# Spin transport in nanocontacts and nanowires

###### Abstract

In this thesis we study electron transport through magnetic nanocontacts and nanowires with ab initio quantum transport calculations. The aim is to gain a thorough understanding of the interplay between electrical conduction and magnetism in atomic-size conductors and how it is affected by different aspects as e.g. the atomic structure and the chemical composition of the conductor. To this end our ab initio quantum transport program ALACANT which combines the non-equilibrium Green’s function formalism (NEGF) with density functional theory (DFT) calculations has been extended to describe spin-polarized systems. We present calculations on nanocontacts made of Ni as a prototypical magnetic material. We find that atomic disorder in the contact region strongly reduces the a priori high spin-polarization of the conductance leading to rather moderate values of the so-called ballistic magnetoresistance (BMR). On the other hand, we show that the adsorption of oxygen in the contact region could strongly enhance the spin-polarization of the conduction electrons and thus BMR by eliminating the spin-unpolarized s-channel. Finally, we show that short atomic Pt chains suspended between the tips of a nanocontact are magnetic in contrast to bulk Pt. However, this emergent nanoscale magnetism barely affects the overall conductance of the nanocontact making it thus difficult to demonstrate by simple conductance measurements. In conclusion, we find that spin-transport through atomic-scale conductors is quite sensitive to the actual atomic structure as well as to the chemical composition of the conductor. This presents both, opportunities and challenges for the realization of future nanoscale spintronics devices.

## Acknowledgments

I am especially grateful to Prof. Juanjo Palacios and Dr. Joaquin Fernández-Rossier for directing this work. The discussions on physical problems were always both, enlightening and enjoyable. Their enthusiasm for the physics was really inspiring and motivating. I also would like to thank Dr. Carlos Untiedt, Dr. María José Caturla, Prof. Enrique Louis, Prof. José Antonio Vergés and Dr. Guillermo Chiappe for fruitful discussions. Special thanks to James McDonald who built the Beowulf cluster facility here in the Applied Physics Department without which this work would not have been possible. I thank the MECD for financial support under grant No. UAC-2004-0052. I feel grateful to all my colleagues, Cristophe, Deborah, Eladio, Federico, Fernando, Giovanni, Igor, Loïc, Martin, Natamar, Pedro, Reyes and Richard which have made my life here in Alicante really enjoyable by sharing a lot of beers, barbecues, paellas, tapas, and a lot more with them. I would like to thank my family for their constant support. Finally but most importantly, I would like to thank María for her love and support, and especially for her patience during the last months.

###### Contents

- \thechapter Introduction
- \thechapter Quantum theory of electron transport
- \thechapter Ab initio quantum transport
- \thechapter Spin transport
- \thechapter Ni nanocontacts
- \thechapter NiO chains in Ni nanocontacts
- \thechapter Transport through magnetic Pt nanowires
- \thechapter Summary and Outlook
- \thechapter Representation of operators in non-orthogonal basis sets
- \thechapter Partitioning method
- \thechapter Self-energy of a one-dimensional lead
- \thechapter Bethe lattices
- \thechapter Non-Collinear Unrestricted Hartree-Fock Approximation
- List of abbreviations
- List of publications

## Chapter \thechapter Introduction

The invention of the integrated circuit (IC) in 1959 by J. Kilby (Nobel price in 2000 together with H. Kroemer and Z. I. Alferov) triggered the stunning and still ongoing development of computer technology which has lead to ever faster, cheaper, and smaller computers. An IC is an electronic circuit where all electronic components (transistors, capacitors, interconnects) are integrated on a single silicon chip. The successive improvement of the fabrication techniques and the introduction of new materials allowed to constantly decrease the sizes of the IC components so that an increasing number of them could be integrated on a single IC. This development has lead to increasingly powerful and faster computer chips. The other basic ingredient of modern computer systems are non-volatile mass storage devices presented in today’s computers by hard drives which store data in form of magnetic bits on magnetic disks (hard disks). Here the improvement of fabrication techniques, introduction of new materials and new concepts for the the read and write mechanism of the hard drives allowed to constantly decrease the minimum area needed to record a magnetic bit making them faster and increasing their data capacities by several orders of magnitude since the invention in 1979.

