This content has been downloaded from IOPscience. Please scroll down to see the full text.

Download details:

IP Address: 87.246.247.135

This content was downloaded on 29/01/2017 at 08:20

Please note that terms and conditions apply.

DOS cones along atomic chains

View the table of contents for this issue, or go to the journal homepage for more

2017 J. Phys.: Condens. Matter 29 095304

(http://iopscience.iop.org/0953-8984/29/9/095304)

Home Search Collections Journals About Contact us My IOPscience

You may also be interested in:

Electronic properties of a quantum wire interacting with a surface: the role of periodically placed

impurities

T Kwapiski

Charge fluctuations in a perfect and disturbed quantum wire

T Kwapiski

Phase-dependent electron transport through a quantum wire on a surface

T Kwapiski

Spin and charge pumping in a quantum wire: the role of spin-flip scattering and Zeeman splitting

T Kwapiski and R Taranko

Correlation between morphology, electron band structure, and resistivity of Pb atomic chains on the

Si(5 5 3)-Au surface

M Jaochowski, T Kwapiski, P ukasik et al.

Conductance oscillations of a quantum wire disturbed by an adatom

T Kwapiski

Local currents in a 2D topological insulator

Xiaoqian Dang, J D Burton and Evgeny Y Tsymbal

Conductance oscillations and charge waves in zigzag shaped quantum wires

T Kwapiski

The KKR-Green's function method

H Ebert, D Ködderitzsch and J Minár

http://iopscience.iop.org/page/termshttp://iopscience.iop.org/0953-8984/29/9http://iopscience.iop.org/0953-8984http://iopscience.iop.org/http://iopscience.iop.org/searchhttp://iopscience.iop.org/collectionshttp://iopscience.iop.org/journalshttp://iopscience.iop.org/page/aboutioppublishinghttp://iopscience.iop.org/contacthttp://iopscience.iop.org/myiopsciencehttp://iopscience.iop.org/article/10.1088/0953-8984/25/41/415304http://iopscience.iop.org/article/10.1088/0953-8984/25/41/415304http://iopscience.iop.org/article/10.1088/0953-8984/18/31/024http://iopscience.iop.org/article/10.1088/0953-8984/24/5/055302http://iopscience.iop.org/article/10.1088/0953-8984/23/40/405301http://iopscience.iop.org/article/10.1088/0953-8984/28/28/284003http://iopscience.iop.org/article/10.1088/0953-8984/28/28/284003http://iopscience.iop.org/article/10.1088/0953-8984/19/17/176218http://iopscience.iop.org/article/10.1088/0953-8984/27/50/505301http://iopscience.iop.org/article/10.1088/0953-8984/22/29/295303http://iopscience.iop.org/article/10.1088/0034-4885/74/9/096501

1 © 2017 IOP Publishing Ltd Printed in the UK

1. Introduction

Atomic chains are the ultimately small conductors and may be used to connect various nanodevices such as nanotransis-tors, qubits or quantum gates. Their electronic properties are of crucial interest and many interesting effects were found in these systems like conductance oscillations [1, 2], conduc-tance quantization [3, 4], spin-charge separation [5], photon-assisted tunneling [6–8], electron and spin pumping [9, 10], charge density waves [11, 12], spin–orbit effects [13, 14] or Majorana states [15, 16]. These properties strongly differ from those of bulk conductors.

Freely suspended chains of metallic atoms can be fabricated using the mechanically controlled break junctions [1–5], how-ever, such chains are unstable and can easily brake. Very long and stable atomic chains were investigated on various vicinal (stepped) surfaces e.g. single, double or triple atomic chains on Si(335), Si(557) or Si(5512) [17–20]. Self-assembling chain of atoms can also grow on other reconstructed surfaces like Ge(0 0 1) [21]. The electronic properties and the atomic

geometry of these chains can by easily investigated with the scanning tunneling microscope (STM) technique [18–22]. This technique concentrate on the perpendicular transport—from the STM tip through the central atomic system to the surface. For a tip with the atomic resolution the transport properties are mainly determined by the local density of states (DOS) related to the atom localized exactly below the tip. In general, the local DOS in STM experiments is determined by normalizing the differential conductance ( )/I V Vd d with I(V)/V [23–25]. Note, that the spectral function related to the scanned atom contains also information about the rest atoms coupled with it which is also reflected in the electron transport properties [26].

The lack of the translation symmetry in quasi-two- dimensional systems can lead to the existence of special states localized perpendicularly to the surface—so called surface states. In analogy to these states, at both ends of a chain (1D structure) so called zero-dimensional end states can be observed, [25, 27]. Such states are characterized by the wave function localized at the end atoms which should decay

Journal of Physics: Condensed Matter

DOS cones along atomic chains

Tomasz Kwapiński

Institute of Physics, M. Curie-Skłodowska University, Pl. M. Curie-Skłodowskiej 1, PL-20-031 Lublin, Poland

E-mail: tomasz.kwapinski@umcs.pl

Received 13 October 2016, revised 18 December 2016Accepted for publication 21 December 2016Published 27 January 2017

AbstractThe electron transport properties of a linear atomic chain are studied theoretically within the tight-binding Hamiltonian and the Green’s function method. Variations of the local density of states (DOS) along the chain are investigated. They are crucial in scanning tunnelling experiments and give important insight into the electron transport mechanism and charge distribution inside chains. It is found that depending on the chain parity the local DOS at the Fermi level can form cone-like structures (DOS cones) along the chain. The general condition for the local DOS oscillations is obtained and the linear behaviour of the local density function is confirmed analytically. DOS cones are characterized by a linear decay towards the chain which is in contrast to the propagation properties of charge density waves, end states and Friedel oscillations in one-dimensional systems. We find that DOS cones can appear due to non-resonant electron transport, the spin–orbit scattering or for chains fabricated on a substrate with localized electrons. It is also shown that for imperfect chains (e.g. with a reduced coupling strength between two neighboring sites) a diamond-like structure of the local DOS along the chain appears.

Keywords: atomic chain, charge waves, density of states, electron localization, spin–orbit coupling

(Some figures may appear in colour only in the online journal)

Tomasz Kwapiński

DOS cones along atomic chains

Printed in the UK

095304

JCOMEL

© 2017 IOP Publishing Ltd

29

J. Phys.: Condens. Matter

CM

10.1088/1361-648X/aa5540

Paper

9

Journal of Physics: Condensed Matter

IOP

2017

1361-648X

1361-648X/17/095304+10$33.00

doi:10.1088/1361-648X/aa5540J. Phys.: Condens. Matter 29 (2017) 095304 (10pp)

mailto:tomasz.kwapinski@umcs.plhttp://crossmark.crossref.org/dialog/?doi=10.1088/1361-648X/aa5540&domain=pdf&date_stamp=2017-01-27publisher-iddoihttps://doi.org/10.1088/1361-648X/aa5540

T Kwapiński

2

exponentially into the chain. In many STM experiments the wave ampl itude is maximized at the chain ends and decays away towards the chain center. According to [12] this decay increases as the chain length (N) extends and for a given N it decreases exponentially from both end sites into the bulk of the chain. The charge wave along the chain is related with the Fermi wave vector, kF [28, 29], which depends on the chain electron energies. However, for shorter chains the wave period is comparable to the chain length and the results strongly depend on the ratio of theses two length parameters, which was confirmed experimentally [12]. Note, that in the presence of impurities or empty sites in the chain local changes in the carrier concentration take place and they lead to the charge waves known as the Friedel oscillations [29, 30]. The ampl-itude of these oscillations decreases with the distance from the impurity, r, according to ( )/ δk r rcos 2 F , where δ is the spatial decay exponent. Also the Majorana quasiparticles can appear at the chain edges e.g. in the presence of strong spin–orbit couplings and a superconducting substrate [15, 16, 32].

