Molecular electronics, a brief introduction (Lecture notes for a course on nanoelectronics, fall 2009)

Janne Viljas March 6, 2013

Contents 1 Introduction

2

2 Experiments and basic concepts

4

3 Elastic transport theory 3.1 Landauer-B¨ uttiker formalism . . . . . . . . . . . 3.2 Non-equilibrium Green function (NEGF) method 3.3 Surface Green’s functions (*) . . . . . . . . . . . 3.4 Simple applications of the “NEGF” theory . . . . 3.4.1 Breit-Wigner resonances . . . . . . . . . . 3.4.2 Fano resonance (with an antiresonance) . 3.4.3 Linear chain (*) . . . . . . . . . . . . . .

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9 10 11 13 14 14 15 16

4 Some further reading

18

A Time-evolution operators and Green’s functions (**)

19

B Derivation of the current formula (**) 21 B.1 Formal scattering theory . . . . . . . . . . . . . . . . . . . . . . . 21 B.2 Application to an LCR system . . . . . . . . . . . . . . . . . . . 22 B.3 Current formula for an LCR system . . . . . . . . . . . . . . . . 25 C Problems

26

1

Figure 1: A metal-molecule-metal junction formed of a dithiolated terphenyl derivative bonded to gold [111] tips. Methyl side groups are used to control the molecular conformation.

1

Introduction

As the size of electronic components keeps decreasing, one must face the reality that the miniaturization cannot be continued indefinitely. The ultimate limit is set by the atomic scale. However, with the conventional “top-down” fabrication methods and the usual semiconductor materials the practical limits may be reached well before that. It is then reasonable to ask how, if at all, progress may be continued. There are many suggestions for at least temporary solutions, including the use of carbon nanotubes or graphene. One of the visions of the field of molecular electronics, however, is to develop some completely new kinds of methods, by which the electronic circuits are constructed in a “bottom-up” fashion, with specifically designed molecules taking the roles of both the active components and of the interconnects. These could furthermore be brought together by some form of self-assembly. Another goal of this field is to be able to interface electronics with biological systems and to be able to perform tasks like electronic single-molecule recognition. These visions are presently mostly science fiction, though. Much more realistically, present-day molecular electronics could be described simply as the study of electron transport at the single-molecule scale. Furthermore, most theoretical and experimental studies so far have concentrated on the prototypical metal-molecule-metal junctions, such as the one illustrated in Fig. 1. Below we describe how such junctions may be constructed in practice, and have a look at what kinds of basic theoretical concepts and methods can be used for describing the transport of electrons in them. From the point of view of physics the molecules in such metal-molecule-metal (MMM) contacts are specific kinds of quantum dots. As very small objects they have discrete electronic spectra and single-electron charging energies reaching the electron volt range. In some cases it is possible to apply gate voltages to the molecule, and due to the high charging energy one can observe the characteristic

2

Figure 2: A donor-bridge-acceptor system for electron transfer.

Coulomb blockade phenomena even at room temperature. However, molecules are more than “just” quantum dots. The word “dot” carries with it the impression of a relatively structureless, even quasi-zero-dimensional object. Yet, the long-term goal of molecular electronics is precisely to take advantage of the endless variability of chemical compounds to design molecules having just the right properties for use as single-molecule electronic components. Thus molecular electronics is to a considerable degree chemistry, something that physicists tend to know little about. In particular the synthesis of the desired molecules can only be made by trained synthetic chemists. On the other hand, most chemists are not traditionally used to dealing with transport studies, and thus successful molecular electronics research often requires the close interaction between physicists and chemists. This makes molecular electronics a good example of an interdisciplinary field. Molecular electronics is usually seen as a rather young field, but it has some closely related predecessors. By our definition, these are not molecular electronics because they typically involve macroscopic numbers of molecules. They may be collected under terms like organic electronics or conducting polymers. These fields have already reached relative maturity, and even some important real-life applications have emerged, such as organic light-emitting diodes. In addition to these fields, for several decades already chemists have been studying so-called electron transfer reactions [1, 2]. In this context one is often concerned with a donor-bridge-acceptor (DBA) system, shown in Fig. 2. The donor is some molecule (reducdant) that donates an electron to an acceptor (oxidant) via a third molecule that is called a bridge. The conceptual leap involved in coupling the donor and the acceptor to macroscopic electrodes through which a constant charge current may be driven (and thus turning “transfer” into “transport”) is not very big. Often the birth of molecular electronics is placed in 1974, when the theoretical chemists Aviram and Ratner proposed to use a metal-D-B-Ametal system to rectify currents, analogously to a semiconductor p-n junction [3]. Rectifiers of this type are now known as Aviram-Ratner diodes. Such ideas remained theoretical curiosities up until the mid-1990s, when it finally became possible to experimentally fabricate contacts where a single molecule, or at most a few, bridges the gap between two metallic electrodes. We briefly discuss these methods below. Recently molecular rectifiers of the Aviram-Ratner-type have been realized experimentally,1 although their rectifying properties are so far not very good. Also other types of single-molecule electronic components such as 1 See

for example Ref. [4] and the “further reading” in Sec. 4.

3

Figure 3: Some experimental methods for forming atomic-size metal contacts: (a) when STM is used, the tip and surface metals may differ. (b) In MCBJ the contact is formed by breaking a thin metal wire by pulling the ends apart. (c) In electromigration a narrow metal contact is prepared on a substrate and the contact is then broken by driving a strong current through it. (d) Atomic contacts may also be formed electrochemically, by immersing the electrodes together with a counterelectrode in an electrolyte solution. In (a) and (b) it is easy to open or close the contact many times. In (c) and (d) it is easy to use the substrate as a “gate electrode” for molecules introduced in the contact. In (d) also electrochemical gating may be possible.

switches [5] have been studied, but they are equally far from applications at this point. More sophisticated experimental techniques must be developed before we can even hope to turn molecular electronics into a viable technology. On the other hand, the theoretical understanding of charge transport on the molecular scale is also facing big challenges. However, considerable progress has already been made in the past 10-15 years. In these notes we only touch upon some of the most basic issues.

2

Experiments and basic concepts

Vital to the experimental study of metal-molecule-metal junctions involving only a single molecule is the ability to form atomic-size metallic contacts. The most common methods for achieving this involve the scanning tunneling microscope (STM) and other scanning-probe techniques, mechanically controllable break junctions (MCBJ), and electromigration (see Fig. 3). Junctions may also be formed for example by electrochemical means, with the contact immersed in an electrolyte solution. The bare atomic contacts themselves, without any molecules, are interesting systems in their own right, exhibiting some hints of conductance quantization in units of the conductance quantum G0 = 2e2 /h. Since the Fermi wavelength in most metals is on the scale of 0.1 nm, atomicscale junctions are needed to see such effects. On the other hand, in atomic contacts it is possible to see the quantization effects even at room temperature, in contrast to semiconductor quantum point contacts. Indeed, many of the experiments in the field are carried out at room temperature, although the stability of the contacts is better in a cryostat. Atomic contacts of some metals (Au, Pt, Ir) are also known to form few-atom-long and single-atom-thick chains, called atomic wires. From the point of view of molecular electronics, atomic contacts act as important test systems.

4

Figure 4: A molecular contact may be formed by applying a solution containing the molecules to the space between the metallic electrodes. The molecules are designed so that when they encounter the metal (gold), the protection groups (squares) detach and the linker groups (circles) bond to the surface. In this way, sometimes a molecular bridge is formed between the two electrodes.