Thus miniaturization has become the leading paradigm of today’s
computer industry and a lot of effort in industrial research and
development is dedicated to further decreasing the minimum feature
size of semiconductor chips or the areas of the magnetic bits on
hard disks. However, this miniaturization trend
cannot go on forever since the atomic scale presents an ultimate
limit which can never be surpassed and possibly not even reached.
This is not a kind of hypothetical scenario but rather relevant for
the development of microprocessor chips in the near future as is
impressively demonstrated by today’s most advanced microprocessors
and memory chips whose smallest features (the gate length) have
already reached a size in the order of 30 nm, and IC transistors
with a gate length of only 10nm are currently under investigation
[1]. The miniaturization of IC components has in fact
reached a level where atomic structure effects like *electro-migration*
processes [2] become an issue because they can seriously
limit the usability and lifetime of ICs. Another serious problem are the so-called
subthreshold leakage currents between the electrodes of the transistor which increase
considerably as the size of the transistors shrinks and are responsible for
a grand part of the power consumption loss in microchips [3].
Similar problems arise when reducing the dimensions of magnetic bits.
Moreover, when finally reaching the nanoscale, inevitably quantum
effects will come into play and seriously challenge current silicon
based IC technology possibly demanding radically different approaches.
Molecular electronics and spintronics are two promising
examples of radically new strategies for information processing.

Molecular electronics aims at employing single molecules as the ultimate electronic components for realizing nanoscale electronic circuits. Indeed the use of single molecules as rectifiers for electrical current has been proposed as early as 1974 by Aviram and Ratner [4]. But not until the advent of the scanning probe microscopes was it possible to study and manipulate material properties at the molecular or atomic level let alone electrically contact individual molecules. The invention of the scanning tunneling microscope (STM) by G. Binning and H. Rohrer in 1981 (Nobel price in 1986 together with E. Ruska) [5] made it for the first time possible to study (metallic) surfaces and molecules adsorbed on them with atomic resolution. By contacting an individual molecule adsorbed on a metal surface with the STM tip it is possible to measure the conductance of this molecular conductor. This technique was first employed to measure the conductance of individual C molecules [6, 7]. Since then more conductance measurements of individual molecules have been reported in the literature [8, 9, 10, 11, 12]. However, establishing electrical contacts with individual molecules and measuring their conductance remains a formidable task and care must be taken to assure that a molecule has indeed been contacted [12].

Using an STM it is also possible to fabricate atomic-size nanocontacts where two sections of a metal wire are connected via a constriction of just a few atoms in diameter which thus represent the ultimate electrical conductors with respect to size [13]. Nanocontacts are formed when an STM tip is pressed into the substrate and then slowly retracted until an atomic-size neck is formed [14]. Another technique to fabricate very stable and reproducible nanocontacts in an efficient and cheap way is the mechanically controllable break junction technique (MCBJ)[15] where a notched thin wire mounted on a bending beam is broken in controlled way. The bending can be fine-controlled by a piezo element allowing a very precise control over the separation of the two sections of the wire. Just a third method to fabricate nanocontacts is by electrodeposition [16, 17] where the nanocontacts are electro-chemically deposited between two macroscopic electrodes. Interesting phenomena for atomic-size nanocontacts have been observed in experiments like the formation of monatomic chains suspended between Gold or Platinum contacts, see e.g. [18].

Another approach that promises to revolutionize conventional electronics is the field of spintronics [19] which aims to combine the traditionally separated fields of magnetic information storage and semiconductor electronics in order to build more powerful electronic devices that exploit the electron spin in addition to the electron charge. The example that best illustrates the spintronics philosophy is the so-called Magnetic Random Access Memory (MRAM) [20] which combines the advantages of conventional magnetic data storage (hard drives) and conventional electronic random excess memory (RAM) into a single device in order to achieve at the same time non-volatile memory cells that are as fast as conventional RAM cells.

An essential ingredient for spintronics is the generation of spin-polarized electron currents which is typically accomplished by passing the electrical current through a ferromagnetic metal. The other basic ingredient are spin-valves which are devices that change their resistivity depending on the polarization of the spin-current and thus allow to detect the spin-polarization of an electrical current. An example for an actual spin-valve device are the giant magneto-resistance (GMR) devices consisting of alternating magnetic and non-magnetic metal multilayers that display a strong sensitivity of the electrical current to the relative orientation of the magnetizations of the magnetic layers [21]. Soon after its discovery in 1988, the GMR effect was exploited to improve the sensitivity of read heads in hard drives which before had been based on simple magnetic induction or the much smaller anisotropic magneto-resistance effect displayed by bulk metals. This in turn allowed to decrease the size of the magnetic bits and thus to increase the data storage density of the hard disks dramatically. Consequently, GMR read heads can now be found in the hard drives of every modern computer.