In this paper we focus on the transport properties of an atomic chain on different substrates with localized and delocalized electrons. The substrate model considered here includes the spatial separation of atoms with two important parameters: the lattice constant and the Fermi wavelength. It allows us to consider two limits (i) each atom couples individually to a substrate with localized electrons which corresponds to an insulating substrate, and (ii) the substrate represents a reservoir of delocalized electrons (conducting substrate) [33, 34]. In particular, we concentrate on electron states behaviour along the chain and find the condition for linear-dependence of the local DOS at the Fermi level—so called DOS cones which are characterized by a linear decay towards the chain with more or less constant (length-independent) slope coefficient. Since the linear behaviour of DOS in one-dimensional structures is a novel effect, in this paper we investigate the origin and modifications of this effect due to different external perturbations. These studies are important in the context of end states, charge density waves (CDW) or Friedel oscillations in 1D systems which do not predict a linear decreasing of charge waves. Moreover, as was shown in [22], the chain end states can move deeper into a chain in the presence of bias voltage, which leads to modu-lation of the effective chain length. For relatively long chains non-vanishing charge waves (with periodic spatial modula-tion) can be found—this effect is related to the conductance oscillations with different periods [35]. It is interesting if DOS cones can appear also in the presence of non-vanishing waves (CDW) along the chain. We study this effect in the paper and instead of a single cone we have found multi-cone structures of the spatial DOS. Additionally, we consider more realistic system i.e. a dis ordered chain on a substrate with a reduced coupling strength between two neighboring sites, and analyze the local DOS cones modifications in this case. To our best knowledge, so far, the cone-like, double-cone or diamond-like structures of the spatial DOS have not been detailed in the literature. Note, that in the presence of DOS cones one can determine the system param eters (like

the site-site couplings or on-site energies) and the conduct-ance through the system, which are important from the experimental point of view.

Particularly interesting are the transport properties of low-dimensional conductors in the presence of the spin–orbit coupling [13, 14, 36–38]. The electron spin and orbital degrees of freedom are coupled which is essentially a rela-tivistic effect. This effect is important for electrons in the vicinity of the nuclei and is often applied in a sphere around the atom. In consequence an electron can change its spin at a given site (on-site spin-flip processes) which modifies the spin-polarized current flowing in the system. In low-dimen-sional systems electrons are often confined by not symmetric potential well with nonzero gradient, which can produce the precession of the electron spin due to spin–orbit couplings. It leads to the Rashba term in the Hamiltonian which, in the tight-binding representation, describes the hopping between neighboring sites paired with a spin flip (inter-site spin-flip scattering). Thus the spin–orbit couplings can be effectively modelled by spin-flip hopping terms in a usual tight-binding approach [16, 38–40]. The Rashba spin–orbit parameter depends on the product of the nuclear number of the atom and a param eter describing the asymmetry of the electron wavefunction (related to the potential gradient) [40, 41]. The tuning of the Rashba coupling parameter is possible by an external gate voltage and can lead to oscillatory behaviour of the ballistic spin conductance [39, 42]. Moreover, in the presence of spin–orbit couplings the conductance oscillates as a function of the chain length [39]. This effect should be also reflected in the local DOS at the Fermi level. Thus in this paper we investigate the role of the spin–orbit (Rashba) terms on the structure of the local DOS along the chain. It is expected that the intrinsic or extrinsic spin-flip processes can induce DOS cones which can be confirmed in STM experiments.

We describe the atomic chain by a tight-binding Hamiltonian and obtain the spectral density function within the Green’s function approach [26, 31, 43]. The paper is organized as follows. In section 2 we present briefly the model and theoretical description of the local DOS along the atomic chain. In section 3 we present and discuss the main results of the paper: formation of DOS cones in the resonant and non-resonant cases (sections 3.1 and 3.2), DOS cones due to the spin–orbit scattering (section 3.3), DOS cones in disordered chains on a substrate (section 3.4). We conclude with a short summary.

2. Model and theoretical description

The system under consideration consists of a linear atomic chain of N sites connected to two metallic electrodes (L and R) and coupled with a substrate (surface electrode), which is schematically shown in figure 1. The model Hamiltonian for this system can be written in the following form: = + +H H H H0 int SO (in the standard second-quantized nota-

tion), where:

J. Phys.: Condens. Matter 29 (2017) 095304

T Kwapiński

3

→ → →→∑ ∑ε ε= +

σασα σα σα

σσ σ σ

=

+

=

+H a a a ak L R S

k k ki

N

i i i0, , , 1

(1)

describes the electrons in the leads and in the chain with energies →ε σαk and εσi, respectively. The operators ( )σ σ

+a ai i annihilate and

create an electron with spin σ at site i = 1, ..., N and ( )→ →σα σα+a ak k

are the according electrode operators. Electron transitions between chain sites (inside the chain) or between the chain and the leads are established by the interchanged Hamiltonian

∑ ∑

∑

= +

+ +

σσ σ σ

σσ σ σ

σσ σ

+ +

=

−

++

+

H V a a V a a

V a a h.c.

k L Rk L R k L R N

i k Sik S k S i

i

N

i i i i

int 1,

, 1

1

, 1 , 1

( )( ) ( ) ( )

→ → → →→ →

(2)

where the neighboring sites are coupled via the tunnel matrix elements Vi,i+1 and the first (and the last) site of the chain is coupled to the left (right) lead via ( )→σVk L R element. We assume that the spectral density of the wire-lead cou-pling, ( )/ / /π δ ε εΓ = ∑ | | −σ σ σV2L R k k L R k L R2 , is energy and spin independent within a wide-band approximation. The chain-substrate coupling for equal distances a0 between neighboring sites of the chain can be written in the form

( )→ → σ=σ σV V ik jaexpjk S k S 0 , [33, 34]. Thus the relevant spectral density depends on the substrate Fermi wavelength (only sub-strate electrons close to the Fermi surface play a role) and the spatial separation of the atomic sites and reads:

Γ = Γ| − || − |

k a i j

k a i j

sinijS

SF 0

F 0 (3)

where ΓS stands for the effective chain-substrate coupling strength.