The single-molecule or few-molecule contacts may be formed in different ways. First, one may use an STM tip to just probe single molecules lying on a metal surface, without forming a bond between the tip and the molecule.2 It is also possible to prepare a self-assembled monolayer of molecules on the surface and then form a bond to some of them by approaching with the tip. Otherwise one may introduce the molecules to the atomic-scale gap of a broken atomic contact by applying a solution where the molecules are immersed (see cartoon in Fig. 4). In the MCBJ and STM methods it is then possible to adjust the gap between the electrodes until a contact with one of the molecules is formed. Choosing the correct materials for making this work in practice is a matter of experience in chemistry or surface science. Typically the “backbone” molecule is some organic compound, and at the ends of the molecule there are linker (anchor) groups that are designed to bond to the metal surface (see again Fig. 1). With Au electrodes the most often used linkers are thiol (-S-) and amino (NH2 -) groups. When the molecules are in the solution, the linkers are protected by some protection groups that detach upon contact with the metal. The problem with such methods of forming the molecular contact is that there is generally no way to see if the molecules are really there and forming a contact.3 Their presence must be inferred indirectly via the transport measurements, for example via some inelastic signatures in the current-voltage characteristics, as discussed further below. There is also no way to control the way in which the contact is formed, and so the geometrical arrangement of the atoms and the position of the molecule will be slightly different every time. This leads to strong statistical fluctuations in the measured transport properties, and it has become customary in the field of atomic contacts and molecular electronics to characterize the junctions in terms of their conductance histograms, obtained by repeating the contact formation many times. A sketch of such a histogram using Au electrodes is shown in Fig. 5. Here the peaks on the scale of G0 are signatures of the atomic contacts. Au single-atom contacts and atomic wires typically exhibit a conductance very close to 1G0 , which yields 2 There

may also be a thin insulating layer between the metal surface and the molecule. In this way it is possible to “image” molecular orbitals [6]. 3 Metallic atomic wires have been imaged with a transmission electron microscope (TEM).

5

Figure 5: Schematic illustration of a typical conductance trace obtained when breaking an MCBJ (left) and a corresponding histogram (right) for contacts based for example on Au or Ag, which have a single outer-shell (valence) electron. The large-scale structure is that of pure atomic point contacts and exhibits approximate conductance quantization in units of G0 = 2e2 /h. The high peak at G0 is due to single-atom contacts or atomic wires. If molecules are introduced in the junction, small peaks at conductances G  G0 corresponding to molecular contacts may be seen (inset).

the sharp peak. Signatures of molecules are usually seen as smaller peaks at conductances much below 1G0 . Multiple peaks may be visible, corresponding for example to multiple molecules conducting in parallel, or various molecular conformations [7, 8]. Indeed, the conductance of most MMM junctions based on organic molecules is very low. This may be understood based on the electronic structure of the molecules, which are typically “insulators” (insofar as it makes sense to use such terms for finite systems having a discrete spectrum.) A simplified energy-level diagram of a typical MMM junction is illustrated in Fig. 6. The electronic spectrum of a typical organic molecule has a gap of some electron volts between the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO). When the molecule is coupled to the metallic electrodes, some charge is transferred between the molecule and the electrodes, such that in equilibrium the chemical potential, or Fermi energy EF , of the electrodes lies somewhere in the HOMO-LUMO gap. The amount of charge transfer required for this is usually quite small, only some fraction of an electron charge e. Moving one of the molecular levels to resonance with EF , would be associated with a charge transfer on the order of e, but this is energetically very costly because the charging energy in molecular-scale systems in usually also on the order of electron volts.4 Thus, in the absence of a gate electrode by which the level alignment can be changed, the simplest approximation is to view the low-bias transport through the molecule as off-resonant tunneling.5 Let us illustrate the concept of molecular orbitals for the case of a hydrogen 4 Note however, that image charge effects may reduce the charging energies from those of the isolated molecule. 5 As we have mentioned in the introduction, gating has also been accomplished. Also, there are molecules where the HOMO or the LUMO can be very close to resonance.

6

unoccupied levels

LUMO

occupied levels

HOMO

EF

Figure 6: Energy diagram of a metal-molecule-metal junction. The Fermi energy EF lies between the “frontier orbitals” HOMO and LUMO.

molecule, H2 . A hydrogen atom has one electron in the 1s atomic orbital (AO) φ, with an energy  = −1 Ry= −13.6 eV below the vacuum level. When two ˆ j i of the atoms 1 and 2 are brought together, the Hamiltonian Hij = hφi |H|φ total system expressed in the two-AO basis φ1 , φ2 is of the form    ∆ HH2 ≈ (1) ∆  where ∆ < 0 is the “hopping” element between the AOs 1 and 2. The use of a local AO basis in this way is called a tight-binding (TB) approximation in solidstate physics, and linear combination of atomic orbitals (LCAO) in molecular physics. The Hamiltonian has the eigenvalues EI =  + ∆, EII =  − ∆ and eigenvectors     1 1 1 1 ψI = √ , ψII = √ (2) 2 1 2 −1 Here the symmetric eigenvector ψI with EI < EII corresponds to the “bonding” molecular orbital (MO) (σ) and the antisymmetric ψII to the “antibonding” molecular orbital (σ ∗ ). In this case these bonding and antibonding orbitals are also the HOMO and the LUMO, respectively. While understanding the absolute magnitudes of the conductances of MMM junctions theoretically has proved very difficult, there are many other aspects that are understood. These at the very least provide some proof that it is indeed the desired molecules that are being measured and not just a metal contact with some impurities. One possibility is to systematically change some chemical properties of the molecule and to repeat the measurement [9]. In this way it has for example been demonstrated that if the molecule is a short “oligomer”, with some repeated units (like the oligophenylene molecule of Fig. 1), then the conductance decays exponentially with the number of the units, i.e. the length of the molecule [10, 11]. This is consistent with the idea that the transport through the molecule may be described with quantum-mechanical 7

(b)

(a)

(c) eV

W

eV

G/G0

1/2

e

G/G0

G/G0

W

1/2 W

eV

e+2W

e e+W

eV

Figure 7: Three different cases for inelastic signals in the differential conductance G(V ) = dI/dV , depending on the lead-coupling strength of the molecular orbitals and their alignment with respect to the bias window: (a) Off-resonant, (b) resonant strong-coupling (or “metallic”), and (c) resonant weak-coupling. The gray areas denote the closest molecular orbitals and their lifetime broadening. The red arrows represent the amplitudes of the most important transport processes, which may be elastic (horizontal), involve the physical emission of one or more vibrations of energy Ω (step down), or involve a vibration only in a virtual intermediate state (step down followed by a step up). In low- and highconductance cases (a) and (b) the conductance steps at eV = Ω are in different directions. In case (c), assuming that the voltage only shifts the left chemical potential and that the energy  of the resonant level is measured relative to the right one, there is a main resonance peak at eV = , and additional satellite peaks at voltages eV =  + nΩ (n = 1, 2, 3, . . .). A single elastic transmission channel is assumed in all cases.

(off-resonant) tunneling.6 Similar length-dependence experiments have been made also for the so-called thermopower [13], and the agreement with theory seems to be good [14]. Systematic studies of this kind may also be performed by modifying the conformation or the electronic structure of the molecule with the help of functional side groups on the molecular backbone. They can act as “dopants” which tend to donate electrons to or withdraw them from the backbone. In this way the level alignments shown in Fig. 6 can be tuned to some extent even without an external gate electrode [15]. Finally, let us mention a very important diagnostic tool, which is based on the coupling of electrons to the local vibrations of the molecule. When the differential conductance G(V ) = dI/dV is measured up to high enough bias voltages V , one can often see signatures that originate from the inelastic transmission of electrons through the contact. The three most basic cases are explained in Fig. 7. These are usually referred to as inelastic electron tunneling spectroscopy 6 For long enough molecules, the exponential decay can turn into power-law decay, because then incoherent thermally activated processes may start dominating the transport [12].

8

(IETS), point-contact spectroscopy (PCS), or resonant IETS (RIETS), respectively. For the first two the following useful rule of thumb exists. If there is only one elastic transmission channel and the zero-bias conductance is roughly less than half of the conductance quantum, i.e. G(V = 0) . G0 /2 = e2 /h, then the signature of the emission of vibrations with energy Ω is an upward step in G at V = Ω/e [case (a), IETS]. This situation occurs if all the molecular orbitals are far from resonance, i.e. far from the voltage window. If G(V = 0) & G0 /2, then the step is downward [16, 17, 18, 19] [case (b), PCS]. This is the relevant case when one or more of the molecular orbitals are in resonance and very strongly coupled to the electrodes so that they are strongly broadened. A similar situation occurs in completely metallic point contacts, which explains the name PCS.7 The last case [case (c), RIETS] corresponds to a resonant, but weakly coupled level in the presence of strong electron-vibration coupling. In this case there are peaks rather than steps in the conductance, which are due to the resonant transmission of electrons while emitting n vibration quanta (see caption). Inelastic tunneling signals can be very useful for identifying the molecules or their presence in a contact. For example, when the molecular contact is for example stretched, or if varying isotopes are used for some of the atoms, the energies of the vibration modes change in a predictable way. The vibrational modes of molecules typically have energies in the range Ω = 1 − 100 meV and there are 3N modes if the number of atoms is N . Not all of these modes are seen in the conductance, however, but some selection or rather “propensity” rules determining the signal strengths have been developed [21]. It is not possible to go into further details of the phenomenology here. Some further reading is listed in Sec. 4.

