Matthew G. Reuter, Thorsten Hansen, Tamar Seideman and Mark A. Ratner- Molecular Transport Junctions...

8/3/2019 Matthew G. Reuter, Thorsten Hansen, Tamar Seideman and Mark A. Ratner- Molecular Transport Junctions with Se

1/12

Molecular Transport Junctions with Semiconductor Electrodes: Analytical Forms for

One-Dimensional Self-Energies

Matthew G. Reuter, Thorsten Hansen, Tamar Seideman, and Mark A. Ratner*

Department of Chemistry, Northwestern UniVersity, EVanston, Illinois 60208-3113

ReceiVed: December 30, 2008; ReVised Manuscript ReceiVed: January 29, 2009

Analytical self-energies for molecular interfaces with one-dimensional, tight-binding semiconductors are derived,along with analytical solutions to the electrode eigensystems. These models capture the fundamental differencesbetween the transport properties of metals and semiconductors and also account for the appearance of surfacestates. When the models are applied to zero-temperature electrode-molecule-electrode conductance, junctionswith two semiconductor electrodes exhibit a minimum bias threshold for generating current due to the absenceof electrode states near the Fermi level. Molecular interactions with semiconductor electrodes additionallyproduce (i) non-negligible molecular-level shifting by mechanisms absent in metals and (ii) sensitivity of thetransport to the semiconductor-molecule bonding configuration. Finally, the general effects of surface stateson molecular transport are discussed.

IntroductionThe transfer of charge between an adsorbed molecule and a

bulk substrate is a key process in a wide range of chemicalreactions. For example, charge separation in dye-sensitized solarcells occurs at the interface of dye molecules and titaniumdioxide nanoparticles.1 Electrochemical processes additionallyoccur at metal surfaces.2 Substrate-molecule interactions, usefulfor both their fundamental importance and their potentialapplications, clearly need to be understood.

A typical scanning tunneling microscopy experiment bringsa metal tip into the vicinity of a surface-adsorbed molecule,where, under bias, current flows through the molecule as ittraverses the tip-substrate gap. The past decade has witnessedimmense interest in such molecular transport situations,3-7 andmost work in molecular electronics has emphasized the criticalimportance of the molecule-electrode interfaces.8-14 Not onlydoes the electronic coupling between the molecule and thesurface control the magnitude of the current, it also influencesthe electronic structure of the molecule.

To date, theoretical descriptions, and to a lesser extentexperimental efforts, have largely focused on systems with twometal electrodes,8-11,13-17 often gold. Interesting physics, how-ever, appears when one or both leads are replaced withsemiconductors,18-25 including negative differential resistance19,20

and rectification.18 The industrial dominance of semiconductors,notably silicon, suggests the development of hybrid electronics,where single molecules or organic thin films are integrated withconventional silicon technology.26-29 Semiconductor surfaceswith adsorbed molecules often lend themselves to near-perfectcharacterization and, with clever choices of semiconductormaterials and dopants, permit customization of the band structureand transport properties to the specific application. Semiconduc-tors also introduce mechanisms for interesting molecularvibrational dynamics, along with approaches for optical con-trol.30

A detailed understanding of molecule-

electrode couplingrequires more than the simple tunneling barrier view of thetransport junction. In the coherent tunneling regime, thetunneling current is often well-described by the Landauer-Imryequation,31-33 where the molecule is treated using Greensfunctions. The molecule-surface interactions are indirectlydescribed by the impact they have on the molecule, which isformally accomplished by a self-energy,

that modifies the molecular Greens functions. The real part ofthe self-energy, (E), describes the shift of a molecular energylevel due to hybridization with the electrode. This molecularlevel shifting is negligible for metals11 in the wide band limit;however, recent studies have shown molecule-semiconductorinteractions to be more significant.22,25 The spectral density,(E), represents the broadening of a molecular level inducedby the electrode and is effectively the density of electrode statesweighted by the square of the molecule-electrode coupling.The self-energy also captures the effects of surface states.

Contemporary computational approaches use electronic struc-ture calculations of the self-energies for investigating moleculartransport junctions.8,10,11,15,23,24,34,35 These approaches, whilebroadly applicable, can be complicated by the treatment of

electron correlation and by basis set errors, and additionally byband bending and by doping when semiconductors are consid-ered. At the expense of quantitative accuracy, much physicalinsight can be gained from analytical models of the self-energy.Newns36 derived a simple analytical model for the self-energyof an interface between a molecule and a one-dimensional, tight-binding metal electrode, building on earlier work by Anderson.37

The introduction of alternating site energies or intersite couplingshas generalized this metal model to one-dimensional semicon-ductors, where such alternations yield band gaps in the modelsdensities of states.38-40

This is the starting point for our work. Following Newns,we derive an analytical expression for the self-energy of a

Part of the George C. Schatz Festschrift.* To whom correspondence should be addressed. E-mail: ratner@

northwestern.edu.