In the presence of local or external electromagnetic fields the spin–orbit coupling is introduced as an additional spin-dependent flux [44] which modifies the hopping matrix ele-ments in the Hamiltonian and mixes spin states at the same or different sites. Thus within the tight-binding Hamiltonian the spin–orbit effects can be modelled by spin-flip hoppings [16, 38, 39]. The effective spin–orbit Hamiltonian introduces both intrinsic (λSF) and extrinsic (Rashba-like) spin-flip scat-tering (λSO) and can be written as follows:

ˆ ( ˆ )∑ ∑ ∑λ σ λ σ= +σ σ

σ σσ σ

σ σ+

=

−+

+′

′′

′H a a a i ai x ii

N

i y iSO,

SF, 1

1

SO 1 (4)

where ˆ /σx y are the Pauli matrixes. The spin-flip effective param-eters are relatively small in comparison with the site-site cou-pling energy and can be controlled by an external applied gate voltage [42]. We assume that in our system electron-electron interactions do not play an important role and can be captured by an effective shift of the onsite energies.

In order to describe the electron transport properties of the system with the above Hamiltonian we concentrate on the local spectral density function along the chain, which is crucial in the STM geometry (perpendicular transport). In this case, for small STM voltage, the tunnelling current depends on the tip and substrate DOS, as well as the spectral density of the central system like single atoms or atomic chains. Assuming that the tip and substrate DOS are tip-position independent the net cur-rent (which reflects the STM topography) depends mainly on the local DOS related to the scanned site of the chain. In order to obtain the chain spectral density function one needs to compute the Green’s function for the total tight-binding Hamiltonian. For ith chain site with spin σ one can write the spectral density function ( ) ( )Iε ε= −σ π σGDOSi ii

r1 , where the retarded Green’s function can be found from the following equation of motion, [43]:

Figure 1. Schematic view of the system for the present study: N-site linear atomic chain between the left (L) and right (R) electrodes and coupled with the surface. The bottom panels show the local DOS as a function of energy for three short chains between L and R leads, i.e. for N = 1, N = 2 (left panel) and for N = 3 (right panel). The DOS lines with different colours correspond to the appropriate chain sites shown in the auxiliary schemes. Here the parameters are: 00ε = , V = 1, 0SΓ = , 1L RΓ = Γ = (energy unit), 0SO SFλ λ= = .

J. Phys.: Condens. Matter 29 (2017) 095304

T Kwapiński

4

( ) ⟨[ ] ⟩ ⟨⟨[ ] ⟩⟩ε ε = +β β β β β β ε++ −

+G a a a H a, , ;r1 2 1 1 1 2 (5)

where [...]−/+ means the commutator (or anticommutator), ( )εβ βG

r1 2

is the Green’s function for β1 and β2 states, and the bracket ⟨⟨ ⟩⟩... corresponds to new Green’s function obtained for the appropriate states. The state β represents all electron states (including spin degeneration) in the considered system. Fortunately, in the wide-band approx imation the leads’ states are captured via the effective / /ΓL R S parameters and the total Green’s function depends only on the chain states

( )α σ σ σ σ σ σ= − − −N N1 , 2 , ..., , 1 , 2 , ..., . Thus from the equation  of motion one obtains 2N linear dependent equa-tions  on α αG

r1 2

functions. These equations  can be written in the general matrix form: ˆ⋅ =G C Ir N2 , where I is the iden-tity matrix and the ×N N2 2 dimension matrix, Ĉ N2 , can be expressed by means of four specific submatrixes ( ×N N dimension each):

ˆ ( ) ( )ˆ ( ) ˆ

ˆ ˆ ( )

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥ε

ε

ε≡ =

σ σ σ σ

σ σ σ σ

− −

− − − ×

C GA B

B A

N r i jN

i jN

i jN

i jN

N N

2 1 , ,

, , 2 2

(6)

Here the on-diagonal matrixes can be written as follows:

ˆ ( ) ( ) ( )ε ε ε δ δ δ

δ δ δ δ

= − − +

+Γ

+Γ

+Γ

σ σ σ + + +A V

i2

i2

i2

i jN

i i j i j i j i j

Li j

Ri N N j

ijS

, , , 1 , 1 1,

,1 1, , ,

(7)

and similar for ˆ ( )εσ σ− −Ai jN

, . For the chain decoupled from the

substrate, Γ = 0S , the above matrix becomes a tri-diagonal type and its determinant can be expressed analytically by the Chebyshev polynomials of the second kind [31, 45]. The off-diagonal elements in equation (6) are always tri-diagonal matrixes related to the spin-flip and Rashba terms:

ˆ ( )λ δ λ δ δ= − − −σ σ− + +Bi jN

i j i j i j, SF , SO , 1 1, (8)

and similar for ˆ σ σ−Bi jN

, .

3. Results and discussion

In our calculations we assume the energy unit Γ = Γ = Γ = 1L R (the left and right leads couple equally strongly to the wire), the Fermi energy of unbiased leads stands for the energy ref-erence point, =E 0F , and the temperature T = 0K. Moreover, the on-site electron energies in the regime of linear transport, as well as the intra-chain hopping integrals are position and spin independent, ε ε ε= =σ σ−i i 0, Vij = V (regular wire) unless we consider a disordered chain and reduce the coupling strength between two neighboring sites. In the calcul ations the parameters have been chosen in order to satisfy the real-istic situation in many experiments, e.g. assuming Γ = 0.5 eV, the hopping integral within a chain is V = 1 = 0.5 eV, the spin–orbit parameters: λ λ= = =0.05 25SF SO meV, and k aF 0 for metallic materials is in the order of few, see [22, 29, 34, 39]. One must keep in mind that the tight-binding approach is relatively simple in comparison with e.g. DFT

methods and shows general physical trends rather than con-centrate on quantitative description of real systems.

3.1. DOS cones in the resonant case

In this section we analyze the local DOS along a chain assuming the resonant case i.e. the on-site electron energies in the chain correspond to the Fermi energy of the system, ε = =E 00 F . Here the spin–orbit couplings are neglected, λ λ= = 0SO SF and the spin index is suppressed. We investigate a linear chain placed on different substrates. The chain-substrate coupling is described by the matrix elements, equation  (3), with the effective coupling strength ΓS and the spatial term. Similarly to [34] one can consider two limiting cases i.e. �k a 1F 0 and �k a 1F 0 . In the first case the chain-substrate coupling van-

ishes unless i = j and thus δΓ = ΓijS

S ij, which corresponds to a model in which each chain atom is coupled to an individual additional electrode. Here an electron that tunnels from a par-ticular atom to the localized substrate state can re-enter only at the same chain site. The situation is different in the second limit for �k a 1F 0 , where the spectral function Γ = Γij

SS for all

i and j. Now the electrons in the chain tunnel to delocalized substrate orbitals and can re-enter at arbitrary site of the chain. The general analytical formulas for the retarded Green’s func-tion do not exist in this case because both matrixes Â

N in

equation (6) are composed of nonzero complex elements (and are not tri-diagonal matrixes).