Magnetic tunnel junctions (MTJs) are similar to GMR devices but feature an insulating layer instead of the non-magnetic metal layer separating the ferromagnetic metal layers which presents a tunnel barrier for the electrons flowing between the ferromagnetic layers. MTJs display a MR known as tunneling magneto-resistance (TMR) which was first demonstrated by Julliere [22]. The TMR spin-valve is a crucial ingredient for an efficient realization of the above described MRAM device [23]. In the earlier experiments TMR was found to be much smaller than GMR, but very recently new material combinations (Fe-MgO-Fe) for the MTJs motivated by theoretical studies [24, 25] have lead to a dramatic increase of the TMR which now actually exceeds GMR values [26, 23].

GMR and TMR spin-valves are nanoscale devices in the sense that the thicknesses of the layers making up the spin-valve structures are on the nanometer scale, and can be made as thin as 5Å with modern growth methods, so that electron transport is governed by quantum effects. In fact, these devices actually only work because of quantum effects, i.e. the coherent transmission of a spin-polarized current through the non-magnetic layer. Moreover, when finally shrinking the other two dimensions of a spin-valve to the nanoscale, quantum and atomic structure effects will become even more important. Thus it is of fundamental importance to gain a thorough understanding of the interplay between magnetism and electrical conduction at the atomic scale. Nanocontacts and nanowires made from ferromagnetic metals allow to study this interplay between magnetism and electrical conduction or in more fundamental terms the interplay between electron spin and charge flow in the smallest possible magnetic conductors.

An important question is whether GMR or TMR effects survive when the other two device dimensions are scaled down to the nanoscale, or whether other magneto-resistance (MR) effects emerge at the nanoscale which could be exploited for the realization of nanoscale spintronics devices. Therefore, measuring the MR of ferromagnetic nanocontacts has recently attracted a lot of interest [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38]. Indeed, some groups have found a huge MR (exceeding even the GMR effect) for Ni nanocontacts which was coined ballistic magneto-resistance (BMR) for its supposed origin in the ballistic scattering of spin polarized electrons on a sharp domain wall (DW) which should form at the atomic neck of the nanocontact[39, 40, 41]. However, the possibility of huge BMR in ferromagnetic nanocontacts has been a controversial topic since its discovery, and is one of the principal topics of this thesis (See Ch. \thechapter and Ch. \thechapter).

Integrating the fields of molecular electronics with that of spintronics is another promising approach for realizing nanoscale spintronics devices because of the expected very long spin-decoherence times in organic molecules as compared to bulk metals and semiconductors. Strong indications on the possibility of integrating the two fields come from a few recent experiments indicating e.g. a long spin-flip scattering length for carbon nanotubes [42], spin injection from strongly spin polarized materials into carbon nanotubes [43], and spin transport through organic molecules [44]. Since ferromagnetic nanocontacts are a basic ingredient for molecular spintronics for injecting spin-polarized currents into the molecules, it is also in this context important to gain a solid understanding of electrical conduction and magnetism in nanocontacts.

In this thesis electron transport through magnetic nanocontacts and nanowires is investigated theoretically. The starting point for the theoretical investigation of electron transport through nanocontacts and nanowires is the Landauer formalism, which is introduced in Ch. \thechapter. The Landauer approach assumes that electron transport through nanostructures is phase coherent, i.e. decoherence by phase-breaking scattering processes is neglected. This turns out to be a reasonable assumption at low temperatures and for small bias voltages. Indeed, the Landauer approach has been successfully applied for studying electrical transport through metallic nanocontacts, so that now we have a good understanding of atomic scale conductors [45, 46].