(E) ) (E) -i

2(E) (1)

J. Phys. Chem. A 2009, 113, 46654676 4665

10.1021/jp811492u CCC: $40.75 2009 American Chemical SocietyPublished on Web 03/26/2009


2/12


3/12

where fL(R)(E;V) is the Fermi function of the left (right) electrode.Each Fermi function depends on the electrodes chemicalpotential, which in turn relies on the applied bias. Two limitsallow simplifications of eq 7. First is the limit of zerotemperature, fL(R)(E;V) f (-E + EF ( eV/2), where (x) isthe Heaviside (step) function. The use of plus/minus signs herearises from the arbitrary left and right labels applied to theelectrodes. The second is the small bias limit, where we neglectthe transmission functions dependence on the bias, T(E;V) fT(E). Applying these limits,

Newns-Anderson (NA) Metal Model. A concise reviewof the Newns-Anderson model will prove indispensable in thederivations presented later. Representing a chain ofNidenticalmetal sites in a tight-binding picture, the Newns-Anderson

Hamiltonian is

All of the atomic levels here have the same energy, allowingus to take Ri ) 0 without any loss in generality ( ) 0). FromBlochs theory, we expect the eigenstates ofHNA to be

for constants A and B. Clearly, j|k ) Aeikj + Be-ikj.To determine A and B, we create virtual sites 0 and N + 1

such that the eigenstate amplitude disappears at these sites, 0|k) N+1|k ) 0. From these conditions, we see that k isquantized, k ) kn ) n/(N + 1), with n ) 1, 2, ..., N, that

once normalized, and that kn ) 2 cos(kn) is the eigenvalue of|kn.

The eigenstates, with Vkn ) 21/2(N+ 1)-1/2 sin(kn), and eq 2

are used to determine NA(E). In the limit of Nf , where knessentially becomes continuous,

for |E| e 2||. NA(E) ) 0 for all other E. The center of themetal band is , and the bandwidth is 4||.

We now use the electrode Greens function formulation ofNA(E) to determine NA(E). Since all sites in this one-dimensional chain are identical, eq 4 becomes

which produces (dropping the superscript)

As foreshadowed earlier, we obtain GNA(E) up to a choice of

sign. GNA(E) is complex when E2

- 42

< 0 (|E| < 2||), andthe negative sign is chosen to make NA(E) g 0; see eqs 1 and2. Parenthetically, this choice yields eq 12 once we introducethe two factors of from V and V.

When the molecular site level is energetically far from themetal band, we expect the surface to induce negligible moleculareffects, that is, NA(E) f 0 as |E| f . Only one sign choicefor each interval, E < -2|| and E > 2||, satisfies thisrequirement, allowing the total specification

where

We note that NA(E) specifies the sign choices and that, forthis simple model, the Hilbert transform of NA(E) can beanalytically evaluated, yielding the same result. Figure 2 showsNA(E) and NA(E).

Semiconductor Models. Several perturbations of the Newns-Anderson metal model have been used for modeling semi-conductors.38-41,47,48 One such model, introduced by Kouteckyand Davison (KD),38 alternates both the site energies and

intersite couplings, doubling the unit cell of the one-dimensionalcrystal. This model may consider each unit cell as an atom,

I(V) )2eh

-

dET(E;V)[fL(E;V) - fR(E;V)] (7)

I(V) )2eh

EF-eV/2

EF+eV/2 dET(E) (8)

HNA ) j)1

N-1

|jj+1| + h.c. (9)

|k ) A|k+ + B|k-

)A j)1

N

eikj|j + B j)1

N

e-ikj|j(10)

|kn ) 2N + 1 j)1

N

sin(knj)|j (11)

NA(E) ) limNf

2n)1

N22

N + 1sin2(kn)(E - 2 cos(kn))

) 420

dksin2(k)(E - 2 cos(k))

)2

242 - E2

(12)

Figure2. Theself-energy,NA(E),foramoleculeandaNewns-Andersonmetal. NA(E) (green, solid line) peaks at E ) 0 to a value of 22/||,and NA(E) (blue, dashed line) realizes its extrema of (2/|| at E )(2||.