In the beginning in figure 2(a) we study the total DOS of atomic chains composed of an even or odd number of sites fab-ricated on an insulating substrate, Γ = 0S . In the resonant case the total DOS (normalized sum of the local DOS at each chain site) is characterized by a local minimum (maximum) at EF for the even (odd) chain length, N (figure 2(a) for N = 10 and N = 11, respectively). The oscillatory behaviour of the total DOS at the Fermi level is related to the chain parity, which is responsible for the even-odd conductance oscillations effect, [1, 2, 35]. Note, however, that the total DOS of the system does not reflect perfectly the local transport properties—the STM current is determined by the local density function. A local minimum in the total DOS at the Fermi level appears due to the minima of the spectral functions at all sites in the chain (it is always true for even-length chains—see the spectral func-tions at the Fermi level for N = 2 in figure 1, left panel). This conclusion, however, is not valid for maxima of the total DOS at the Fermi energy. In that case the local DOS at each chain site can be different e.g. for a wire composed of N = 3 atoms, there is a local maximum in the spectral density at the first and at the third sites, but the second site is characterized by a local minimum at the Fermi level—this situation is shown in figure 1 in the right bottom panel. Thus the STM current flowing through the second site of this chain should vanish (due to the minimum in the local DOS) although the total DOS of this system is characterized by a local peak at the Fermi level. It is the reason that in this paper we concentrate on the local DOS characteristics.

In figures  2(b)–(d) we investigate the local DOS at the Fermi level along the chain decoupled form the surface (panel

J. Phys.: Condens. Matter 29 (2017) 095304

T Kwapiński

5

(b)) and weakly coupled with the substrate (panels (c) and (d)). For the chain composed of the even number of atoms, N = 50, and Γ = 0S the spectral density at the Fermi level is very low and position-independent, figure 2(b) (see also the local DOS for N = 2 in figure 1). However, the odd-length chain, N = 51, reveals non-vanishing DOS oscillations along the chain due to finite chain length—every second site in the chain has low local DOS value. It results from the structure of the local DOS at each site of the chain—for N = 3 the local DOS curves as a function of the energy are shown in figure 1 and two maxima (for red atoms) and one minimum (for green atom) at the Fermi level are well visible. The situation changes in the presence of chain-substrate coupling, Γ = 0.02S , figure 2(c). As one can see the oscillation amplitude of the local DOS( )EF decreases for odd N, and more important, for the even number of sites the spectral density at the Fermi level starts to oscillate along the chain with maximal amplitude at both ends. Thus a kind of cone-like structure of DOS is observed in this case. In fact, there are two coupled V-shaped structures (called here the cones) with common vortex in the middle of the chain. In figure 2(d) we analyze the local DOS cones for finite ΓS and the even-length chains (N = 26, 50, 100, 150 and 200). As a main feature it is found that the substrate can generate the cones of DOS along the chain. The oscillation amplitude of DOS always vanishes in the middle of the chain and increases with N at both ends (which is related to the zero-dimensional end states).

Note, almost linear dependence of both cone lines and constant cone apertures. For odd-length chains such DOS cones are not observed as the spectral density function still oscillates (also in the middle of these chains) with hardly vanishing ampl itude. The formation process of DOS cones with analytical calcul-ations will be analyzed in details in the next subsection.

In order to investigate further the role of the chain-

substrate coupling strength, ΓijS, on the local DOS structure

we consider the resonant case, ε = 00 , and assume localized electrons in the substrate, δΓ = Γij

Sij S, figures 3(a) and (b), or

delocalized electrons with finite values of k aF 0 parameter, figures 3(c) and (d). For the chain decoupled from the sub-strate, Γ = 0S (circles), the spectral density function does not oscillate along the chain for even N, (panel (a)). In the pres-ence of small chain-substrate coupling with localized elec-trons in the substrate the cone-like structure of DOS appears (crosses, figure 3(a), see also figures 2(c) and (d)). However, for larger ΓS these oscillations (cone aperture) decrease and the oscillation amplitude vanishes especially in the middle part of the chain (dotted line for Γ = 1.0S ). For odd number of sites in the chain, figure 3(b), the amplitude of the local DOS oscillations is maximal for Γ = 0S and decreases rapidly with ΓS. To study this effect in more details, in the right panels in figure 3 we analyze the oscillation amplitude at the chain ends as a function of the chain-substrate coupling, ΓS, for substrate with different electron localizations. The case for relatively large (small) k aF 0 corresponds to localized (delocalized) elec-trons in the substrate. As one can see for the even-length chain the oscillation amplitude is very small for ⩽k a 1F 0 (substrate with delocalized electrons), figure  3(c). On the other hand, for �k a 1F 0 (strongly localized electrons) the oscillation amplitude increases with ΓS (surface-induced DOS cones) and then slowly decreases which is in accordance with the results shown in the left panels. The situation is different for the odd-length chain (panel (d)). Here the oscillation ampl itude is

Figure 2. (a) Total density of states (TDOS) of a chain composed of N = 10 (solid curve) and N = 11 (broken curve, shifted down by 0.1 for better visualization) atomic sites. (b) DOS at the Fermi level as a function of chain sites, i, for N = 50 and N = 51 and for 0SΓ = . (c) The same as in (b) but for small chain-substrate coupling, 0.02SΓ = . (d) DOS cones (DOS at the Fermi level along a chain) for N = 26, 50, 100, 150, 200 and 0.02SΓ = . The cones for N < 200 are shifted for better presentation and the lines serve as a guide to the eye. The other parameters are: V = 1, 00ε = ,

1L RΓ = Γ = (energy unit), 0SO SFλ λ= = .

Figure 3. (a) and (b) Spatial DOS at the Fermi level along a chain for N = 50 and N = 51, respectively, and for different chain-substrate coupling, 0.0SΓ = (circles), 0.1SΓ = (crosses) and

1.0SΓ = (dotted lines), k a 10F 0 = . (c) and (d) Amplitude of DOS oscillations at chain ends as a function of SΓ for N = 50 and N = 51, respectively, and for k a 1.0, 1.5F 0 = and 5.0. The other parameters are the same as in figure 2.

J. Phys.: Condens. Matter 29 (2017) 095304

T Kwapiński

6

finite for the chain decoupled from the substrate and decreases monotonically with ΓS for all considered electron localiza-tions. Note that for larger k aF 0 ( >k a 5F 0 ), the oscillation amplitude decreases more rapidly. These results show that the end states (related to the DOS oscillations at both ends) in a chain appear mainly for substrates with localized states and can be controlled by the chain-substrate coupling parameters.

3.2. Single and double cones in biased chains

It is interesting whether the cone-like structure of DOS along the chain can be also observed for the non-resonant case. Therefore, we have calculated the corresponding spectral den-sity for biased chain, i.e. the on-site electron energies do not correspond to the Fermi level, ε ≠ E0 F. The results for the even N are shown in figure 4(a) and for the odd number of atoms in figure 4(b). As one can see for very small ε0 (triangles) the structure of DOS( )EF slightly differs from the resonant case, ε = 00 . However, for larger ε0 and even N, DOS cones along the chain are well visible (crosses and circles in figure 4(a)). Contrary to the even N, for odd number of sites the amplitude of DOS oscillations slightly decreases with ε0 for all sites and no DOS cones are observed, figure 4(b).