In order to predict the transport properties of atomic scale nanocontacts and nanowires it is important to have a realistic description of the electronic and magnetic structure of the nanocontact taking into account its actual atomic structure. This can be achieved most conveniently by ab initio electronic structure methods based on localized atomic orbitals like e.g. GAUSSIAN [47] or SIESTA [48]. Ch. \thechapter shows how ab initio quantum transport calculations based on density functional theory (DFT) [49] are implemented in the ALACANT package [50, 51] which as a part of this thesis was extended in order to describe spin transport in magnetic systems and to incorporate one-dimensional leads calculated from first principles in addition to the semi-empirical Bethe lattice electrodes [52]. Other DFT based quantum transport methods are developed by various groups around the world [53, 54]. In Ch. \thechapter, the Landauer formalism is generalized to describe transport of spin-polarized electrons (or in short “spin transport”) in magnetic nanostructures. In order to understand basic aspects of spin transport in nanoscopic conductors like the scattering of spin-polarized electrons on domain walls as well as the formation of domain walls in nanoscopic conductors, the spin-resolved transport formalism is applied to simplified models of magnetic materials.

In Ch. \thechapter spin transport through Ni nanocontacts is investigated theoretically with ab initio quantum transport calculations using the afore mentioned ALACANT package. In order to asses the above discussed possibility of huge BMR in Ni nanocontacts, we calculate the magneto resistance due to the formation of a DW at the atomic neck of the nanocontact for different contact geometries. We find that BMR of pure Ni nanocontacts is rather moderate [55], i.e. much smaller than the famous GMR effect. This is contrary to the claims of huge BMR found in the first experiments on Ni nanocontacts but in agreement with some recent experiments measuring very clean samples under very controlled conditions [37, 38].

In Ch. \thechapter, we study the electronic and magnetic structure, and transport properties of atomic NiO chains, both ideal infinite ones and short ones suspended between the tip atoms of Ni nanocontacts. It is found that the presence of a single oxygen atom between the tip atoms of a Ni nanocontact increases the spin-polarization of the conduction electrons dramatically, converting the nanocontact into an almost perfect half-metallic conductor, and leading to huge values of the BMR [56]. It is also discussed to what extent these results could explain the huge MR of Ni nanocontacts obtained in the experiments mentioned above.

In Ch. \thechapter, the electronic structure and transport properties of magnetic Pt nanowires is studied. It had been shown before that atomic Pt nanowires can actually become magnetic in contrast to bulk Pt due to the lower coordination of the Pt atoms in the atomic chain compared to the bulk [57, 58]. We reproduce this result for infinite chains, and find that also short Pt chains suspended between the tips nanocontacts become magnetic. However, the overall conductance of the Pt nanocontact is barely affected by the magnetism of the chain, so that simple conductance measurements of Pt nanocontacts cannot probe the magnetism [59].

Finally, Ch. \thechapter concludes this thesis with a general discussion of the obtained results, and their significance in the broader context of spin transport and spintronics in nanoscopic systems. Moreover, we will point out open questions and give an outlook on possible future lines of work.

## Chapter \thechapter Quantum theory of electron transport

The description of electrical conduction through a nanoscopic conductor like a molecule bridging the tips of two metal electrodes or a nanoscopic constriction in a metal wire as shown in Fig. 1 is a challenging problem. In systems of such small size the dimension of the conductor becomes comparable to the Fermi wavelength of the conduction electrons, so that the transport properties of the conductor are governed by quantization effects demanding a full quantum treatment of the transport process. Moreover, for molecular- and atomic-size conductors the actual atomic structure of the conductor has a strong effect on the electronic structure and transport properties.

Our starting point for a quantum description of electrical conduction is the Landauer formalism [60, 61, 62, 63, 64] which is introduced in Sec. 1. In Sec. 2 we describe the Landauer formalism within the framework of non-equilibrium Green’s functions (NEGF) formalism. In Sec. 3 we illustrate the derived formalism by applying it to simple model systems. Finally, in Sec. 4 we discuss the validity of the Landauer approach, pointing out its problems and limitations, and indicating ways of improving it as well as alternative approaches.

### 1 Landauer formalism

In the Landauer formalism electron transport is considered as a scattering process where the nanoscopic conductor
acts as a quantum mechanical scatterer for the electrons coming in from the leads. It is further assumed that the
electrons scatter only elastically
on the nanoscopic sample, i.e. inelastic scattering e.g. by phonons or by other electrons is neglected so that
transport becomes phase coherent. Thus electron transport through a nanoscopic conductor is described in terms of
*non-interacting* quasi particles coming in from the leads and being scattered elastically on the
nanoscopic device.