GNAi (E) )

1

E - 2GNAi (E)

GNA(E) )E ( E2 - 42

22(13)

NA(E)

2)

E

22+ NA(E)

E2 - 42

22(14)

NA(E) ) (-2||-E) - (E - 2||)

Semiconductor-Molecule Transport Junctions J. Phys. Chem. A, Vol. 113, No. 16, 2009 4667


4/12

with the two sites states corresponding to s and p orbitals. Thecoupling between the s and p orbitals in the same unit celldescribes their overlap, and the coupling across unit cellsdescribes the interatomic bonding. Neighboring s-s and p-pinteractions are neglected in the simple tight-binding model.For convenience, we will assume there are 2N atomic sites inthe system, a moot distinction in the limit of N f . TheHamiltonian for this system is

The restriction ) 0 here allows a reduction in the number ofparameters; only one (R) is needed to describe the disparatesite energies (R and -R), as opposed to the more general caserequiring two.

The KD model has three important limits. First is thecombined limit R f 0 and 1 f 2 , which produces the

Newns-Anderson model. Second is the limit 1 f 2 .This alternating site (AS) model has been used to describetitanium dioxide,40 where the site energies (R and -R) cor-respond to the different atoms. The AS Hamiltonian ( ) 0) is

Lastly is the limit Rf 0, where the model alternates bonds (AB).

The AB model has been used to model silicon and germanium,39

where the bond disparities (1 and2) relate to orbital hybridization(the s and p orbitals become degenerate, as is the case when theyhybridize). The AB Hamiltonian ( ) 0) is

Surface States. When a crystal is cleaved into two noninter-acting parts, symmetries break and surfaces form. The surfacescan exhibit dangling bonds, potentially leading to reconstructions,

and surface states, with densities localized near the surface. Twoprincipal types of surface states, Tamm49 and Shockley50 states,are very similar in effect but caused by different physics. Tammstates appear when the site energies and intersite couplings nearthe surface are sufficiently perturbed from their bulk values. Theseperturbations are collectively called the surface potential. Alter-natively, Shockley states result from band crossings and appearin the gap between crossed bands. Additional contrasts betweenTamm and Shockley surface states are discussed in refs 38 and47. With no surface potential in the KD model (also AB and AS),any surface states here will be of the Shockley type.

Surface states are properties of the electrode alone, as opposedto the molecule-electrode interface, and can be identified by anypoles of the electrode Greens function.51 Koutecky calculated that

a surface state appears in the KD model at E ) R when |1| e|2|.38 It follows that a surface state also appears at E ) 0 when

|1| < |2| in the AB model. In these cases, the localized bondbetween sites 0 and 1 is of the stronger bond type and creates asurface state of nonbonding character when it ruptures duringsurface formation.38 Furthermore, this surface state moves to a bandedge at E ) R in the AS model (1 )2), representing the pointof band crossing. A surface state does not appear when a weakerbond is broken (|1| > |2|). As a small digression, the surface statesfor these one-dimensional models can also be termed end states.Just as two-dimensional surface states appear for three-dimensional

solids, these end states are zero-dimensional, only having densitiesat the ends of the chain. Experiments have recently observed suchend states.52

Semiconductor-Molecule Self-Energies

The derivation of the self-energy for the Koutecky-Davison(KD) semiconductor-molecule interface will follow the meth-odology of the Newns-Anderson (NA) metal-molecule inter-face: we solve the Hamiltonian eigensystem and use its solutionsto calculate the spectral density. We then obtain KD(E) by theelectrode Greens function method. We finally take the limitsR f 0 and 1 f 2 to explore AB and AS junctions,respectively.

Koutecky-Davison (KD) Model. The two-fold alternationof the sites suggests that two Bloch states are required fordescribing the system. If we first restrict our attention tooutgoing states (scattering states obeying retarded boundaryconditions, denoted by a + superscript), these two states are

and

where |k+ ) Co|k+,o + Ce|k+,e. Since we want |k+ to be aneigenstate ofHKD, we solve the secular equation,

to find that

The sign function in eq 18 forces both the density of states (notshown) and the spectral density (vide infra) to remain non-negative for all E, as required by definition and eq 2,respectively. Using these eigenvalues, the secular equation, and

the normalization condition |Co|2 + |Ce|2 ) 1, we determine theoutgoing eigenstates as

HKD ) R j)1

N

|2j-12j-1| - R j)1

N

|2j2j| +

[1 j)1

N

|2j-12j| + 2 j)1

N-1

|2j+12j| + h.c.] (15)

HAS ) R j)1

N

|2j-12j-1| - R j)1

N

|2j2j| +

[ j)1

2N-1

|jj+1| + h.c.] (16)

HAB ) 1 j)1

N

|2j-12j| + 2 j)1

N-1

|2j+12j| + h.c.