To confirm analytically the cone-like behaviour of the local DOS we will show that under certain conditions, there are two straight lines which include all values of the spec-tral density functions at EF (along the chain). It means that one of these lines passes through ( )EDOSi F , i = 1, 3, 5, ... and the second is related to i = 2, 4, 6, .... We consider the system described by the retarded Green’s function, Gr, equa-tion (6), where the matrix Âij

N is a tri-diagonal one for Γ = 0S .

The spectral density function at site i depends on the ( )Gr ii matrix element, ( ) ˆ / ˆ=G A Acof detr ii

Nii

N, where Âcof is the

cofactor of Â. Note that the denominator of ( )Gr ii (i.e. Âdet ) does not depend on the chain site i and we can analyze only

the corresponding numerators i.e. the cofactors, Âcof iiN

and ˆ+ +Acof i i

N2, 2. We expect that both cofactors can differ only by a

constant value from each other to satisfy the linear condition,

i.e. ˆ ˆ= ++ +A Acof cof const.i iN

iiN

2, 2 for all i. After some algebra

the above cofactors can be written in the following form:

( )ˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ

= −Γ

+Γ

+

− − − − −

− − − − −

A A A A A

A A A A

cof det det4

det det

i2

det det det det

iiN i N i i N i

i N i i N i

01

0

2

02

01

02

0 01

01

(9)

and similar for ˆ + +Acof i iN

2, 2. Here Ân0 is a tri-diagonal matrix

( ×n n dimension) corresponding to the chain decoupled from the leads, i.e. ( ˆ ) ( ) ( )ε ε δ δ δ= − − ++ +A V

nij i j i j i j0 0 , , 1 1, .

The determinant of this matrix satisfies the recursion rela-

tion ˆ ˆ ˆ= −ε ε−− −

A A Adet det detn

V

n n0 0

10

20 , with ˆ =Adet 100

and

ˆ ( )/ε ε= −A Vdet 01

0 . For the physical case ( )/ε ε| − |

T Kwapiński

7

In figure 4(c) we analyze DOS cones for different chain length N and for the non-resonant case ε ≠ E0 F (ε = 0.020 ). As before, almost ideal linear behaviour of cone lines is observed especially for small N. For larger N small deviation from the linearity is visible at both chain ends. To further study this interesting effect we obtain the oscillation amplitude at the chain ends as a function of N, figure 4(d) (here we consider only even-length chains). For small ε0 the amplitude slowly increases which means that the cone-like structure of DOS appears in the system. However, for larger ε0 the oscilla-tion amplitude of the spectral function reaches its maximum (e.g. for ε = 0.040 and N = 60) and then it oscillates with N. It means that end states can move deeper into the chain and, for certain N, they disappear. For the chain length N corre-sponding to the vanishing amplitude of DOS at both ends, maximal oscillations of the local DOS are observed for i = N/4 and i = 3N/4. In this case the estimation of the chain length, defined as a distance between two endmost maxima in the STM topography profile, is not valid. The spatial position of end states depends also on the STM bias voltage [22]. Note that for greater N we observe again the local DOS oscillations at both chain ends. The variation of the oscillation amplitude leads to regular non-vanishing local DOS oscillations along the chain related to the conductance oscillations and charge density waves [31, 35, 46]. It is worth noting that the results shown in [12] could be the STM experimental confirmation of DOS cones but more detailed analyze of the local DOS behaviour should be performed in this case.

The conductance along an atomic chain depends on whether the number of atoms is even or odd (so called even-odd conductance oscillations) which was predicted theor-etically and confirmed experimentally [1, 2, 47, 48]. The conductance oscillations with larger periods may occur as well in the stationary and driven chains [7, 12, 35, 49]. For a regular chain of atoms the period M of the conduct-ance oscillations can be find from the analytical relation:

( / ) ( )/π ε= −l M E Vcos 2F 0 , where l = 1, ..., M − 1, [31, 35]. Moreover, for a given number of atoms, the formation of charge waves inside the chain was investigated [31, 34, 46]. Thus we predict that the local DOS along the chain should also oscillate i.e. ( ) ( )= +E EDOS DOSi i MF F , with the period of M atoms. To corroborate this effect in figure 5 we analyze the local DOS along the chain for selected electron energies ε0. We have found that the general condition on the local DOS oscillations for a given N is exactly the same as for the conductance oscillations (where the number of atoms in the chain changes). This important result allows us to estimate the system param eters from experiments on the local DOS oscillations (e.g. from the STM topography images). Here, for ε = V0 one obtains the oscillation period of the spectral density inside the chain M = 3 sites. Similarly, for ε = V20 or ε = V30 the oscillation period is M = 4 or M = 6, respectively—these oscillations of the local DOS are shown in figure  5(a), for N = 50. Note that the oscillation periods do not depend on the parity of N in contrary to the results for the resonant case, figures 2 and 3. However, in real systems it is difficult to change accurately the position of ε0 versus the

Fermi energy to reach the condition on DOS oscillations. Thus in figure 5 (panels (b) and (c)) we consider the influence of small deviation of ε0 from this exact condition. In the middle panel the on-site electron energy is shifted by 0.02 from the value ε = V1.00 and clear cone-like structure of the local DOS is visible (instead of the regular oscillations with the period M = 3 shown in the upper panel). However, there is also one non-cone bottom branch—one has always M branches of DOS points and here two of them form the cone and the third one stands for a free bottom branch. Note that the chain parity does not play a role and for odd N similar behaviour of the local DOS is observed (not shown here). In the bottom panel in figure 5 the energy shift is 0.01 from ε = 20 and because the oscillation period is M = 4, we observe double-cone structure of DOS along the chain (four branches of DOS points). For odd length chains (not shown here) there are also four branches of DOS points but only one DOS cone is visible with two upper and bottom free (non-crossing) branches. For greater oscillation periods, M, more DOS cones appear but they are characterized by smaller oscillation amplitude and their detailed analysis is not as transparent as for smaller M. Note that for larger ε0, ε > V20 , the Fermi energy is outside the chain energy band and the local DOS at EF vanishes. To

Figure 5. (a) Regular oscillations of DOS at the Fermi level along a chain for N = 50 and 00ε = , V1.00ε = , V20ε = (shifted down by 0.2) and V30ε = (shifted down by 0.3). (b) and (c) DOS cones for small deviation from the resonance condition:

V1.0 0.020 ( )ε = + , N = 50, and V2 0.010 ( )ε = + , N = 100, respectively. The other parameters are the same as in figure 4.

J. Phys.: Condens. Matter 29 (2017) 095304

T Kwapiński

8

conclude this section, we have found that non-resonant posi-tions of the on-site electron energies lead to single or multi-cone structures of the local DOS which do not depend on the chain parity.