3

Elastic transport theory

The theoretical treatment of electron transport in atomic-size systems is quite complicated. The main difficulty stems from the combination of an infinitely extended quantum system, which has very few symmetries (no periodic lattice except in the bulk of the leads), and which is being driven out of equilibrium. On top of this, electron-electron correlations (Coulomb interactions) and electronvibration coupling may be strong. It is impossible at present to take all of these aspects into account in a single theoretical framework. In the rest of these notes we view the transport as completely elastic and phase-coherent, in which case the theory reduces essentially to the scattering theory familiar from mesoscopic physics. Thus we consider the atomic nuclei to be fixed to their equilibrium positions, which is a reasonable first approximation at least at low temperature and low voltage bias.8 Description of gating and charging effects is only possible within a “mean-field” approximation, however. In terms of the quantum-dot language, the method is thus applicable only if the molecule is strongly enough 7 The difference between the cases G(V = 0) ≷ G /2 is sometimes called a “1/2 rule”. It 0 has recently been observed [20]. However, it is not exact. 8 This is the Born-Oppenheimer approximation.

9

coupled to the electrodes, so that the level broadening (or inverse lifetime) Γ exceeds the single-electron charging energy U , even though the thermal energy kB T would not. The method we will introduce below is often applied in connection with abinitio electronic structure methods, most commonly density functional theory (DFT). Such methods involve self-consistent iterations to determine the effective single-particle Hamiltonian of the interacting system. We will not go into any such details, but rather just illustrate the use of the general method with simple noninteracting “toy models”, where the parameters of the Hamiltonian are given.

3.1

Landauer-B¨ uttiker formalism

The standard language for phase-coherent elastic transport is the LandauerB¨ uttiker formalism, developed in the context of mesoscopic physics. In this formalism the current through a phase-coherent conductor connected to two particle reservoirs may generally be written in the form (see appendices) Z 2e dE τ¯(E)(fL (E) − fR (E)), (3) I= h where fL,R (E) = {exp[(E − µL,R )/kB T ] + 1}−1 are Fermi functions and τ¯(E) is the transmission function.9 The latter is the sum of the transmission P probabilities for all the transmission eigenchannels at energy E: τ¯(E) = n τn (E). From the knowledge of the transmission function, various experimentally measurable transport properties may be calculated. Most importantly, the lowtemperature zero-bias conductance G = (dI/dV )|V =0 is given by G=

2e2 2e2 X τ¯(EF ) = τn (EF ). h h n

(4)

where EF is the Fermi energy. Another example is the low-temperature thermopower S = −(∆V /∆T )I=0 , which is of the form of Mott’s law S=−

2 π 2 kB T τ¯0 (EF ) . 3e τ¯(EF )

(5)

where the prime denotes a derivative. Since the slope of τ¯(E) is negative at E just above the HOMO and positive just below the LUMO (see below), the sign of S may at least in principle be used to obtain information on the location of EF [22, 14]. Similar information may be obtained from the second derivative of the current. 9 In general τ ¯(E) depends also on the applied voltage V = (µL − µR )/e. We may neglect this if we only consider linear response.

10

Figure 8: Metal-molecule-metal junction: division of the geometry and the local basis into parts. Each atom (circles) may have several basis functions attached to it, for example s, p, and d-like functions.

3.2

Non-equilibrium Green function (NEGF) method

The transmission function may be calculated by using the scattering-matrix methods described elsewhere in this course. However, in order to calculate the scattering matrix itself for some given scattering potential, Green function methods and the so-called Fisher-Lee relation are typically employed. In this chapter we eliminate the scattering matrix altogether by employing a Greenfunction method, which is in widespread use in the field of molecular electronics. It is generally called the non-equilibrium Green function method (NEGF). However, we only present it in a simplified, single-particle form. The advantage of this method is that it may be quite straightforwardly generalized to include interactions effects, at least approximately. Furthermore, we view the problem in terms of a local TB (LCAO) basis instead of plane-wave like states. This is the most natural choice in atomic-scale systems that lack periodicity and cannot be described in terms of effective-mass models. It is also very convenient for developing general-purpose numerical methods and allows for a completely atomistic treatment of the electrodes. Assume the Hamiltonian of the system to be of the single-particle form 2

2

ˆ = − ~ ∇~r + Uef f (~r), H 2m

(6)

where the effective potential Uef f includes the nuclear potentials and the effects of other electrons on some self-consistent mean-field (for example Hartree) level. ˆ Below we are only concerned with its matrix elements Hjk = hj|H|ki, where the basis functions h~r|ji are assumed to be atomic-orbital-like real-valued functions, which are all localized around some atom. There may be several basis functions corresponding to one atom. In practice such a basis will almost always be nonorthogonal (hi|ji = Sij 6= δij ), but for simplicity we treat the basis as an orthonormal (and complete if needed) set. In the absence of magnetic fields (i.e. 11

assuming time-reversal invariance) the matrix elements Hij are real-valued, such that the corresponding matrix is symmetric: H T = H. Now let us partition the basis into L, C, and R parts (see Fig. 8). The Hamiltonian matrix may be partitioned accordingly, and assuming that there are no direct “hopping integrals” between L and R, we may write   HLL HLC 0 H = HCL HCC HCR  . (7) 0 HRC HRR Here HCC is of size M × M , where M is the total number of basis functions in the C region. For simplicity we do not include spin in the basis here. Spin will only appear as a degeneracy factor 2 in the current. The retarded Green function corresponding to the matrix H is defined as G(E) = [E + i0+ − H]−1 (see Appendix). It may be shown (see exercise) that the CC submatrix of G may be written as GCC (E) = [E − HCC − Σ(L) (E) − Σ(R) (E)]−1 ,

(8)

where Σ(X) (E) = HCX gXX (E)HXC ,

X = L, R

(9)

gXX (E) = [E + i0+ − HXX ]−1 ,

X = L, R.

(10)

and

Here gXX (E) is the Green function of lead X uncoupled to C, and the nonHermitian matrices Σ(X) (E) are called lead “self energies”.10 In the appendices it is shown that the transmission function of the system may now be calculated as follows: τ¯(E) = Tr[GCC (E)Γ(R) (E)G†CC (E)Γ(L) (E)] (11) Here Γ(X) (E) = i[Σ(X) (E) − Σ(X)† (E)] = −2Im[Σ(X) (E)],

X = L, R

(12)

are matrices describing the escape rates of electrons from C to the leads (see exercise). In the second form we made use of the symmetricity of Σ(L,R) (E). Since the matrices Γ(L,R) (E) are positive-definite (see exercise), we may rewrite X τ¯(E) = Tr[t† (E)t(E)] = τm (E), (13) m

where we define the matrix t(E) = [Γ(L) (E)]1/2 GCC (E)[Γ(R) (E)]1/2

(14)

10 This is because the equation has the form of the Dyson equation, appearing in the context of many-body perturbation theory.

12

Figure 9: Calculation of the surface Green function is generally based on assuming the surface to consist of repeated superlayers, and that the C part of the system is only coupled to the surface superlayer (here 0).

and where the τm (E) are the eigenvalues of the matrix t† (E)t(E). They are the transmission probabilities of the transmission eigenchannels mentioned above.11 In the exercises we study the properties of all these quantities further. Note that, although we assumed a local atomic-orbital basis above, the trace expression in Eqs. (11) and (13) is independent of the basis set, as long as the basis may be divided into the L, C, R parts. Although the matrices HXX , X = L, R are of size ∞ × ∞, the coupling elements HXC typically have a limited range and thus the inverse in gXX (E) is only needed for a few atomic layers near the surface. If the electrode is assumed to be a perfectly periodic, semi-infinite crystal, there are efficient numerical methods for calculating the surface Green function.