(17)

|k+,o )1

Nj)1

N

eik(2j-1)|2j-1

|k+,e ) 1N

j)1

N

eik2j|2j

0 ) |k+,o|HKD|k

+,o - k+,o|HKD|k

+,e

k+,e|HKD|k

+,o k

+,e|HKD|k+,e

- |

) | R - 1eik

+ 2e-ik

1e-ik

+ 2eik

-R - |

k2

) R2

+ 12

+ 22

+ 212 cos(2k)

k ) sign[cos(k)]R2 + 12 + 22 + 212 cos(2k)(18)

4668 J. Phys. Chem. A, Vol. 113, No. 16, 2009 Reuter et al.


5/12

|k+ )(1e

ik+ 2e

-ik)|k+,o + (k - R)|k+,e

(k - R)2 + 12 + 22 + 212 cos(2k))

1

Nj)1

N (1eik

+ 2e-ik)eik(2j-1)|2j-1 + (k - R)e

ik2j|2j

(k - R)2 + 12 + 22 + 212 cos(2k)(19)

We obtain the incoming eigenstate equivalent,

from a very similar process.Following the Newns-Anderson approach, we write the Hamiltonian eigenstates as |k ) A|k+ + B|k-, where

Again introducing the virtual sites 0 and 2N + 1 such that (0 |k ) (2N+1 |k ) 0, we see that

where k is quantized by

with n ) 1, 2, ..., 2N. An additional solution with ) R can arise from 0|k ) 0, indicating the possible presence of the surfacestate.

The Hamiltonian eigenstates are then

where Akn is the normalization constant. For the case where 1 ) 2, the kn satisfy kn ) n/(2N + 1) and Akn ) 2/(2N + 1)1/2 (an

equivalent quantization condition to the Newns-Anderson case for 2N sites, after simplification).53 Even though changes in 1 and2 vary the spacings between the kn, we will assume they are uniform (and hence the normalization constants are invariant) for the

spectral density calculation.54

This assumption is needed for the conversion of a discrete sum to a continuous integral.Having the eigenvalues and the eigenvectors, the spectral density can be calculated using eq 2, as detailed in the Appendix.Accordingly

for [R2 + (1 - 2)2]1/2 e |E| e [R2 + (1 + 2)2]1/2. KD(E) ) 0 for all other E.Here, we see that the center of the band gap is at E ) 0, suggesting represents the center of the band gap. We also note that

KD(E) simplifies to NA(E) in the combined limit of R f 0 and 1 f 2 , as desired. For parametrization purposes, the bandgap is given by 2[R2 + (1 - 2)2]1/2, and the valence (or conduction) band width is given by [R2 + (1 + 2)2]1/2 - [R2 + (1 -2)2]1/2. Having three parameters and two nonlinear conditions, there may be flexibility in choosing values, and a third criterion from

the band structure may be prudent. One immediate limitation of this model is that the valence and conduction bands have equalband widths.

|k- )1

Nj)1

N (1e-ik

+ 2eik)e-ik(2j-1)|2j-1 + (k - R)e

-ik2j|2j

(k - R)2 + 12 + 22 + 212 cos(2k)

2j-1|k )A(1e

ik+ 2e

-ik)eik(2j-1) + B(1e-ik

+ 2eik)e-ik(2j-1)

N(k - R)2 + 12 + 22 + 212 cos(2k)

2j|k )(k - R)(Ae

ik2j+ Be-ik2j)

N(k - R)2 + 12 + 22 + 212 cos(2k)

2j-1|kn )2A

N

1 sin[2knj] + 2 sin[2kn(j - 1)]

(kn - R)2

+ 12

+ 22

+ 212 cos(2kn)

2j|kn )2A

N

(kn - R) sin(2knj)

(kn - R)2

+ 12

+ 22

+ 212 cos(2kn)

1 sin[2kn(N + 1)] + 2 sin[2knN] ) 0 (20)

|kn ) Akn j)1

N (1 sin[2kn j] + 2 sin[2kn(j - 1)])|2j-1 + (kn - R) sin(2kn j)|2j

(kn - R)2

+ 12

+ 22

+ 212 cos(2kn)(21)

KD(E) )

2

22

[R2 + (1 + 2)2

- E2][E2 - R2 - (1 - 2)

2]

(E - R)2(22)



6/12

KD(E) is also asymmetric with respect to the interchange of1 for 2 and of R for -R. For this disparity, a careful review ofthe Hamiltonian shows that, for 2N sites, there is one more 1 bond than there is 2, demonstrating the difference between the twobond types. The sensitivity to R stresses the importance of the surface. As we will thoroughly explore later, these asymmetries hintat the importance of the molecule-semiconductor bonding configuration to the molecule-electrode interactions.