3.3. DOS cones due to the spin–orbit couplings

In this section  we consider the spin–orbit couplings in the chain which can significantly modify the structure of DOS and the electron transport properties. These couplings are responsible for the intrinsic and extrinsic spin-flip processes, λSF and λSO, respectively. As before, the retarded Green’s function needed to obtain the local DOS at the ith site of the chain can be find from the relation, ˆ / ˆ=G C Ccof detii

rii

N N2 2,

where the matrix Ĉ is defined in equation (6) and because the spin–orbit parameters are relatively small, λ λ � V,SO SF , the spin index was omitted. The determinant of the block matrix Ĉ can be written in the form of the product of two tri-diagonal

matrixes: = − +C A B A Bdet det detN N N N N2ˆ ( ˆ ˆ ) ( ˆ ˆ ), thus after some algebra the diagonal elements of the retarded Green’s functions can be found from:

( )

ˆ

ˆ ˆ ( ˆ ) ( ˆ )

=+ Γ −

×−

+ Γ −

+ +− Γ

+−

−− − −

− −− Γ

−−

GA

S S S

A X A X

S S S

det

det i det det

det

det i det det

iir

N

N N N

iN

iN N N

iN N T

N N i i

14

2

1, 1 1,

42

2

2

(11)

where ˆ = + Γ −− Γ −A A i A Adet det det detN N N N0 01

4 022 , and ˆ−A i

N

is related to the matrix AN formed by deleting the ith column and ith row. The matrix ±S

N corresponds to ÂN0 but for the fol-

lowing substitutions: →ε ε λ±0 0 SF, and for the coupling parameters → λ±V V SO (for the upper off-diagonal elements) and → λ∓V V SO (for the lower elements), respectively. Thus the determinant of ±S

N can be obtained analytically [45]. The

rectangular matrix ˆ −XiN N1,

(dimension − ×N N1 ) is formed from the square matrix B̂

N, equation (8), by deleting the ith

row and it includes only the spin–orbit parameters. For the case of vanishing inter-site spin–orbit couplings, λ = 0SO , this matrix can be written as follows:

( ˆ )  

 

 

⎧⎨⎪

⎩⎪

λ δλ δ=− <− >

−+X

n in i

, for, for

0 elsewhereiN N

nm

n m

n m1,

SF

SF 1 (12)

In this case the determinant of the off-diagonal matrix B̂ijN

is

simply ˆ ( )λ= −Bdet N NSF , and ( ) /φ φ= +± ± ±S Ndet sin 1 sinN ,

where φ = ε ε λ±− ±arccos

V20 SF. One can find that in this case the

retarded Green’s function related to the ith site of the chain can be obtained form:

λ

λ=

−

⋅ −

−−

−− −−

−⎛⎝⎜

⎞⎠⎟

G A

A A

A A Idet

det

detiir N

iN A

Ai

N

N N N

1SF

2 det

det

1 1

SF2

iN

N

1

ˆˆ ( ) ( ˆ )

( ˆ ˆ ( ) ˆ)

ˆ

ˆ (13)

where Î is the identity matrix.To study the spin–orbit couplings we consider the atomic

chain composed of the even number of sites in the resonant case i.e. ε = =E 00 F (figure 6, panels (a) and (b)) for which the flat (no-oscillating) structure of DOS along the chain was analyzed in figure 2(b) for N = 50. Now it is desirable to consider two special cases: (i) there are only on-site spin-flip scattering terms in the chain, λSF (upper panels, λ = 0SO ) or (ii) there are only inter-site spin-flip processes, λSO (bottom panels, λ = 0SF ). In the first case for small λSF the cone-like structure of DOS due to the spin-flip couplings is very well visible (crosses, panel (a)). For small value of λSF the cone aperture increases linearly with λSF. This effect can be explain as follows: in the presence of the spin-flip scattering the on-site electron energies are split and instead of a single state with energy ε = 00 , two effective states appear with the energies ε λ=± ≠± ESF F. Thus here a non-resonant transport takes place for which the local DOS cones appear, similar to the case discussed in figure 4(a). One expects that also the inter-site spin-flip couplings can generate the cone structure of the spectral density function along the chain. However, as we show in the bottom panel in figure 6(b), for λ = 0SF and λ = 0.05SO (or even 0.3) the local DOS cones do not appear at all. Note that the λSO parameter is related to the off- diagonal elements of the matrix B̂ and thus it influences only the hopping integrals between chain sites. In many sys-tems the site-site coupling parameter, V, is much larger than λSO so the effective coupling parameter, Veff, slightly differs from V. It is the reason that in the bottom panel (figure 6(b)) the results almost do not depend on λSO and only flat structures of DOS are visible.

The situation is different in the non-resonant case i.e. ε ≠ E0 F (panels (c) and (d)) where in the absence of the spin–orbit couplings the cone-like structure of DOS was analyzed in figure 4. In the presence of the on-site spin-flip scattering, λSF, the local DOS cones are still visible with small modifications (panel (c)). These changes are related to

Figure 6. Spatial DOS at the Fermi level along a chain composed of N = 50 sites with the on-site electron energies 00ε = (left panels, (a) and (b)) and 0.040ε = (right panels, (c) and (d)). The upper panels correspond to 0SOλ = and 0.02SFλ = (crosses) or 0.05 (circles). The results in the bottom panels are obtained for 0SFλ = and 0.05SOλ = (crosses) or 0.3 (circles). The other parameters are the same as in figure 4.

J. Phys.: Condens. Matter 29 (2017) 095304

T Kwapiński

9

the effective on-site electron energies which now depend on the λSF parameter, ε ε λ= ±± 0 SF. Also for nonzero extrinsic spin-flip scattering, λSO (panel (d)), the structure of DOS cones slightly differ from the case of λ = 0SO . It means that the extrinsic spin-flip processes do not disturb significantly the cone-like structure of DOS. It is also the reason that for both intrinsic and extrinsic spin-flip processes the local DOS depends mainly on the intrinsic scattering. Generally, cone-like structures of the spectral density function can be observed in the Rashba chain due to the on-site spin-flip pro-cesses. It is very important results because, in contrast to the resonant case, the spin–orbit couplings can be controlled exper imentally [42] which allows one to tune the system parameters to obtain the cones of DOS.

3.4. Imperfect chains on a substrate

In the last part of this paper we consider more realistic case and investigate the local DOS for an imperfect chain placed on a substrate. Thus, let us analyze a linear chain of atomic sites where one of the internal hopping integral is reduced and can break the chain into two pieces. Very long chains of atoms fabricated on different surfaces often contain a missing atom which leads to multiple short chains with end-states weakly coupled to each other or completely separated parts [22, 25, 27]. Long chains can be also disturb by protrusion, adatoms, surface defects or others. These imperfections can locally modify the chain param eters especially the site-site coupling strength, Vij. In figure 7 we show the spectral density func-tion along the chain weakly coupled only with the substrate electrode (Γ = Γ = 0L R ). For a perfect chain very clear cone-like DOS structure with minimum amplitude in the middle of the chain is visible (panel (a), upper curve). Here the DOS

cones appear due to the localized states in the substrate, see also figure  2. However, for the reduced coupling strength, V50,51, between the internal 50th and 51st sites the DOS cone is modified and a diamond-like structure of the local DOS appear. Finally for V50,51 = 0 the chain is split (bottom curve) and two identical abutted DOS cones are observed. They are related to two separate chains of the even length = =N N 501 2 each. The situation changes if the chain is split into two odd-length pieces which is shown in figure 7(b) for N = 100 and reduced V49,50 coupling parameter. Here the end states are still visible but the amplitude of the local DOS in the middle of the chain increases with decreasing V49,50, and the DOS cone vanishes (compare the results in panel (b) with the upper cone in panel (a) obtained for V49,50 = 1). For V49,50 = 0 there are two separate chains (N1 = 49 and N2 = 51) thus we observe two almost symmetrical DOS structure along each chain, circles in figure 7(b). Because the system for odd and even N behaves differently it is interesting to consider an odd-length chain with the reduced hopping integral in the middle site, figure 7(c). In this case the total chain length is N = 101 and for V49,50 < 1 two subsystems appear with the even (N1 = 52) and odd (N2 = 49) number of atomic sites. As one can see even for lower finite value of the hopping integral, V49,50 = 0.4, both subsystems are separated enough to observe the cone-like structure of DOS along the even length chain and no cones for the part with odd N. The reduced hopping integral breaks the chain into two (or more) weakly coupled subsystems, so that, depending on the chain length, different DOS structures can be observed and the local DOS cones can emerge. It is also true for biased chains on various substrates and for greater oscillation periods, M.