3.3

Surface Green’s functions (*)

If the leads in L and R regions are assumed to consist of ideal surfaces, the Green functions may be calculated as follows. Let us consider only one lead, say R, and drop the lead index. The surface is divided into repeated superlayers (Fig. 9), so that the Hamiltonian for the isolated surface is of the form   H0 H1 0 · · ·  †  H1 H0 . . . . . .    H= (15) . .. ..  0  . .   .. .. . . For simplicity we are assuming the surface to have a finite transverse extent, so that the matrices H0 and H1 are finite. (Otherwise a Fourier transformation could be performed in the transverse directions.) The full Green function of the lead is now g(E) = [E + − H]−1 , but we only want to know the surface block [g(E)]00 . Let us isolate the surface superlayer 11 The eigenchannels themselves are certain linear combinations of scattering states. See appendices for some discussion.

13

by writing H = Hsb + V where  H0 0 0   0 H0 H1  Hsb =   0 H† . . . 1  .. .. .. . . .

 ··· ..  .  , ..  . 



0

 † H1  V =  0  .. .

H1

0

0

0 .. .

0 .. .

..

 ··· ..  .  . ..  . 

(16)

.

Writing now [E + − H]g(E) = 1 and defining g˜(E) = [E + − Hsb ]−1 , we have [˜ g −1 (E) − V ]g(E) = 1 or g(E) = g˜(E) + g˜(E)V g(E).

(17)

Taking the 00 component of this yields the first of g00 (E) = g˜00 (E) + g˜00 (E)H1 g10 (E) g10 (E) = g˜11 (E)H1† g00 (E),

(18)

while the second one is needed to close the set of equations for g00 (E). Now the important point is to notice that here g˜11 (E) is the surface Green function for the surface where the first layer was removed. But since the surface is semi-infinite, g˜11 (E) = g00 (E). Thus we find g00 (E) = g˜00 (E) + g˜00 (E)H1 g00 (E)H1† g00 (E).

(19)

Here g˜00 (E) = [E + − H0 ]−1 . This is a second-order matrix equation for g00 (E), which may in principle be solved by numerical iteration. Below we solve it in a very simple case analytically. For numerical calculations also much more efficient methods than a direct iteration of Eq. (19) exist [23, 24, 25, 26].

3.4 3.4.1

Simple applications of the “NEGF” theory Breit-Wigner resonances

Let us demonstrate here the method for a simple model which considers a molecular junction as two resonant levels, one for the HOMO, and another for the LUMO (see Fig. 10). Thus, instead of a local basis, let us use the basis where   1 0 HCC = , (20) 0 2 with 1,2 being the HOMO and LUMO energies. Instead of specifying the other components of the Hamiltonian, let us further assume that the self-energies have the diagonal forms Σ(X) (E) = −idiag(γ1X /2, γ2X /2) (X = L, R) so that Γ(X) (E) = diag(γ1X , γ2X ) (X = L, R), where γ1L , γ2L , γ1R , γ2R are positive constants. The latter assumption is usually called a wide-band approximation, since it assumes the electrodes to have a flat density of states. (Real parts of 14

τ(E)

1

0.1

0.01 -2

-1.5

-1

-0.5

0 0.5 E / |ε1|

1

1.5

2

Figure 10: Two-level system and transmission with parameters γ1L = γ2L = γ1R = γ2R = 0.12 , 1 = −2 . the self energies were dropped because they may be absorbed into 1,2 .) Using these results Eq. (11) yields τ¯(E) =

X α=1,2

=

2 1 γαL γαR E − α + i(γαL + γαR )/2

γαL γαR . 2 (E − α ) + (γαL + γαR )2 /4 α=1,2

(21)

X

Thus in this model each of the two levels 1, 2 gives rise to its own, independent transmission channel, whose transmission probabilities have Lorentzian resonances, illustrated in Fig. 10. This type of resonance is often referred to as a Breit-Wigner resonance [27]. It should be noted that in practice this model only works if the two resonances are well-separated, such that the broadenings are much smaller than |1 − 2 |. Below, in Sec. 3.4.3, a similar double-peak structure is obtained from a more detailed calculation, but there the result cannot be written as a sum of two Lorentzians. 3.4.2

Fano resonance (with an antiresonance)

There is another type of resonance that we are now in a position to discuss, called a Fano resonance. This type of resonance appears in many contexts where there is an interference between two transmission paths, one via a “continuum” and another via a discrete level. Consider the situation of Fig. 11, which can effectively occur in some molecular junctions with certain types of side groups attached to the backbone of the molecule, for example. Here a single level is coupled to another “impurity” level, which does not necessarily have a direct 15

1

τ(E)

0.1 0.01 0.001 0.0001

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

E / |ε|

Figure 11: Level system giving rise to a Fano resonance and transmission with parameters γL = γR = 0.1i ,  = −i , and t = i . contact to the electrodes. Now  HCC =

 t t i

 (22)

and we couple only  to wide-band leads by assuming Σ(X) (E) = −idiag(γX /2, 0) (X = L, R) and so Γ(X) (E) = diag(γX , 0) (X = L, R), where γL , γR are positive constants. The calculation of the transmission function is left as an exercise, but the result is illustrated in Fig. 11. We see that for E ≈  there is a normal Breit-Wigner-type resonance. However, at E → i there is a combination of both a peak and a dip. In the peak the transmission probability goes to 1, and in the dip it goes to zero! These two features are called a resonance and an antiresonance, respectively. Such asymmetrical peak-dip structures are usually called Fano resonances [28]. In general neither the resonance nor the antiresonance need be perfect. For some examples, see Refs. [29, 30, 31, 32]. 3.4.3

Linear chain (*)

Perhaps the simplest example of using an atomic-orbital basis is that of a linear atom chain within a phenomenological nearest-neighbor tight-binding model [33]. Chain models of this type may be used to describe for example atomic gold chains [17] or polyacetylene [34]. The specific model that we consider below describes, with a suitable choice of parameters, a metal-H2 -metal system [35]. The system under consideration is depicted in Fig. 12. Here an infinite atom chain is divided into two semi-infinite “leads” and a central two-atom chain (a “dimer”). Let us assume that the on-site energies (diagonal elements of H) are everywhere , and the hopping elements t are otherwise the same, except 16

Figure 12: A dimer connected to semi-infinite atomic chains.

between the L, C, R subsystems, where they are denoted by u. The Hamiltonian of the center is    t HCC = d d , (23) td d the coupling Hamiltonians between the subsystems are    ··· 0 u 0 0 HCL = , HCR = ··· 0 0 u 0

 ··· , ···

(24)

T T with HLC = HCL and HRC = HCR , and the lead Hamiltonians are

 HLL

   = .  .. ···

.. ..

.

. 0

..

.

..

.

 t

 .. .  0 ,  t 

 HRR



t

0 .. .

 t  = 0  .. .

 ..

.

..

.

..

.

 ··· ..  .  .  

(25)

(L)

The self energies for the two leads are now of the form Σmn (E) = σL (E)δ1m δ1n (R) and Σmn (E) = σR (E)δ2m δ2n , with σX (E) = u2 [gXX (E)]00 (X = L, R), and where 0 refers to the first (or “surface”) atom of the semi-infinite chain. Correspondingly, the CC component of the Green function is found to be of the form  −1 E − d − σL (E) −td GCC (E) = −td E − d − σR (E) 1 = (26) (E − d − σL (E))(E − d − σR (E)) − t2d  −1 E − d − σR (E) td × . td E − d − σL (E) Using Eq. (11) and dropping the subscript“CC”, the transmission function is X τ¯(E) = Gkl (E)γR (E)δ2l δ2n G∗nm (E)γL (E)δ1m δ1k (27) klmn = γL (E)γR (E)|G12 (E)|2 ,

17

1

0.5

0.8 τ(E)

Re(g00)t, Im(g00)t

1

0

0.6 0.4

-0.5

0.2

-1

0 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 (E-ε) / (2t)

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 (E-ε) / (2t)

Figure 13: Left: Real and imaginary parts of the Green function on the surface (0) atom of a semi-infinite wire. Right: Transmission probability for a perfectly periodic chain (t = u, solid line) and for a “dimer” chain (u = 0.5t, dotted line). In both cases d = , td = t > 0. where γL,R (E) = −2ImσL,R (E). Assuming σL,R (E) = σ(E) and γL,R (E) = γ(E), we may write τ¯(E) =

t2d γ 2 (E) . |E − d − td − σ(E)|2 |E − d + td − σ(E)|2

(28)

For the calculation of σ(E), consider lead R for instance. The Hamiltonian HRR given above is a special case of Eq. (15). The surface component of the Green function is thus determined from Eq. (19), which simplifies to the secondorder scalar equation 2 t2 g00 (E) − (E + − )g00 (E) + 1 = 0.