We now use eq 4 to deduce an analytic form for KD(E). Due to the alternations, there are two electrode Greens functions: onefor the odd-indexed sites and another for those even,

We only need GKD1 (E) for calculating KD(E) since the molecule solely couples to site 1 in the tight-binding Hamiltonian (V is 0 forthe other sites). Correspondingly

Following the discussion for the NA model, we choose the positive radical for -[R2 + (1 + 2)2]1/2 e Ee -[R2 + (1 - 2)2]1/2

and the negative for [R2

+ (1 - 2)2

]1/2e Ee [R

2

+ (1 + 2)2

]1/2

to make KD(E) g 0. Similarly, the negative branch ensuresthat KD(E) f 0 as |E| f .The only remaining sign selection pertains to the band gap region. From Koutecky,38 we expect a surface state at E ) R only

when |1| e |2|. Translating to requirements of the Greens function, GKD1 (E) should have a pole at E ) R when |1| e |2| and bewell-behaved when |1| > |2|. The denominator of GKD1 (E) is irrefutably 0 at E ) R, meaning GKD1 (E) has a removable singularitywhen its numerator is 0 and a first-order pole otherwise. Furthermore, the radical simplifies to |12 - 22| at E ) R, making thepositive branch the desired choice here for all combinations of 1 and 2. Finally

where

As with KD(E), KD(E) f NA(E) in the combined limit of R f 0 and 1 f 2 .55

Alternating Site (AS) and Alternating Bond (AB) Models. The above procedure could be repeated in the analysis of the ASand AB models; however, it is far easier to take the limits 1 f 2 and R f 0, respectively. For the AS model, we get

for |R| < |E| e (R2 + 42)1/2, and AS(E) ) 0 otherwise. Similarly

with

Given the reduction in the number of parameters, a semiconductor material can be represented by choosing 2| R| to be the bandgap and (R2 + 42)1/2 - |R| to be the valence (conductance) band width. We note that R is only specified in magnitude. Asexpected from Koutecky,38 the bulk band edges have extended to (R, and the pole in AS(E) at E ) R indicates the surface

state. Figure 3 displays AS(E) for positive and negative R, highlighting the dependence of the surface states location on thesite energy of the surface.

GKD1 (E) )

1

E - R - 1

2G

KD

2 (E)

GKD2 (E) )

1

E + R - 22GKD

1 (E)

GKD1 (E) )

E2

- R2

- 12

+ 22

( [E2 - R2 - (1 - 2)2][E2 - R2 - (1 + 2)2]22

2(E - R)(23)

KD(E)

2)

E2

- R2

- 12

+ 22

+ KD(E)[E2 - R2 - (1 - 2)2][E2 - R2 - (1 + 2)2]22

2(E - R)(24)

KD(E) ) (R2

+ (1 - 2)2

- E2) - (E2 - R2 - (1 + 2)

2)

AS(E)

2)

1

2(R2 + 42 - E2)(E + R)

E - R(25)

AS(E)

2)

E2

- R2

+ AS(E)(E2

- R2)(E2 - R2 - 42)

22(E - R)(26)

AS(E) ) (R2

- E2) - (E2 - R2 - 42)



7/12

Turning to the AB model,

for |1 - 2| e |E| e |1 + 2| and AB(E) ) 0 otherwise.Additionally

where

We can choose 1 and 2 such that 2|1 - 2| is the materialband gap and |1 + 2| - |1 - 2| is the valence (conduction)band width. Similar to the AS model, these specificationsambiguously determine two values for the material, and ,with two ways to assign them to 1 and 2. Furthermore, thesurface state expectedly appears at E ) 0 when |2| > |1|. Theexistence of the surface state is the primary difference when 1and 2 are interchanged, although the self-energy also scaleswith 2-2. Finally, Figure 4 depicts AB(E) for both |1| > |2|and |1| < |2|.

Comparison to Previous Results. A previous article41

reported significantly different spectral densities for the AS andAB models from those reported here. In that derivation, thecorrect Bloch state energies were obtained; however, it was thenassumed that Vkn sin(kn) for the spectral density calculations,as was the case in the NA model. This assumption results inbands with the correct widths and gaps but incorrect structures.For comparison, these forms were

and

We correct typographical errors in ref 41 by replacing E with|E| in the numerators. Figures 5 and 6 display the old formsalong with the new results presented here. In most cases, theold versions tend to overestimate the spectral densities, which

could lead to inflated conductances, and they completely missthe surface states. The old versions are also symmetric withrespect to the interchange of R for -R and 1 for 2.

