust against the mixing of Bi and Pb.

For x = 0.5, the measurement is performed slightly

below the crossing point of K1. We observe the conven-

tional situation, i.e., a straight cut from¯Γ to the sur-

face Brillouin zone (SBZ) boundary crosses two bands

with opposite spin polarization vectors. This can be seen

in the spin-resolved spectra of Fig. 3(a), which are ob-

tained from the fits of the corresponding spin polarization

data shown in Fig. 3(c). The spin polarization vectors of

the bands are opposite for all adjacent bands. The cor-

responding qualitative FSST are drawn in the inset of

Fig. 3(c).

For x = 0.6, an unconventional FSST is observed. Fit-

ting the spin polarization data of Fig. 3(d) clearly shows

that, for positive and negative kx, both bands crossing

the Fermi energy have nearly parallel spin polarization

vectors. The corresponding spin-resolved spectra are dis-

played in Fig. 3(b). Due to strong transition matrix el-

ement effects, the inner band on the left side of normal

emission is only visible as a weak shoulder of the Iy,dn

curve. When E0> EF> E¯Γ, the FSST match qualita-

tively those shown in the inset of Fig. 3(d). A cut from

¯Γ to the SBZ boundary crosses two bands with parallel

spin polarization vectors.

We have thus established that varying x between 0.5

and 0.6 induces a topological transition in the shape of

the Fermi surface of K1 surface states with an impact on

their spin texture and on their DOS that is qualitatively

captured by the NFERB model. Intuitively, one could

expect a spin filtering effect due to unconventional FSST,

since states with parallel k-vectors posses identical spin

polarization vectors. However, it is the group velocity

which determines electronic transport and this remains

the same for anti-parallel spin directions. We will now

argue that the transport of spins across an ideal one-

dimensional boundary separating a spin-degenerate two-

dimensional electron gas from a RB hole gas is sensitive

to this topological transition.

In principle, a two-dimensional scattering geometry, as

depicted in Fig. 4(a), could be realized by the deposi-

tion of BixPb1−xon Ag(111) through a shadow mask.

We denote with x and y the coordinates of the two-

Page 4

4

kx

-ky

y

x

Ag(111)

surface state

(Rashba)

metal

-ky

kx

(a)(b)

0.8

0.4

0.0

-0.050.05

(c)(d)

0.0

kx (Å-1)

-0.050.05 0.0

Polarization

Polarization

EF - EΓ (eV)

EF - EΓ (eV)

PAg x

PAg x0.8

0.4

0.0

0.05

0.0

-0.05

EF - EΓ (eV)

0.0-0.1 0.1

FIG. 4: (color online) (a) Incoming and outgoing surface plane

waves from the interface at x = 0 between a spin isotropic

(x < 0) and a RB (x > 0) metal. (b) Dispersions of the

Ag(111) and RB surface states. (c) PAg←x as a function of

EF− E¯Γas defined in the text. (d) PAg→x as a function of

EF− E¯Γas defined in the text.

dimensional Ag(111) surface.

tron gas with an effective mass corresponding to that

of Ag(111) surface states meets the states from the K1

band of BixPb1−x/Ag(111) at the ideal one-dimensional

boundary x = 0. We imagine driving a small charge cur-

rent through the boundary by applying an infinitesimal

voltage difference across the interface. The polarity of

this applied voltage defines whether the charge current

is from the left to the right, i.e., from the Ag(111) to

the RB side, or from the right to the left, i.e., from the

RB to the Ag(111) side. In the Drude limit, the charge

current can be calculated from the reflection coefficients

Rσfor an incoming surface state of energy EFwith spin

quantum number σ along some quantization axis, which

is here chosen to be the y-axis.

We denote the spin current by PAg→xwhen the current

is from the Ag(111) to the RB side or by PAg←xother-

wise. To quantify the transport of spin across the bound-

A spin-degenerate elec-

ary, we divide, on the Ag(111) side, the spin current nor-

mal to the boundary (the difference between the spin up

and spin down current) by the particle current normal to

the boundary, i.e. PAg↔x= (jup− jdn)/jtot. We use the

parameters m∗

Ag(111) side [21] and m∗

E¯Γ,x= 0, on the RB side [15]. The Ag(111) and RB

dispersions are shown in Fig. 4(b).

We plot in Fig. 4(c) PAg←x, and in Fig. 4(d) PAg→xfor

different band fillings as described by the value of EF(see

Ref [22] for computational details). In the absence of RB

coupling, the spin current across the boundary vanishes.

The breaking of the spin-rotation symmetry by the RB

coupling induces a spin current on the Ag(111) side in

Figs. 4(c) and 4(d). This induced spin current is strongly

enhanced by the onset of an unconventional FSST when

the Fermi level triggers a topological transition of the

RB Fermi surface and νivanishes. Thus, the RB metal

acts as a spin injector or a spin acceptor depending on

the polarity of the applied voltage difference across the

boundary. Finally, even for non-ideal systems, such spin

currents might lead to local spin accumulation that could

be detected with magnetic STM.