This is solved by  " # r 2   1 E− E−  − 1 , |E − | > 2|t|  2|t|  |t| 2|t| − sgn(E − ) " # r g00 (E) =  2   1 E−  1 − E− − i , |E − | < 2|t|  2|t|  |t| 2|t|

(29)

(30)

which may be simplified by defining (E −)/2|t| = cos φ. The real and imaginary parts of g00 (E) are shown in Fig. 13 together with the transmission function calculated from Eq. (28). If d =  and td = u = t, then the transmission is exactly 1 inside the band |E − | < 2|t|. This is the case of a completely periodic, ballistic wire. Otherwise, Breit-Wigner-like resonances may be seen at approximately the energies d ± td of the bonding and antibonding orbitals.

4

Some further reading

By now there is a vast literature on molecular electronics, but very few elementary discussions of the physics in book form. A clear and simple discussion 18

of the theoretical concepts is presented in Ref. [36]. Another book, covering both theory and experiment has recently appeared [37]. Some general review articles are given as Refs. [38, 39, 40, 41, 42, 43] and metallic atomic contacts are reviewed in Ref. [44]. Some specific topics are listed below. • Reviews about electron transfer and its relation to molecular electronics may be found in Refs. [1, 2]. The original paper on molecular rectifiers by Aviram and Ratner is Ref. [3]. • The non-equilibrium Green function method in the local-basis representation was introduced in Refs. [45, 46]. Elementary discussions of the method are given in Refs. [36, 47]. A thorough discussion of non-equilibrium Green-function methods may be found for example in Ref. [48]. The details of one density-functional implementation of the elastic-transport theory may be found in Ref. [49]. The method may also be extended for describing inelastic (electron-vibration) effects [50, 17, 51, 18]. • We have completely neglected many important topics, such as Coulombblockade effects in molecular quantum dots. Plenty of literature exists, of course [52, 53, 54]. Another interesting subtopic concerns the effects of light on molecular contacts [5, 55, 56, 57]. • One of the fancier new subtopics is DNA sequencing with transverse transport measurements. This goes in the direction of molecular recognition and other potential biochemical applications. A review is given by Ref. [58].

A

Time-evolution operators and Green’s functions (**)

In this appendix a brief introduction is given to the concept of Green functions in nonrelativistic, single-particle quantum mechanics. This concept is very closely related to the time-evolution operator. A very thorough treatment of Green functions is given in Ref. [59]. For simplicity let us consider a system with a time-independent Hamiltonian H. (Note: we drop all hats on operators.) Then the time evolution of a Schr¨ odinger-picture state vector |ψ(t)i is given by |ψ(t)i = U (t, t0 )|ψ(t0 )i,

(31)

where the time-evolution operator U (t, t0 ) satisfies the homogeneous equation of motion ∂ (32) i~ U (t, t0 ) − HU (t, t0 ) = 0. ∂t Since we must have U (t, t) = 1 (the unit operator), the solution is 0

U (t, t0 ) = e−iH(t−t )/~ . 19

(33)

This expression for U (t, t0 ) is valid also for t < t0 and thus it describes also the evolution (or propagation) of the system backward in time. Let us now define two new operators 1 G± (t, t0 ) = ± θ(±(t − t0 ))U (t, t0 ), i

(34)

where θ(t) is the Heaviside step function. They satisfy the inhomogeneous equation of motion i~

∂ ± G (t, t0 ) − HG± (t, t0 ) = ~δ(t − t0 ), ∂t

(35)

where we used (d/dt)θ(t) = δ(t). Quantities satisfying such an equation with a delta function source term are called Green’s functions, or in this case Green’s operators. We furthermore call G± the retarded and advanced Green operators or propagators, because G+ only propagates forward in time (and hence effects caused by sources are always “retarded”, i.e. they occur later in time, which is known as causality) and G− propagates backward. Note that due to U † (t, t0 ) = U (t0 , t) the two Green functions are Hermitian conjugates of each other: [G+ (t, t0 )]† = G− (t0 , t).

(36)

Note furthermore that in the case of the time-independent Hamiltonian (i.e. stationary state), U (t, t0 ) depends on the separation of the two times, U (t, t0 ) = U (t − t0 , 0), and therefore also G± (t, t0 ) = G± (t − t0 , 0).

(37)

Solving the equation of motion is now simplified by making a Fourier transformation. We define the pair of forward and backward transformations for any function f (t) as (the integrals are from −∞ to ∞) Z dt +iEt/~ e f (t) F (E) = ~ Z (38) dE −iEt/~ e F (E). f (t) = 2π When transforming the functions G± (t − t0 , 0), the time integrand is oscillatory in one of the two integration limits and we have to introduce damping factors 0 e∓η(t−t ) to the functions G± (t − t0 , 0) to make the integrals convergent. At the end of the calculation η → 0+ . So, setting t − t0 → t: Z dt +i(E±iη)t/~ ± ± e G (t, 0) G (E) = ~ Z (39) dE −iEt/~ ± e G (E). e∓ηt/~ G± (t, 0) = 2π In this way the equation of motion (35) becomes [(E ± iη)1 − H]G± (E) = 1. 20

(40)

Solving this for G± (E) we finally have G± (E) = [(E ± iη)1 − H]−1 .

(41)

The symmetry (37) now becomes [G+ (E)]† = G− (E). Due to the existence of the symmetry, it is only necessary to introduce one of the functions. Elsewhere in these notes we choose G+ and call it simply G(E). Furthermore, we only need the definition (41) in the energy representation and simplify notation so that G(E) = [E + − H]−1 = [E + i0+ − H]−1 . Note that this quantity is essentially what in mathematics is called the resolvent of H.

B

Derivation of the current formula (**)

Here we present a simple derivation for the Landauer formula, based on singleparticle scattering theory. A similar simplified derivation could also be made based on the so-called generalized Golden Rule. The more rigorous derivation could be given in terms of many-body non-equilibrium (Keldysh) techniques. However, that is outside the scope of the course. Note that we will use a simplified notation where for example H may refer to the Hamiltonian operator or one of its matrix representations.

B.1

Formal scattering theory

For details on the methods presented in this section, see for example Ref. [59]. Consider a Hamiltonian of the form H = H0 + V.

(42)

Here H0 is the “unperturbed” Hamiltonian and V some perturbation. Below we specify what these are in the specific problem that we are interested in. Both H0 and H may have both discrete and continuous parts in their spectra, but here we only consider the continuous spectrum, represented by the continuous energy variable E. Further, we assume that if E belongs to the continuous part of the spectrum of H0 , then it will also belong to the continuous part of the spectrum of H. (The discrete part could be treated with the usual time-independent “bound-state perturbation theory” [60].) Let us denote the eigenstates of H0 having energy E with |φl (E)i, i.e. H0 |φl (E)i = E|φl (E)i. Here the index l enumerates all the states that are degenerate, i.e. having the same energy E. Similarly, the eigenstates of the full Hamiltonian H are |ψl (E)i, i.e. H|ψl (E)i = E|ψl (E)i. These eigenstates are connected via the so-called Lippmann-Schwinger equation |ψl (E)i = |φl (E)i + G0 (E)V |ψl (E)i,

(43)

where G0 (E) = [E + i0+ − H0 ]−1 is the Green function (operator) of the unperturbed system. This may be rewritten |ψl (E)i = [1 − G0 (E)V ]−1 |φl (E)i, 21

(44)

or |ψl (E)i = (1 + G0 (E)V + G0 (E)V G0 (E)V + · · · )|φl (E)i = (1 + G(E)V )|φl (E)i = (1 + G0 T (E))|φl (E)i,