Electronic Transport

Having computed the self-energies associated with thesevarious models, we proceed to investigate the effects ofsemiconductor electrodes on electronic transport. The AB andAS models have been parametrized for various materials in thepast, and values of R, , and for gold, silicon, and titaniumdioxide are listed in Table 1. We take to be the materialsFermi level by assuming the metal band is half-filled56 (we notedthat is the center of the band gap, the conventional Fermi

level for semiconductors). Since all junctions have electrodesof the same material, we rescale the energy coordinate to EF )

Figure 3. The self-energy, AS(E)/2, for an interface with an ASsemiconductor. The surface state at E ) R manifests as a pole in bothAS(E) (green, solid line for R > 0 or purple, dotted line for R < 0) andAS(E) (blue, dashed line for R > 0 or red, dot-dashed line for R < 0).The band edges are denoted, in magnitude, by E- |R| and E+ (R2

+ 42)1/2.

AB(E)

2)

1

22

[(1 + 2)2 - E2][E2 - (1 - 2)2]

E2

(27)

AB(E)

2)

E2

- 12

+ 22

+ AB(E)(E2 - (1 + 2)2)(E2 - (1 - 2)2)22

2E

(28)

AB(E) ) ((1 - 2)2

- E2) - (E2 - (1 + 2)

2)

AS

old

(E)2

) 2|E|2

1 - [E

2

- R

2

- 22

22 ]2

(29)

Figure 4. The self-energy, AB(E)/2, for an interface with an ABsemiconductor. When |1| < |2|, the surface state presents a pole inAB(E) (red, dot-dashed line) at E ) 0. AB(E) is well-behaved at E) 0 when |1| > |2| (blue, dashed line). The interchange of1 and 2simply rescales the spectral densities (the green, solid line for |1| >|2| and the purple, dotted line for |1| < |2|). E- |1 - 2| and E+ |1 + 2| denote the band edges, in magnitude.

Figure 5. Comparison ofASold(E)/2 (green, solid line) with AS(E)/2

for R < 0 (blue, dashed line) and R > 0 (purple, dotted line). As in

Figure 3, E- t |R| and E+ t (R2 + 42)1/2.

ABold(E)

2)

2|E|121 - [

E2

- 12

- 22

212 ]2

(30)



8/12

0 ( ) 0), highlighting the effects of R, , and . We followthe convention of choosing /2 for TiO2,40 and similarlyfor Si. We pick for gold such that the magnitude of the metalspectral density is comparable to that in previous studies. Before

applying the models, we warn against the quantitative inter-pretation of the ensuing results due to the qualitative selectionof these critical parameters. To this end, the currents reportedlater will be normalized to the maximum current for a givenset of electrodes in the particular calculation.

One paramount issue is the effect of semiconducting elec-trodes on the transport: how do semiconductors change thecurrent-voltage profiles of electrode-molecule-electrode junc-tions? As a necessary basis for comparison, we briefly reviewthe transport across a gold-molecule-gold junction, within theNA model and the Landauer-Imry limit. Figure 7 displays thetransmission and current through the junction for variousmolecular-level energies, electron injection energies, and appliedbias voltages. Figure 7a shows the formation of a resonance at

each , as indicated by the ridge of high transmission. Theresonances are present near the molecular state energies,

although careful inspection reveals the high-transmission ridge(HTR) to be slightly perturbed from the E ) diagonal. Theinset of Figure 7a shows the transmission function when weneglect the molecular-level shiftings caused by adsorption,NA(E). With essentially unchanged results (the HTR remains

linear, although now coincident with the E)

diagonal), wesee that the weak molecular-level shiftings caused by adsorptionto metals can be omitted to a good approximation (in agreementwith ref 11). In Figure 7b, current appears for all positivevoltages when the molecular site level is positioned at the Fermienergy, with increased voltages widening the range of molecularsite levels able to function as current channels.

Two Semiconductor Electrodes. Having examined ametal-molecule-metal junction for comparison, we nowconsider semiconductor-molecule-semiconductor junctionswithin the AS and AB models. We first examine two siliconelectrodes in the alternating bond framework. The semiconductorband gap is the most noticeable feature in the transmission plots(left column) of Figure 8, as evidenced by the vertical strip of

zero transmission in each plot. In these regions, the absence ofstates in the donor electrode prevents electrons from injecting

Figure 6. Comparison ofABold(E)/2 (green, solid line) with AB(E)/2

for |1| < |2| (blue, dashed line) and |1| > |2| (purple, dotted line). Asin Figure 4, E- t |1 - 2| and E+ t |1 + 2|.