Fig. 2 shows how the transport problem is modeled in the Landauer formalism: The central scattering region () containing the nanoscopic conductor is connected via two ideal semi-infinite leads of some finite width to two electron reservoirs that are each in thermal equilibrium but at different chemical potentials and . The reservoirs are assumed to be reflection-less, i.e. incoming electrons are not reflected back to the leads, so that the two reservoirs are independent of each other. Due to the finite width of the leads the motion of the electrons perpendicular to the direction of the leads is quantized giving rise to a finite number of propagating modes or bands for a given energy . This is illustrated on the right hand side of Fig. 2 for the case of a two-dimensional electron gas confined in the vertical direction but moving freely in the horizontal direction. In that case a propagating wave is given by a plane wave in the -direction modulated by a transverse wavefunction :

In the more general case of a periodic potential, , generated e.g. by the atomic nuclei in the direction of the lead the propagating modes are Bloch waves

where the sum goes over all unit cells of the lead and is a localized wavefunction centered in unit cell . From the fact that , it follows that the Bloch functions also have the periodicity of the potential, apart from a trivial phase factor:

We define as the wave vector corresponding to the band at the energy that gives rise to a current
in the *positive* -direction. The current associated with the propagating mode is given by its group velocity
which in turn is given by the derivative of the dispersion relation for that band:

(1) |

where is the length of the conductor. This is trivial to proof for the case of free electrons where the group velocity is directly proportional to the wave vector, . For Bloch waves the situation is more complicated, and the group velocity can even have the opposite sign of the wave vector. A proof can be found e.g. in the book by Ashcroft and Mermin [65].

Elastic scattering means that an electron with some energy coming from one of the reservoirs will be scattered
to some out-moving state with the *same* energy of one of the leads, so that the process is phase coherent.
Thus an incoming wave at some energy on the left lead will give rise to a coherent superposition with outgoing
states of the same energy on both leads:

(2) |

where is the probability amplitude for an incoming electron on mode at energy to be reflected into the outgoing mode of the left lead, and is the amplitude for the electron to be transmitted into the mode of the right lead. Thus an incoming electron on mode of the left lead will be transmitted with a probability of to the right lead giving rise to a current density in that lead of magnitude

(3) |

The left electron reservoir injects electrons into the *right*-moving modes of the left lead up to the chemical
potential . Thus the transmission of electrons through the nanoscopic conductor gives rise to a current to the
right in the right electrode.

In the second step we have converted the integral over the wave vectors into an integral over the energy by making use of the density of states (DOS) projected onto band of the right lead. For one-dimensional systems the DOS is given by the inverse derivative of the dispersion relation of the band, , so that it cancels exactly with the group velocity of the band:

(4) |

where we have defined the transmission per conduction channel as

(5) |

On the other hand the right electron reservoir injects electrons into the *left*-moving
modes of the right lead up to the chemical potential and the transmission of electrons
through the device region gives rise to a left-directed current in the left lead:

(6) |

where now is the transmission probability of channel of the right lead:

(7) |

and is the transmission amplitude for a mode of the right lead to be transmitted into mode of the left lead. Because of time inversion symmetry the amplitude is the same as the the amplitude for the transmission from the left to the right electrode apart from a trivial phase factor. Hence the total transmission probability from the left to the right lead is equal to the total transmission probability from the right to the left lead :

(8) |

Furthermore the summed reflection probability for all electrons injected from the
right reservoir at some energy is . Therefore
the current composed of *backscattered* electrons (originating from the right reservoir)
and *transmitted* electrons (originating from the left reservoir) cancels exactly the
current of the *incoming* electrons coming in from the right electron reservoir. Analogously
the same holds true for the left lead. Thus the net current at some energy is zero when the current
there is current injection from *both* electrodes at that energy.

Thus assuming a positive bias voltage , so that , only electrons above give a net contribution to the total current. Since for energies above the only contribution to the net current through the right lead is the transmission current of electrons coming from the left, the total current for a given bias voltage is given by the famous Landauer formula:

(9) |

Taking the derivative with respect to the bias voltage one obtains the corresponding conductance:

(10) |

where is half the fundamental conductance quantum . The spin-degree of freedom of the electrons is contained in the index for the channels. Assuming a spin-degenerate system the transmissions for up- and down channels are equal, so that a factor of two appears when summing transmissions over the spin-degree of freedom, and one obtains the usual Landauer formula with as the proportionality constant. Here however, we are interested in magnetic systems, so the transmissions are spin-dependent.