(45)

where G(E) = [E + i0+ − H]−1 is the Green function of the perturbed system, T (E) = V + V G0 (E)T (E) = V + V G(E)V is the so-called transition matrix or “T matrix”, and W (E) = 1 + G(E)V is sometimes referred to as the “Møller wave operator”. Note the one-to-one correspondence between the unperturbed and the perturbed eigenstates. In the following we may assume the states |φl (E)i and |ψl (E)i to be “energy-normalized” such that for example P R hψl (E)|ψl0 (E 0 )i = δll0 δ(E − E 0 ) and 1 = l dE|ψl (E)ihψl (E)|. If this formalism is applied to the standard scattering problem, where H0 is the free-particle Hamiltonian and V a localized scattering potential, the eigenstates |φl (E)i are the plane-wave-like “asymptotic” states for various incoming momentum directions l, while |ψl (E)i include also the outgoing scattered waves. The states |ψl (E)i are therefore called scattering states. Below, the identification of H0 , V , and the states is made a bit differently. Finally, let us introduce the spectral representation of G(E) as XZ |ψl (E 0 )ihψl (E 0 )| . G(E) = dE 0 (46) E + i0+ − E 0 l

Defining also the spectral function (operator) A(E) = i[G(E) − G† (E)] we have, using 1/(E − E 0 + i0+ ) = P/(E − E 0 ) − iπδ(E − E 0 ), X A(E) = 2π |ψl (E)ihψl (E)|.

(47)

(48)

l

This like a density-of-states operator. In particular, hx|A(E)/2π|xi = P is something 2 |hx|ψ (E)i| is the (local) density of states projected onto state |xi. l l

B.2

Application to an LCR system

Now let us apply the above formal notation to the problem of calculating the current through some mesoscopic or atomic-scale junction. Before actually deriving the current formula, we need some definitions. Let us assume that the basis {|ji} may be divided into there distinct sets corresponding to the left (L) and right (R) leads and a central system (C). We will further assume the basis to be orthonormal, although not necessarily complete. We then choose H0 to be the Hamiltonian corresponding to an “uncoupled” junction. Its matrix representation in the basis is of the block form   HLL 0 0 HCC 0 , H0 =  0 (49) 0 0 HRR 22

where HLL and HRR are ideally “∞ × ∞” matrices, while HCC is finite. Similarly the perturbation V is then made up of the coupling elements   0 HLC 0 0 HCR  V = HCL (50) 0 HRC 0 and H = H0 + V . We may also formally write Eqs. (49) and (50) as H0 = HLL + HCC + HRR

(51)

V = HCL + HLC + HCR + HRC ,

(52)

and † HCX

with HXC = (X = L, R) and H = H0 + V independently of the basis. The expansions of the terms of Eqs. (51) and (52) in the |ji basis are of the form XX HXY = hjk |jihk|, X, Y = L, C, R, (53) j∈X k∈Y

with the matrix elements hjk = hj|H|ki. Here we may identify the projection operator Pj = |jihj| to basis state |ji. Let us also define projectors X PL,C,R = |jihj|. (54) j∈L,C,R † They are Hermitian (PL,C,R = PL,C,R ) and satisfy PL + PC + PR = 1 as well 2 = PL,C,R . We also have relations like PR HRC = HRC , PR HCR = 0 as PL,C,R and so on. The unperturbed lead eigenstates |φl (E)i are now eigenstates of HLL or HRR . Thus they are localized on one of the leads, i.e. for example PR,C |φl (E)i = 0 for an eigenstate of HLL . In principle there are no restrictions on the type of the leads i.e. of H0 , but they should support a continuous spectrum of eigenstates in the energy range of interest. This is only possible if the dimension of HLL and HRR is infinite, which is, of course an idealization of reality. The spectrum of HCC , on the other hand, is discrete. In this description only the perturbed system (H) is capable of transmitting particles between the L, C, and R parts, whereas in the unperturbed system (H0 ) all electrons arriving from the leads are reflected at the uncoupled surfaces. Thus the eigenstates |φl (E)i of H0 are not simply the right-propagating or left-propagating free-particle states, but ones that include the reflected waves (i.e. they are “standing waves”). It is thus as though the unperturbed system would include impenetrable potential barriers between the subsystems and the perturbation, given by V , would correspond to the removal of these barriers. This is the opposite of the usual scattering situation described above. Let us now consider the CC component of the Green function

GCC (E) = [E + − HCC − Σ(L) (E) − Σ(R) (E)]−1 , 23

(55)

where Σ(X) (E) = HCX gXX (E)HXC ,

(56)

X = L, R

and gXX (E) = [E + − HXX ]−1 ,

(57)

X = L, R.

We call Σ(X) (E) a “self energy”, and gXX (E) is the Green function of lead X uncoupled to C. We also define Γ(X) (E) = i[Σ(X) (E) − Σ(X)† (E)] = −2Im[Σ(X) (E)] = HCX aXX (E)HXC ,

X = L, R,

(58)

where † aXX (E) = i[gXX (E) − gXX (E)] = 2π

X

|φl (E)ihφl (E)|,

X = L, R.

(59)

l∈X

Here the notation “l ∈ X” means that the state |φl (E)i is an eigenstate of HXX . Then we also have ACC (E) = i[GCC (E) − G†CC (E)] = iGCC (E)[(G†CC (E))−1 − (GCC (E))−1 ]G†CC (E) (L)

= GCC (E)[Γ

(E) + Γ

(R)

(E)]G†CC (E)

=

(L) ACC (E)

(60) +

(R) ACC (E),

where ACC (E) = GCC (E)Γ(X) (E)G†CC (E), (X)

X = L, R.

(61)

This is the spectral density corresponding to scattering states originating from lead X = L, R only. This may be shown explicitly by defining A(X) (E) as X A(X) (E) = 2π |ψl (E)ihψl (E)|, X = L, R. (62) l∈X

Then, inserting the Lippmann-Schwinger equation and using PC |φl (E)i = 0 we have X (X) ACC (E) = PC A(X) (E)PC = PC G(E)HCX aXX (E)HXC G† (E)PC l∈X (63) † (X) = GCC (E)Γ (E)GCC (E). Using the projection operators we also have GCC (E) = PC G(E)PC , for example.

24

B.3

Current formula for an LCR system

For the derivation of the current we follow Ref. [61] (see P also Ref. [62]). The probability for the particle be in the R region is ρR (t) = j∈R |cj (t)|2 , where cj (t) are the expansion coefficients of an arbitrary Schr¨odinger-picture state |ψ(t)i in the local basis. Using the projector PR this may be written as ρR (t) = hψ(t)|PR |ψ(t)i. The probability current flowing to/from the R region is the time derivative of this, ρ˙ R (t). ˜ and P˜R (t) = Switching now to the Heisenberg picture with |ψ(t)i = U (t)|ψi † −iHt/~ ˜ ˜ Thus U (t)PR U (t), where U (t) = e , we may write ρR (t) = hψ|P˜R (t)|ψi. ˜ the time derivative in ρ˙ R (t) only acts on PR (t) now. We can define the time derivative of the projector as the operator for the probability current through the C-R interface: i ˜ i ˜ J˜R (t) ≡ P˜˙R (t) = [H, P˜R (t)] = (H CR (t) − HRC (t)). ~ ~

(64)

Here we used Eqs. (51) and (52) and some of the properties following Eq. (54). After having identified the form of the current operator in this way, we can switch back to the Schr¨ odinger picture in the following, if we like.12 Hence the current operator is simply written as JR = ~i (HCR − HRC ). Now let us consider a scattering state |ψl (E)i originating from the left (“l ∈ L”) at energy E. The current carried by this state (per energy unit) is jlL→R (E) = hψl (E)|JR |ψl (E)i.

(65)

Using the Lippmann-Schwinger equation, PR |φl (E)i = 0, and PR G0 (E)PR = gRR (E), we have PR |ψl (E)i = PR |φl (E)i + PR G0 (E)V |ψl (E)i = gRR (E)HRC |ψl (E)i.