TABLE 1: Model Parameters for Au, Si, and TiO2

material model |R| (eV) (eV) (eV) (eV)

Au11 NA -8.95 -0.45Si39 AB -1.60 -2.185 -1.0TiO240 AS 1.6 -2 -1.0

Figure 7. (a) Transmission and (b) current for two gold electrodes;see Table 1 for the parameterizations. The transmission with NA(E)neglected [inset of (a)] is displayed for comparison, showing themolecular-level shifting described by NA(E) to be insignificant formetals. The current is normalized to the maximum current in (b), whichappears at ) EF.

Figure 8. Transmissions (left column) and currents (right column)for junctions with two silicon electrodes. (a, b) |2| < |1| for bothelectrodes (zero contributed surface states); (c, d) mixed 1 and 2 valuesfor the two electrodes (one surface state); (e, f) |2| > |1| for both

electrodes (two surface states). The band gaps cause the strips of zerotransmission around the Fermi level and the minimum bias thresholdsfor the existence of current. The insets of the transmission plots showthe respective transmission functions when the molecular level shifting,AB(E), is neglected. Unlike metals, such shiftings are important forsemiconductors. The currents are normalized to the maximum current[which occurs in (b)], and the numerical parameter values are listed inTable 1.



9/12

into the junction; likewise, there are no states for them to occupyonce transmitted to the acceptor electrode. The existence of abias voltage threshold, V0, is correspondingly the most prominenteffect of semiconductor electrodes on the current. This thresholdis the minimum applied bias voltage needed to access states ineither the valence or conduction band (thereby allowing current)and is independent of the molecular site energy. Figure 9a showsthe current as a function of the voltage for fixed , furtherillustrating this voltage threshold.

Recalling the asymmetry in the self-energy (eqs 27 and 28) when1 and2 are exchanged, we have three possible electrode bondingconfigurations in each junction. Noting that || < || in Table 1,one assignment possibility, where neither electrode contributes asurface state (1,L )1,R )), is displayed in Figure 8a,b. Secondis the case of one surface state (1,L ) and 1,R ) , or viceversa) in Figure 8c,d, and last is the case of two degenerate surfacestates (1,L ) 1,R ) ) in Figure 8e,f.

As expected from the possibility of surface states and asevidenced by Figure 8, the bonding configuration has a dramaticimpact on the transport, particularly when is in the band gap.When neither electrode contributes a surface state (Figure 8a,b),

we observe broad HTRs in both bands. The HTRs are shiftedaway from the E ) diagonal, into the bands, and cause the

double-humped current-voltage profile. With one contributedsurface state (Figure 8c,d), the HTRs edge closer to the bandgap. The surface state in the center of the band gap appears tocontribute transmission, signaling enhanced transport throughmolecular energy levels in the band gap. The HTRs finallyspread into the band gap with the facilitation of two surfacestates (Figure 8e,f). Ignoring the voltage threshold, the semi-conductor transport in Figure 8f is very similar to the metaltransport in Figure 7b; notably, only one hump is present in

the current, centered around the Fermi level. We infer twoprincipal effects of surface states on the transport. First, theyaid transport through nearby molecular site levels (presently inthe band gap). Second, as suggested by the reduction in thebroadening of the HTRs with more surface states, they interferewith molecule-electrode hybridization. This reduction is furtherexplained by the scaling ofAB(E) with 2-2; see Figure 4.

Having insight about the shifting and broadening of the HTRsin Figure 8, we now investigate their bending. The insets ofFigure 8a,c,e each display the transmission through theirrespective junctions when the molecular-level shifting, AB(E),is neglected. Molecular-level shifting through AB(E) is re-sponsible for both the movement of the HTR away from the

diagonal as well as its contorted shape. The nonlinear shape ofAB(E) when AB(E) > 0 explains the amplified shifting nearthe band edges; see Figure 4. Recalling that NA(E) is essentiallynegligible for metals, we see that semiconductors interact morestrongly with the molecule; the electronic transport is influencedby the molecular-level shifts and is also sensitive to the presenceof surface states.