The transmission amplitudes define an in general non-quadratic matrix . The square of this matrix defines a (quadratic) hermitian matrix called the transmission matrix:

(11) |

The channel transmissions are now just the diagonal elements of this transmission matrix, and summing up over all channel transmissions in the Landauer formula now corresponds to taking the trace of the transmission matrix:

(12) |

The transmission matrix is the central quantity in the Landauer formalism since it allows to calculate the electrical conductance and current-voltage characteristics of a nanoscopic conductor. Depending on the actual system a variety of methods exists for obtaining this quantity. For our purpose of describing transport through atomic- and molecular-size conductors the NEGF described in the next section is the most appropriate approach since it can be combined in a straight-forward manner with ab initio electronic structure methods like density functional theory (DFT) or the Hartree-Fock approximation (HFA) as implemented in standard quantum chemistry codes employing atomic orbital basis sets.

### 2 Non-equilibrium Green’s function formalism

In this section we will restate the Landauer approach to quantum transport in the language of one-body Green’s functions [66, 64]. To this end we will first introduce the Hamiltonian and overlap matrices for the transport problem in a basis set of localized atomic orbitals. We will then derive a Green’s function for the finite scattering region connected on both sides to semi-infinite leads. From the Green’s function (GF) one can then obtain the (reduced) density matrix and the transmission matrix. Here we will mainly follow the arguments presented by Paulsson in his introductory paper on the NEGF [67] but generalize them to non-orthogonal basis sets (NOBS) as commonly employed in quantum chemistry packages. A few other derivations of the Landauer formalism within the NEGF framework taking into account non-orthogonality of basis sets can be found in the recent literature [68, 69].

#### 2.1 Hamiltonian and overlap

We divide the system into 3 parts as shown in Fig. 3: The left lead (L), the right lead (R), and the intermediate region called device (D) containing the central scattering region (S). This scattering region can be given by e.g. a molecule coupled to metallic contacts, or simply a nanoscopic constriction as indicated in the figure.

We assume that the leads are only coupled to the scattering region but not to each other. Thus the device region must be chosen sufficiently large for that to be true. The Hamiltonian describing the system is then given by the matrix

(13) |

Furthermore we assume a non-orthogonal localized basis set. Assuming again no overlap between atomic orbitals in different leads the overlap of the atomic-orbitals of the system is given by the following overlap matrix:

(14) |

As indicated in Fig. 3 we subdivide the leads into unit cells (UCs) which must be chosen sufficiently large so that the coupling between non-neighboring unit cells can be neglected. Thus in general a UC consists of several primitive unit cells (PUCs). The Hamiltonian matrix of the left lead can be subdivided into sub-matrices in the following manner:

(15) |

Analogously the Hamiltonian of the right lead is given by the following matrix:

(16) |

In a similar way, the overlap inside the leads is given by the matrices

(17) |

and

(18) |

Furthermore the unit cell of each lead that is immediately connected to the scattering region (unit cell “” for the left and unit cell “” for the right lead) is included into the device part of the system.:

(19) |

In the next subsection we will show how to calculate the GF for the device part of the system as defined by the above Hamilton and overlap matrices.

#### 2.2 Green’s function for the open system

The one-body GF operator of a system is defined as the solution to the generalized inhomogeneous Schrödinger equation (see e.g the book by E. N. Economou, Ref. [66]):

(20) |

where is an (effective) one-body Hamiltonian, and is a complex number. When does not coincide with the eigenvalues of the Hamiltonian , the GF operator has the following formal solution:

(21) |

Obviously, for the GF operator has a pole and is thus not well defined. In this case one can define two GFs which are both solutions to eq. (20) by a limiting process. 1) The retarded GF is defined as:

(22) |

and 2) the advanced GF is defined as the hermitian conjugate of the retarded GF:

(23) |

is a real number which can be any of the energy eigenvalues of the Hamiltonian . When does not coincide with an eigenvalue of the two GFs reduce to the GF operator defined in eq. (21). In the basis of eigenstates of the Hamiltonian the GF operator becomes diagonal:

(24) |

The GF yields the complete information of a one-body system. For example, eq. (24) makes clear that the poles of the GF along the real axis represent the eigenvalues of the Hamiltonian . Thus by plotting along the real axis one can find all the eigenvalues in a certain energy range.