4. Conclusions

Using one-dimensional tight-binding Hamiltonian and the Green’s function approach, we have studied the local DOS oscillations along a biased atomic chain in the presence of the spin–orbit couplings and in contact with a substrate electrode. The spin–orbit effects are captured by the on-site and inter-site spin-flip processes. As a main feature, we have found that a cone-like structure of the spectral density function at the Fermi level appears along the chain. This effect was analyzed analytically and the linearity of the local DOS was confirmed. It is important that (in contrast to the electron end states in many chains as well as the Friedel oscillations in 1D systems) a linear decreasing towards the chain was not predicted in the literature.

The local DOS cone structures were observed for the Rashba chain due to the on-site spin-flip processes. In the presence of electron leakage from the wire to various types of substrates the cones of the local DOS emerge especially for a substrate with localized electrons. Interestingly, the formation of DOS cones depends on the parity of the chain length and are mainly observed for even-length chains. It is reflected in the disordered chains with reduced coupling strength between two neighboring sites. In this case a diamond-like structure of the local DOS can appear along the chain.

Figure 7. The spatially dependent spectral function, DOS EF( ), of atomic chain where the hopping integral between i = 50 and 51 sites (i = 49 and 50) is reduced to V50,51 = 1, 0.6, 0 (V49,50 = 0.6, 0) for the chain length N = 100—panel (a) (panel (b)). Panel (c) corresponds to N = 101 and the reduced hopping integral between i = 49 and 50 sites, V49,50 = 0.4. The results for V50,51 = 1 and 0.6 are shifted by 0.6 and 0.3, respectively. The other parameters are the same as in figure 4 and 0.05SΓ = , k a 10F 0 = , 0L RΓ = Γ = , 00ε = .

J. Phys.: Condens. Matter 29 (2017) 095304

T Kwapiński

10

Note that in the chain with non-resonant positions of the on-site electron energies multi-cone structures of DOS are observed. In comparison with the resonant case, they do not depend on the chain parity and can appear for biased chains, in the presence of the substrate or due to the spin–orbit couplings.

The appearance of DOS cones give some insight into the physical mechanism of the transport properties as well as charge distribution in the chain. Detailed investigations of this effect allow one to estimate many important param-eters of experimental systems. The STM topography of one- dimensional chains in some experiments revels a cone-like behaviour along the investigated object e.g. [12, 22]. It is believed that the results of this paper will stimulate further STM experiments with the thinnest possible electric conduc-tors grown on various surfaces.

Acknowledgments

This work was supported by National Science Centre, Poland, under Grant No. 2014/13/B/ST5/04442.

References

[1] Smit R H M, Untiedt C, Rubio-Bollinger G, Segers R C and van Ruitenbeek J M 2003 Phys. Rev. Lett. 91 076805

[2] Agrait N, Levy Yeyati A and van Ruitenbeek J M 2003 Phys. Rep. 377 81

[3] Ohnishi H, Kondo Y and Takayanagi K 1998 Nature 395 780 [4] Yanson A I, Rubio Bollinger G, van den Brom H E, Agrait N

and van Ruitenbeek J M 1998 Nature 395 783 [5] Auslaender O M, Steinberg H, Yacoby A, Tserkovnyak Y,

Halperin B I, Baldwin K W, Pfeiffer L N and West K W 2005 Science 308 88

[6] van der Wiel W G, De Franceschi S, Elzerman J M, Fujisawa T, Tarucha S and Kouwenhoven L P 2003 Rev. Mod. Phys. 75 1

[7] Kwapiński T 2004 Phys. Rev. B 69 153303 [8] Kohler S, Lehmann J and Hanggi P 2005 Phys. Rep. 406 379 [9] Das S and Rao S 2005 Phys. Rev. B 71 165333 [10] Avishai Y, Cohen D and Nagaosa N 2010 Phys. Rev. Lett.

104 196601 [11] Gruner G 1998 Rev. Mod. Phys. 60 1129 [12] Sung Shin J, Ryang K-D and Woong Yeom H 2012 Phys. Rev.

B 85 073401 [13] Moroz A V and Barnes C H W 1999 Phys. Rev. B 60 14272 [14] de Andrada e Silva E and La Rocca G 2003 Phys. Rev. B

67 165318 [15] Kitaev A Y 2001 Phys.—Usp. 44 131 [16] Maśka M, Gorczyca-Goral A, Tworzydło J and Domański T

2016 arXiv:1609.00685

[17] Zdyb R, Stróżak M and Jałochowski M 2001 Vacuum 63 107 [18] Crain J N, McChesney J L, Zheng F, Gallagher M C,

Snijders P C, Bissen M, Gundelach C, Erwin S C and Himpsel F J 2004 Phys. Rev. B 69 125401

[19] Baski A A, Saoud K M and Jones K M 2001 Appl. Surf. Sci. 182 216

[20] Jałochowski M, Kwapiński T, Łukasik P, Nita P and Kopciuszyński M 2016 J. Phys.: Condens. Matter 28 284003

[21] Oncel N, van Houselt A, Huijben J, Hallback A-S, Gurlu O, Zandvliet H J W and Poelsema B 2005 Phys. Rev. Lett. 95 116801

[22] Krawiec M, Kwapiński T and Jałochowski M 2006 Phys. Rev. B 73 075415

[23] Zhang Z and Bullemer B 1995 Thin Solid Films 264 277 [24] Kwapiński T and Jałochowski M 2010 Surf. Sci. 604 1752 [25] Heimbuch R, Atesci H, Slootheer I and Zandvliet H J W 2016

Surf. Sci. 644 91 [26] Kwapiński T, Krawiec M and Jałochowski M 2008 Phys. Lett.

A 372 154 [27] Crain J N and Pierce D T 2005 Science 307 703 [28] Urban D F, Stafford C A and Grabert H 2007 Phys. Rev.

B 75 205428 [29] Gorczyca A, Maśka M M and Mierzejewski M 2007

Phys. Rev. B 76 165419 [30] Weiss Y, Goldstein M and Berkovits R 2007 Phys. Rev.