To conclude, we have shown that substitutional alloy-

ing does not alter the spin polarization vectors of the

mixed BixPb1−x/Ag(111) surface alloys. Furthermore,

unconventional Fermi surface spin textures were realized

through an adequate choice of the composition and were

measured. Systems with strong RB type spin-orbit split-

ting and EF≈ E¯Γare suggested to function as a spin fil-

ter. One could also envisage using materials with similar

properties as spin injectors for a “classical” RB system.

This could reduce the problems encountered at interfaces

to ferromagnets.

Ag/me= 0.397, E¯Γ,Ag= −63meV on the

x/me= −0.25, E0,x= 94meV ,

Acknowledgments

Fruitful discussions with M. Grioni and G. Bihlmayer

are gratefully acknowledged. We thank C. Hess, F. Dubi,

and M. Kl¨ ockner for technical support. The measure-

ments have been performed at the Swiss Light Source,

Paul Scherrer Institut, Villigen, Switzerland. This work

is supported by the Swiss National Foundation.

[1] S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J.

M. Daughton, S. vom Moln´ ar, M. L. Roukes, A. Y.

Chtchelkanova, and D. M. Treger,

(2001).

[2] Y.A. Bychkov, and E.I. Rashba,

(1984).

[3] S. Datta, and B. Das, Appl. Phys. Lett. 56, 665 (1990).

[4] D. S. Saraga, and D. Loss, Phys. Rev. B 72, 195319

(2005).

[5] E. Cappelluti, C. Grimaldi, and F. Marsiglio, Phys. Rev.

Science 294, 1488

JETP Lett. 39, 78

B 76, 085334 (2007).

[6] E. Cappelluti, C. Grimaldi, and F. Marsiglio, Phys. Rev.

Lett. 98, 167002 (2007).

[7] J. Sinova, D. Culcer, Q. Niu, N. A. Sinitsyn, T. Jung-

wirth, and A. H. MacDonald Phys. Rev. Lett. 92, 126603

(2004).

[8] J. Wunderlich, B. Kaestner, J. Sinova, and T. Jungwirth,

Phys. Rev. Lett. 94, 047204 (2005).

[9] M. Liu, G. Bihlmayer, S. Bl¨ ugel, and C. Chang, Phys.

Rev. B 76, 121301(R) (2007).

Page 5

5

[10] C. R. Ast, J. Henk, A. Ernst, L. Moreschini, M. C. Falub,

D. Pacil´ e, P. Bruno, K. Kern, and M. Grioni, Phys. Rev.

Lett. 98, 186807 (2007).

[11] D. Pacil´ e, C. R. Ast, M. Papagno, C. Da Silva, L. Mores-

chini, M. Falub, A. P. Seitsonen, and M. Grioni, Phys.

Rev. B 73, 245429 (2006).

[12] F. Meier, H. Dil, J. Lobo-Checa, L. Patthey and J. Os-

terwalder, Phys. Rev. B 77, 165431 (2008).

[13] G. Bihlmayer, S. Bluegel, and E.V. Chulkov, Phys. Rev.

B 75, 195414 (2007).

[14] J. Premper, M. Trautmann, J. Henk, and P. Bruno, Phys.

Rev. B 76, 073310 (2007).

[15] C. R. Ast, D. Pacil´ e, L. Moreschini, M. C. Falub, M.

Papagno, K. Kern, and M. Grioni, Phys. Rev. B 77,

081407(R) (2008).

[16] C. R. Ast, G. Wittich, P. Wahl, R. Vogelsang, D. Pacil´ e,

M. C. Falub, L. Moreschini, M. Papagno, M. Grioni, and

K. Kern, Phys. Rev. B 75, 201401(R) (2007).

[17] J. Nitta, T. Akazaki, H. Takayanagi, and T. Enoki, Phys.

Rev. Lett. 78, 1335 (1997).

[18] J. H. Dil, F. Meier, J. Lobo-Checa, L. Patthey, G.

Bihlmayer and J. Osterwalder Phys. Rev. Lett.

266802 (2008).

[19] R. Winkler,Spin-Orbit Coupling Effects in Two-

Dimensional Electron and Hole Systems, Springer Tracts

in Modern Physics Vol. 191 (Springer, Berlin, 2003).

[20] M. Hoesch, T. Greber, V.N. Petrov, M. Muntwiler, M.

Hengsberger, W. Auwaerter, and J. Osterwalder, J. Elec-

tron Spectrosc. Relat. Phenom. 124, 263 (2002).

[21] F. Reinert, G. Nicolay, S. Schmidt, D. Ehm, and S.

H¨ ufner Phys. Rev. B 63, 115415 (2001).

[22] B. Srinsongmuang, M. Berciu, and P. Pairor, Phys. Rev.

B 78, 155317 (2008).

101,

#### View other sources

#### Hide other sources

- Available from J. Hugo Dil · May 29, 2014
- Available from ArXiv