A very important quantity is the density of states (DOS) . The DOS can be calculated from the trace of the imaginary part of the GF along the real axis:

(25) | |||||

Another important quantity of the GF formalism is the spectral density which is defined as the difference between the retarded and the advanced GF:

(26) |

It is straight forward to show that the eigenstate representation of the spectral function on the real axis is given by:

(27) |

so that the trace of the spectral function directly gives the DOS. The spectral function can thus be seen as a generalized DOS.

The grand advantage of the GF formalism is that it allows one to calculate all properties of a one-body system without having to calculate the eigenstates of the Hamiltonian explicitly. Instead the GF can be calculated in any basis set by a matrix inversion for any value , eq. (21). It turns out that in many situations this is more convenient than to solve the whole eigenvalue problem. This is the case for the transport problem defined by the Hamiltonian and overlap matrices given in the previous subsection, eqs. (13) and (14).

The matrix corresponding to the GF operator in a non-orthogonal basis set is given by:

(28) |

where is the overlap matrix of the NOBS: . This, however, is a somewhat inconvenient definition for the calculation of the GF since it involves the inversion of the matrix. Instead, we define a new GF matrix by

(29) |

This GF can be calculated more conveniently from the much simpler equation

(30) |

For convenience we introduce the following energy-dependent matrix:

(31) |

Thus we obtain for the GF for the transport problem the following matrix equation which defines a system of equations for each sub-matrix of the total GF.

(32) | |||||

This matrix equation can be solved for each of the matrix elements (see App. \thechapter). For the device part of the GF we obtain:

(33) |

where we have introduced the *self-energies* of the leads
and
which describe the influence of the leads on the electronic
structure of the device. They can be calculated from the GFs
of the *isolated* left and right semi-infinite lead,
and ,
respectively:

(34) | |||||

(35) |

For later use, we also define the so-called coupling matrices:

(36) | |||||

(37) | |||||

Thus we have expressed the GF of the device region in terms of the device Hamiltonian and the Green’s
functions of the *isolated* leads. The term
can be interpreted as an effective Hamiltonian for the device region, and its energy dependence stems from the fact, that the
lifetime of an electron in the device region is now finite due to the coupling to the leads.
The technique of calculating the GF in parts by dividing the underlying Hilbert space into subspaces
is called partitioning technique and is shown in more detail in App. \thechapter where
we also list the other matrix elements of the GF.

Since the coupling between the left lead and the device is only due to coupling between the last unit cell () of left lead and the unit cell included in the device region, only the surface GF of the left lead, i.e. the GF projected into the unit cell , is necessary for calculating the self-energy . The same holds true for the right lead in an analogous manner. Furthermore, is different from zero only in the -region of the device, and analogously in the -region:

(38) |

where the non-zero matrix elements and of can be expressed in terms of the surface GFs of the left and right lead, and :

(39) | |||||

(40) |

The self-energies can be calculated iteratively by Dyson equations, as shown in App. \thechapter:

(42) |

Now we have achieved a description of the electronic structure of the device region in terms of one-body GFs which take into account the effect of the coupling of the leads to the device region. In the next section we will show how to obtain the (reduced) density matrix and subsequently the electron density of the device region from this GF. Thereafter we demonstrate, how to calculate the transmission matrix from the GF of the device region and the self-energies of the leads.

#### 2.3 Calculation of density matrix and electron number at equilibrium

The reduced density matrix of first order (in quantum chemistry often called charge density matrix) is obtained by tracing out all but one of the one-particle subspaces from the many-body density matrix :

(43) |

where is a set of one-body states.

At zero temperature the system will be in its (many-body) ground state , so that the density matrix becomes

(44) |

It is straight forward to show that in this case the reduced density matrix can be expressed as

(45) |

One can obtain a lot of information from the reduced density matrix about a many-body system in spite of the fact that it is actually a one-body operator. For example, one can calculate the expectation value of any one-body observable from the trace of the product of the reduced density matrix and the observable:

(46) |

The electron density is given by the diagonal elements of the reduced density matrix in the real space representation:

(47) |

Finally, the number of electrons is the trace of the density matrix.