(66)

Then jlL→R (E) = = =

= =

i hψl (E)|(HCR − HRC )|ψl (E)i ~ i hψl (E)|(HCR PR − PR HRC )|ψl (E)i ~ i [hψl (E)|HCR gRR (E)HRC |ψl (E)i ~ † − hψl (E)|HCR gRR (E)HRC |ψl (E)i i hψl (E)|[Σ(R) (E) − Σ(R)† (E)]|ψl (E)i ~ 1 hψl (E)|Γ(R) (E)|ψl (E)i. ~

(67)

12 In fact the assumed “picture” is irrelevant here, because we a dealing with stationary states with the same energy. In Schr¨ odinger picture they are all of the form |ψl (E, t)i = e−iEt/~ |ψl (E)i, where the states |ψl (E)i are eigenstates of the time-independent Schr¨ odinger equation: H|ψl (E)i = E|ψl (E)i. The time-dependences may be dropped since the exponentials cancel in the matrix elements anyway.

25

Now take any orthonormal basis |ni that is complete in the C subspace. Then P we may insert 1 = n |nihn| and rearrange to find jL→R (E) =

X

jlL→R (E) =

l∈L

1 X 2π hψl (E)|Γ(R) (E)|ψl (E)i h l∈L

1X 1 (L) (L) = hn|ACC (E)Γ(R) (E)|ni = Tr[ACC (E)Γ(R) (E)] h n h =

(68)

1 1 Tr[GCC (E)Γ(L) (E)G†CC (E)Γ(R) (E)] = τ¯L→R (E). h h

In the last line we identified the transmission function τ¯L→R (E). A similar equation may be derived for scattering states originating from the right X jR→L (E) = jlR→L (E) l∈R

1 1 = Tr[GCC (E)Γ(R) (E)G†CC (E)Γ(L) (E)] = τ¯R→L (E). h h

(69)

The net charge current from R to L is then obtained by weighting the scattering states with their respective occupation probabilities, i.e. the Fermi functions of the corresponding leads, and by integrating over energy Z 2e I L→R = dE[¯ τL→R (E)fL (E) − τ¯R→L (E)fR (E)]. (70) h Here −e < 0 is the electron charge and we have added a factor 2 for the spin degeneracy of the scattering states. In the case of time-reversal-invariant systems H T = H etc. such that τ¯L→R (E) = τ¯R→L (E) = τ¯(E).

(71)

in which case the current formula simplifies to the usual Landauer formula.

C

Problems • Problem 1 By expanding the Landauer-B¨ uttiker expression for the charge current I with respect to small voltage and temperature differences ∆V and ∆T , derive the expression for the low-temperature Seebeck coefficient S = −(∆V /∆T )I=0 , given in the text. The result is analogous to Mott’s law for bulk conductors. • Problem 2 Derive the result GCC = [E + − HCC − Σ(L) − Σ(R) ]−1 and the forms of Σ(L,R) given in the text. Start from the definition [E + i0+ − H]G = G[E +i0+ −H] = 1, partition the basis into L, C and R parts, and assume † HLR = HRL = 0. 26

• Problem 3 Show that the “escape-rate” matrices Γ(L,R) (E) = i(Σ(L,R) (E)−Σ(L,R)† (E)) are Hint: Use the spectral decomposition gXX (E) = P positive-(semi)definite. + −1 |li(E + i0 −  ) hl|, where |li and l are the eigenvectors and eigenl l values of HXX (X = L, R), and the result 1/(x + i0+ ) = P/x − iπδ(x), where P denotes principal value. Note: For infinitely extended electrodes the spectrum is continuous and the sum should actually involve an integration, but you may neglect this for simplicity. The solution also demonstrates how Γ(L,R) (E)/~ are related to the Golden-Rule escape rates of an electron from C to the leads. • Problem 4 Show that the results ACC = GCC [Γ(L) +Γ(R) ]G†CC = G†CC [Γ(L) +Γ(R) ]GCC hold, where ACC = i(GCC − G†CC ) is the spectral function. Use (one † † of) √ these to√ derive the relation t t + r r = 1, where (for example) t = (L) (R) Γ GCC Γ is the transmission matrix from R to L. What is the expression for the reflection matrix r? Using this relation demonstrate that the transmission eigenvalues satisfy 0 ≤ τm (E) ≤ 1 as probabilities should. • Problem 5 Show that the model discussed in Sec. 3.4.2 leads to a Fano resonance in the transmission function. • Problem 6 Consider a linear single-orbital tight-binding chain with on-site energies , nearest-neighbor hopping element t, and lattice constant a. The Hamiltonian is thus of the form Hij = δi−j,0 + t(δi−j,−1 + δi−j,1 ). Assume first a finite chain length L = N a and periodic boundary conditions and then take the limit N → ∞. (i) Show that the electronic dispersion relation is of the form k =  + 2t cos(ka). (ii) Using this result, calculate P the bulk density of states (DOS) per length, defined by Nb (E) = L1 k δ(E − k ). Compare the result to the local density of states (LDOS) at the end of a semi-infinite chain, which is defined as Ns (E) = − π1 Img00 (E), g00 (E) being the previously calculated Psurface Green function. (iii) Calculate the right-going current I> = − 2e k>0 vk fk and show that the group velocity L k vk = ~1 d and the DOS cancel each other, giving a transmission 1 for any dk energy E within the band −2|t| < E −  < 2|t|. P Note: As a side note, we may also write Nb (E) = − π1 L1 k Im(E + i0+ − k )−1 = − π1 Imgbb (E), where gbb (E) = (E + i0+ − H)−1 bb is the bulk Green function on some bulk site b. • Problem 7 ˆ If the basis used for representing Hij = hi|H|ji is nonorthogonal, so that Sij = hi|ji = 6 δij , how should one modify the expressions for G = [E + − H]−1 and thus for GCC and Σ(L,R) ?

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• Problem 8 Estimate the order of magnitude for the frequency of relative torsional oscillations of a biphenyl molecule H5 C6 -C6 H5 . In this oscillation mode the relative torsion angle ϕ between the two phenyl (benzene) rings oscillates around its minimum-energy value ϕ0 . How low should the temperature be in order not to excite the vibrations? Hints: The Hamiltonian is of the form H = 12 Ired ϕ˙ 2 + 12 k(ϕ − ϕ0 )2 , where Ired is the reduced moment of inertia (analogous to a reduced mass) and k the torsional spring constant. Estimate Ired and use the value k ∼ 1 eV. The C-C bond length is ≈ 0.14 nm and C-H bond length is of the same order. • Problem 9 Estimate the single-electron charging energy e2 /2C for a molecule-sized quantum dot by using a parallel-plate estimate C = 0 A/d for the capacitance. All dimensions may be taken to be on the order of 1 nm. (The use of the parallel-plate formula or even the concept of a purely geometrical capacitance is wrong here, but it gives you an idea of the orders of magnitude.) • Problem 10 Effect of symmetry on transport. The electronic structure of a benzene ring is described by the Hamiltonian 

 γ  γ H= 0  0 0

γ  0 γ 0 0

γ 0  0 γ 0

0 γ 0  0 γ

0 0 γ 0  γ

 0 0  0  γ  γ 

2

4 γ γ γ 1 γ γ 6 γ 3 5

(i) Calculate the eigenvalues and eigenvectors of this Hamiltonian and plot the vector components. (ii) Imagine that this molecule is coupled to electrodes via two of the six atoms. Calculate the transmission function by assuming wide-band self energies first on atoms 1 and 6 (“para” position) and then on atoms 1 and 5 (“meta” position). Is there a difference and why? • Problem 11 Estimate the strength of the magnetic field needed for seeing AharonovBohm oscillations in transport through a metal-molecule-metal contact consisting of a benzene molecule. The length of the C-C bond is 1.4 ˚ A. Hint: Don’t worry if you get a crazy result. It’s probably correct.