B 75 064209 [31] Kwapińnski T 2006 J. Phys.: Condens. Matter 18 7313 [32] Brouwer P W, Duckheim M, Romito A and von Oppen F 2011

Phys. Rev. B 84 144526 [33] Newns D M and Read N 1987 Adv. Phys. 36 799 [34] Kwapiński T, Kohler S and Hänggi P 2010 Eur. Phys. J. B

78 75 [35] Kwapiński T 2005 J. Phys.: Condens. Matter 17 5849 [36] Krawiec M, Kopciuszyński M and Zdyb R 2016 Appl. Surf.

Sci. 373 26 [37] Datta S and Das B 1990 Appl. Phys. Lett. 56 665 [38] Birkholz J E and Meden V 2008 J. Phys.: Condens. Matter

20 085226 [39] Mireles F and Kirczenow G 2001 Phys. Rev. B 64 024426 [40] Peterson L and Hedegard P 2000 Surf. Sci. 459 49 [41] Bihlmayer G, Koroteev Y M, Echenique P M, Chulkov E V

and Blugel S 2006 Surf. Sci. 600 3888 [42] Nitta J, Akasaki T, Takayanagi H and Enoki T 1997 Phys. Rev.

Lett. 78 1335 [43] Datta S 1995 Electronic Transport in Mesoscopic Systems

(Cambridge: Cambridge University Press) [44] Ramos J P, Torres L E F, Orellana P A and Apel V M 2014

J. Appl. Phys. 115 124507 [45] Fonseca C M and Petronilho J 2001 Linear Algebr. Appl. 325 7 [46] Mierzejewski M and Maśka M M 2006 Phys. Rev. B

73 205103 [47] Kim T-S and Hershfield H 2002 Phys. Rev. B 65 214526 [48] Zeng Z Y and Claro F 2002 Phys. Rev. B 65 193405 [49] Thygesen K S and Jacobsen K W 2003 Phys. Rev. Lett.

91 146801

J. Phys.: Condens. Matter 29 (2017) 095304

https://doi.org/10.1103/PhysRevLett.91.076805https://doi.org/10.1103/PhysRevLett.91.076805https://doi.org/10.1016/S0370-1573(02)00633-6https://doi.org/10.1016/S0370-1573(02)00633-6https://doi.org/10.1038/27399https://doi.org/10.1038/27399https://doi.org/10.1038/27405https://doi.org/10.1038/27405https://doi.org/10.1126/science.1107821https://doi.org/10.1126/science.1107821https://doi.org/10.1103/RevModPhys.75.1https://doi.org/10.1103/RevModPhys.75.1https://doi.org/10.1103/PhysRevB.69.153303https://doi.org/10.1103/PhysRevB.69.153303https://doi.org/10.1016/j.physrep.2004.11.002https://doi.org/10.1016/j.physrep.2004.11.002https://doi.org/10.1103/PhysRevB.71.165333https://doi.org/10.1103/PhysRevB.71.165333https://doi.org/10.1103/PhysRevLett.104.196601https://doi.org/10.1103/PhysRevLett.104.196601https://doi.org/10.1103/RevModPhys.60.1129https://doi.org/10.1103/RevModPhys.60.1129https://doi.org/10.1103/PhysRevB.85.073401https://doi.org/10.1103/PhysRevB.85.073401https://doi.org/10.1103/PhysRevB.60.14272https://doi.org/10.1103/PhysRevB.60.14272https://doi.org/10.1103/PhysRevB.67.165318https://doi.org/10.1103/PhysRevB.67.165318http://arxiv.org/abs/1609.00685https://doi.org/10.1016/S0042-207X(01)00177-4https://doi.org/10.1016/S0042-207X(01)00177-4https://doi.org/10.1103/PhysRevB.69.125401https://doi.org/10.1103/PhysRevB.69.125401https://doi.org/10.1016/S0169-4332(01)00412-3https://doi.org/10.1016/S0169-4332(01)00412-3https://doi.org/10.1088/0953-8984/28/28/284003https://doi.org/10.1088/0953-8984/28/28/284003https://doi.org/10.1103/PhysRevLett.95.116801https://doi.org/10.1103/PhysRevLett.95.116801https://doi.org/10.1103/PhysRevB.73.075415https://doi.org/10.1103/PhysRevB.73.075415https://doi.org/10.1016/0040-6090(94)05827-Xhttps://doi.org/10.1016/0040-6090(94)05827-Xhttps://doi.org/10.1016/j.susc.2010.06.026https://doi.org/10.1016/j.susc.2010.06.026https://doi.org/10.1016/j.susc.2015.09.019https://doi.org/10.1016/j.susc.2015.09.019https://doi.org/10.1016/j.physleta.2007.08.017https://doi.org/10.1016/j.physleta.2007.08.017https://doi.org/10.1126/science.1106911https://doi.org/10.1126/science.1106911https://doi.org/10.1103/PhysRevB.75.205428https://doi.org/10.1103/PhysRevB.75.205428https://doi.org/10.1103/PhysRevB.76.165419https://doi.org/10.1103/PhysRevB.76.165419https://doi.org/10.1103/PhysRevB.75.064209https://doi.org/10.1103/PhysRevB.75.064209https://doi.org/10.1088/0953-8984/18/31/024https://doi.org/10.1088/0953-8984/18/31/024https://doi.org/10.1103/PhysRevB.84.144526https://doi.org/10.1103/PhysRevB.84.144526https://doi.org/10.1080/00018738700101082https://doi.org/10.1080/00018738700101082https://doi.org/10.1140/epjb/e2010-10452-xhttps://doi.org/10.1140/epjb/e2010-10452-xhttps://doi.org/10.1088/0953-8984/17/37/020https://doi.org/10.1088/0953-8984/17/37/020https://doi.org/10.1016/j.apsusc.2015.09.219https://doi.org/10.1016/j.apsusc.2015.09.219https://doi.org/10.1063/1.102730https://doi.org/10.1063/1.102730https://doi.org/10.1088/0953-8984/20/8/085226https://doi.org/10.1088/0953-8984/20/8/085226https://doi.org/10.1103/PhysRevB.64.024426https://doi.org/10.1103/PhysRevB.64.024426https://doi.org/10.1016/S0039-6028(00)00441-6https://doi.org/10.1016/S0039-6028(00)00441-6https://doi.org/10.1016/j.susc.2006.01.098https://doi.org/10.1016/j.susc.2006.01.098https://doi.org/10.1103/PhysRevLett.78.1335https://doi.org/10.1103/PhysRevLett.78.1335https://doi.org/10.1063/1.4868902https://doi.org/10.1063/1.4868902https://doi.org/10.1016/S0024-3795(00)00289-5https://doi.org/10.1016/S0024-3795(00)00289-5https://doi.org/10.1103/PhysRevB.73.205103https://doi.org/10.1103/PhysRevB.73.205103https://doi.org/10.1103/PhysRevB.65.214526https://doi.org/10.1103/PhysRevB.65.214526https://doi.org/10.1103/PhysRevB.65.193405https://doi.org/10.1103/PhysRevB.65.193405https://doi.org/10.1103/PhysRevLett.91.146801https://doi.org/10.1103/PhysRevLett.91.146801