In the Landauer approach we consider the electrons as a system of non-interacting quasi-particles, i.e. the Coulomb interaction between the electrons is only taken into account on the mean-field level. In the case of a system of non-interacting particles the many-body ground state of the system is given by a single Slater determinant:

(48) |

where are the eigenstates of the one-body Hamiltonian, is the vacuum ground state, and the chemical potential, so that the Slater determinant consists of all states with energies less than or equal to the chemical potential. In this case the reduced density matrix is diagonal in the one-body eigenstates :

(49) |

where the occupation number of the eigenstate is given by the Fermi distribution function, , and thus is either one for states below the chemical potential or zero for states above the chemical potential at zero temperature. Since we will make use of the reduced density matrix only but not of the full many-body density matrix we will refer to the reduced density matrix for the sake of simplicity from here on simply as the density matrix, unless otherwise stated.

Using the eigenstate representation of the GF, eq. (24), it is straight forward to show that the density matrix can actually be obtained from the GF of the system by integrating the imaginary part of the GF up to the chemical potential of the system:

(50) | |||||

Subsequently, the (standard) density matrix in a NOBS , is obtained by integration of the standard non-orthogonal GF matrix, :

(51) |

Analogously to the GF matrix we can define a new density matrix by

(52) |

which is thus obtained by integration of the non-standard GF :

(53) |

One can obtain the number of electrons from the trace of the density matrix. However, the trace of
an operator has to be taken in an orthogonal basis set, but has been defined in
terms of the density matrix in a NOBS . The standard procedure
for orthogonalizing atomic orbitals is the so-called *symmetric orthogonalization*
[70]. A density matrix in a NOBS will be transformed
to the density matrix in the orthogonalized basis according to:

(54) |

In the last step we have used the definition of the non-standard density matrix to obtain the transformation to the orthogonalized density matrix.

Applying the above transformation for the density matrix we find for the number of electrons the following expression:

(55) | |||||

where we have first exploited the invariance of the trace under commutation of matrices and then applied the above explained division into sub-matrices of the density matrix and the expression (14) for the overlap matrix. In analogy to the Mulliken analysis we can identify the number of electrons in the device, , in the left lead, , and in the right lead as follows:

(56) |

Thus due to the overlap of the device orbitals with the lead orbitals one also has to calculate the off-diagonal elements and of the density matrix in order to calculate the number of electrons in the device part. However, computing the off-diagonal elements of the density matrix would require calculation of the corresponding off-diagonal elements of the GF matrix in terms of the GF matrix of the device and the self-energies (see App. \thechapter) which are quite tedious expressions. Instead we will require charge neutrality only for the scattering region and not for the entire device region. That is the reason why we have included one unit cell of each lead into the device region in addition to the scattering region. Since the scattering region does not have an overlap with the two leads it is sufficient to calculate the density matrix of the device region to obtain the number of electrons inside the scattering region:

(57) |

Finally, we also have to define an appropriate DOS and projected DOS (PDOS) for the case of non-orthogonal orbitals. In order to be coherent with the above definition of the electron numbers in the respective subspaces of the leads and the device, it is best to adapt a similar Mulliken-like definition for the PDOS:

(58) |

where denotes some of the atomic orbitals of the system. If we are interested in the DOS projected onto some orbital of the scattering region, it suffices to calculate the GF and overlap matrix for the device region instead for the entire system, . And the DOS projected onto the entire scattering region is just .

#### 2.4 Non-equilibrium density matrix

In the previous section we have assumed that the system is in thermal equilibrium, i.e. the chemical potentials of the two electron reservoirs are equal: . Although in many cases one can estimate the conductance at small bias from the zero-bias conductance, i.e. in equilibrium, it would be very interesting to study the interplay between an electron current and the electronic structure of the nanoscopic conductor. Thus we have to derive an expression for the density matrix when the system is out of equilibrium, i.e. .

First, we calculate the response of the entire system to an incoming wave on the left lead
which we assume is a solution of the *isolated* left lead, i.e. .

(59) |

Because of the non-orthogonality of the lead orbitals with the device orbitals we now switch to a matrix presentation of the problem as done in the previous section. The incoming wave as a solution of the left isolated lead is given as an expansion over the lead orbitals:

(60) |

The response wave function on the other hand expands over the whole system:

(61) |

Thus in matrix representation the above Schrödinger equation for the response to an incoming wave reads:

(62) |

Since is a solution for the isolated left lead,

(63) |

we obtain finally:

(64) |

The left electron reservoir fills the incoming electron waves on the left lead up to the chemical potential of that reservoir which result in wavefunctions expanded over the whole system . Thus the density matrix due to injection of electrons from the left reservoir is given by:

(65) | |||||