References [1] K. V. Mikkelsen and M. A. Ratner, Chem. Rev. 87, 113 (1987). 28

[2] D. M. Adams, L. Brus, C. E. D. Chidsey, S. Creager, C. Creutz, C. R. Kagan, P. V. Kamat, M. Lieberman, S. Lindsay, R. A. Marcus, R. M. Metzger, M. E. Michel-Beyerle, J. R. Miller, M. D. Newton, D. R. Rolison, O. Sankey, K. S. Schanze, J. Yardley, and X. Zhu, J. Phys. Chem. 107, 6668 (2003). [3] A. Aviram and M. A. Ratner, Chem. Phys. Lett. 29, 277 (1974). [4] M. Elbing, R. Ochs, M. Koentopp, M. Fischer, C. von H¨anisch, F. Weigend, F. Evers, H. B. Weber, and M. Mayor, Proc. Natl. Acad. Sci. 102, 8815 (2005). [5] S. J. van der Molen, H. van der Vegte, T. Kudernac, I. Amin, B. L. Feringa, and B. J. van Wees, Nanotechnology 17, 310 (2006). [6] P. Liljeroth, J. Repp, and G. Meyer, Science 317, 1203 (2007). [7] X. D. Cui, A. Primak, X. Zarate, J. Tomfohr, O. F. Sankey, A. L. Moore, T. A. Moore, D. Gust, G. Harris, and S. M. Lindsay, Science 294, 571 (2001). [8] B. Xu and N. J. Tao, Science 1301, 1221 (2003). [9] L. Venkataraman, J. E. Klare, M. S. Hybertsen, and M. L. Steigerwald, Nature 442, 904 (2006). [10] D. J. Wold, R. Haag, M. A. Rampi, and C. D. Frisbie, J. Phys. Chem. B 106, 2813 (2002). [11] L. Venkataraman, J. E. Klare, I. W. Tam, C. Nuckolls, M. S. Hybertsen, and M. L. Steigerwald, Nano Lett. 6, 458 (2006). [12] S. H. Choi, B. Kim, and C. D. Frisbie, Science 320, 1482 (2008). [13] P. Reddy, S.-Y. Jang, R. A. Segalman, and A. Majumdar, Science 315, 1568 (2007). [14] F. Pauly, J. K. Viljas, and J. C. Cuevas, Phys. Rev. B 78, 035315 (2008). [15] L. Venkataraman, Y. S. Park, A. C. Whalley, C. Nuckolss, M. S. Hybertsen, and M. L. Steigerwald, Nano Lett. 7, 502 (2007). [16] N. Agra¨ıt, C. Untiedt, G. Rubio-Bollinger, and S. Vieira, Phys. Rev. Lett. 88, 216803 (2002). [17] J. K. Viljas, J. C. Cuevas, F. Pauly, and M. H¨afner, Phys. Rev. B 72, 245415 (2005). [18] M. Paulsson, T. Frederiksen, and M. Brandbyge, Nano Lett. 6, 258 (2006). [19] L. de la Vega, A. Mart´ın-Rodero, N. Agra¨ıt, and A. L. Yeyati, Phys. Rev. B 73, 075428 (2006). 29

[20] O. Tal, M. Krieger, B. Leerink, and J. M. van Ruitenbeek, Phys. Rev. Lett. 100, 196804 (2008). [21] N. Lorente, M. Persson, L. J. Lauhon, and W. Ho, Phys. Rev. Lett. 86, 2593 (2001). [22] M. Paulsson and S. Datta, Phys. Rev. B 67, 241403(R) (2003). [23] F. Guinea, C. Tejedor, F. Flores, and E. Louis, Phys. Rev. B 28, 4397 (1983). [24] J. N. Schulman and Y.-C. Chang, Phys. Rev. B 27, 2346 (1983). [25] M. P. L. Sancho, J. M. L. Sancho, J. L. M. Sancho, and J. Rubio, J. Phys. F: Met. Phys. 15, 851 (1985). [26] H. L. Skriver and N. M. Rosengaard, Phys. Rev. B 43, 9538 (1991). [27] G. Breit and E. Wigner, Phys. Rev. 49, 519 (1936). [28] U. Fano, Phys. Rev. 124, 1866 (1961). [29] B. Babi´c and C. Sch¨ onenberger, Phys. Rev. B 70, 195408 (2004). [30] J. Kim, J.-R. Kim, J.-O. Lee, J. W. Park, H. M. So, N. Kim, K. Kang, K.-H. Yoo, and J.-J. Kim, Phys. Rev. Lett. 90, 166403 (2003). [31] P. Jarillo-Herrero, J. A. van Dam, and L. P. Kouwenhoven, Nature 439, 953 (2005). [32] T. A. Papadopoulos, I. M. Grace, and C. J. Lambert, Phys. Rev. B 74, 193306 (2006). [33] V. Mujica, M. Kemp, and M. A. Ratner, J. Chem. Phys. 101, 6849 (1994). [34] W. P. Su, J. R. Schrieffer, and A. J. Heeger, Phys. Rev. Lett. 42, 1698 (1979). [35] R. H. M. Smit and et al., Nature 419, 906 (2002). [36] S. Datta, Quantum Transport: Atom to Transistor (Cambridge University Press, Cambridge, 2005). [37] J. C. Cuevas and E. Scheer, Molecular Electronics: An Introduction to Theory and Experiment (World Scientific, 2010). [38] M. Galperin, M. A. Ratner, A. Nitzan, and A. Troisi, Science 319, 1056 (2008). [39] N. J. Tao, Nature Nanotech. 1, 173 (2006). [40] A. Nitzan and M. A. Ratner, Science 300, 1384 (2003).

30

[41] A. Nitzan, Annu. Rev. Phys. Chem. 52, 681 (2001). [42] U. Peskin, J. Phys. B 43, 153001 (2010). [43] S. A. Claridge, J. J. Schwartz, and P. S. Weiss, ACS Nano 5, 693729 (2011). [44] N. Agra¨ıt, A. Levy-Yeyati, and J. M. van Ruitenbeek, Phys. Rep. 377, 81 (2003). [45] C. Caroli, R. Combescot, P. Nozieres, and D. Saint-James, J. Phys. C: Solid St. Phys. 4, 916 (1971). [46] C. Caroli, R. Combescot, D. Lederer, P. Nozieres, and D. Saint-James, J. Phys. C: Solid St. Phys. 4, 2598 (1971). [47] S. Datta, Electronic Transport in Mesoscopic Systems (Cambridge, Cambridge, 1995). [48] H. Haug and A.-P. Jauho, Quantum Kinetics in Transport and Optics of Semiconductors (Springer, 2008), 2nd edition. [49] F. Pauly, J. K. Viljas, U. Huniar, M. H¨afner, S. Wohlthat, M. B¨ urkle, J. C. Cuevas, and G. Sch¨ on, New. J. Phys. 10, 125019 (2008). [50] C. Caroli, R. Combescot, P. Nozieres, and D. Saint-James, J. Phys. C: Solid St. Phys. 5, 21 (1972). [51] M. Galperin, M. A. Ratner, and A. Nitzan, J. Chem. Phys. 121, 11965 (2004). [52] S. Kubatkin, A. Danilov, M. Hjort, J. Cornil, J.-L. Br´eas, N. Stuhr-Hansen, P. Hede˚ ard, and T. Bjørnholm, Nature 425, 698 (2003). [53] M. Poot, E. Osorio, K. O’Neill, J. Thijssen, D. Vanmaekelbergh, C. van Walree, L. Jenneskens, and H. van der Zant, Nano Lett. 6, 1031 (2006). [54] D. R. Ward, G. D. Scott, Z. K. Keane, N. J. Halas, and D. Natelson, J. Phys. Condens. Matter 20, 374118 (2008). [55] Z. Ioffe, T. Shamai, A. Ophir, G. Noy, I. Yutsis, K. Kfir, O. Chesnovsky, and Y. Selzer, Nature Nanotech. 3, 727 (2008). [56] D. R. Ward, N. J. Halas, J. W. Ciszek, J. M. Tour, Y. Wu, P. Nordlander, and D. Natelson, Nano Lett. 8, 919 (2008). [57] J. K. Viljas, F. Pauly, and J. C. Cuevas, Phys. Rev. B 77, 155119 (2008). [58] M. Zwolak and M. D. Ventra, Rev. Mod. Phys. 80, 141 (2008). [59] E. N. Economou, Green’s functions in Quantum Physics (Springer, 2006), 3rd edition.

31

[60] J. J. Sakurai, Modern Quantum Mechanics (Addison-Wesley, 1994), revised edition. [61] M. Paulsson and M. Brandbyge, Phys. Rev. B 76, 115117 (2007). [62] T. N. Todorov, J. Phys.: Condens. Matter 14, 3049 (2002).

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