A quick glance at the transmission contour plots in Figure 8shows that the HTRs are bent, shifted, and broadened dependingon the presence of surface states. That the HTRs span the sameinjection energy ranges with a single HTR per band may suggesta conservation of transmission through the various junctions.This visual effect is fictitious. Physical systems transmit

electrons through discrete, but broadened, resonances. Only theircorresponding horizontal segments in Figures 7 and 8 havemeaning. For instance, a molecular site level at - EF ) -3.5eV does not display a resonance in Figure 8a when AB(E) isincluded (the main panel); however, a resonance appears at E- EF ) -3.5 eV when the shifting is neglected (the inset). Forthis particular level, the existence of a resonance, and thus themagnitude of transmission, is notconserved, despite appearancesin the contour plots. To further illustrate how the transmissionfunction changes with the molecular site level, Figure 9b showsthe transmission function of Figure 8a for particular .

If we instead use the alternating site model and consider aTiO2-molecule-TiO2 junction, we encounter similar choicesin bonding configuration. Here, we can choose sign(R) sinceonly |R| is parametrized, and three similar bonding configurationsarise. These bonding configurations may be loosely interpretedas the molecule bonding with titanium on both electrodes, withoxygen on both, or with one titanium and one oxygen.

Figure 10 shows that many of the trends observed in the ABmodel are still present. First, the band gap is again manifestedby strips of zero transmission and leads to similar voltagethresholds in the currents. Second, the bonding configurationin these junctions is once again important. When the moleculeis connected to the same atom type on both electrodes (both R> 0 or both R < 0), the degenerate surface states appear to forma system resonance at R for all . This leads to perfecttransmission at E ) R regardless of the molecule site level, as

displayed in Figure 10a,e. While always present, these reso-nances become infinitely narrow as | - EF| f and are

Figure 9. (a) Current-voltage profiles for molecular transport junctions with two gold (red, dot-dashed line) and two siliconelectrodes. The molecular site levels are at the Fermi levels of theirrespective junctions, causing the metal-metal line to rise quicklyfor small voltages. Silicon junctions with zero (green, solid line),one (blue, dashed line), and two (purple, dotted line) contributedsurface states all display a minimum voltage threshold, V0. Electrodestates are unavailable below this threshold, and no current isobserved. The currents are normalized to the maximum current fora given set of electrodes. (b) Transmission functions at various for a silicon-molecule-silicon junction with zero contributed

surface states; see Figure 8a. This illustrates how the bending,shifting, and broadening of the resonances change their interactionswith different molecular site levels. The conservation of transmis-sion suggested by the presentation of Figure 8 is misleading.



10/12

unlikely to be experimentally observed due to realistic averagingeffects. Conversely, when the molecule has mixed bonding tothe electrodes, Figure 10c,d, the surface states at R and -Rappear to destroy transmission at (R. We speculate that themolecule-semiconductor interfaces, recently shown to becritical in transport,25 are responsible for these observations;however, it is not immediately obvious why or how degenerate

surface states enhance transport and nondegenerate surface statesdampen it.Figure 3 shows thatAS(E), similar to NA(E), is linear when

AS(E) > 0. Thus, when we remove AS(E) from the transmis-sion (the insets of Figure 10a,c,e), the HTRs maintain linearity.Even without the contortions of the AB model, this shifting isstill more prominent than it was in the NA model and is seento make the HTRs more horizontal. As with the AB model, theAS model indicates that semiconductor-molecule-semiconductor junctions have strong molecular-level shiftings induced byadsorption [(E)] and are sensitive to the bonding configurationsat the electrode-molecule interfaces.

Conclusions

The study of electronic transport through electrode-mol-ecule-electrode junctions has been increasing in recent years

due to the interest in scanning probe microscopies, solar cells,and general molecular transport junctions. While mosttheoretical studies have focused exclusively on junctions withmetal electrodes, experiments have also been reported forsystems utilizing semiconductor electrodes. Such studies haveobserved negative differential resistance and asymmetriccurrent-voltage profiles, which pose great promise for noveldevice construction.

Inthispaper,weconsideredextensionsoftheNewns -Anderson

model for one-dimensional metals to one-dimensional semi-conductors, where the atomic site energies and/or theinteratomic site couplings were alternated. We derived thespectral densities and ultimately the total self-energies fordescribing molecular interactions with these one-dimensionalsemiconductors, including the presence of surface states.These self-energies (eqs 22 and 24) are the primary contribu-tions of this paper and should be used instead of those fromthe previous paper.41

These models were then applied to several molecular transportjunctions, and the current was calculated in the Landauer-Imry(coherent tunneling) limit. We found that a semiconductor-mol-ecule-semiconductor junction displays a minimum bias thresh-old for generating current across the junction due to theelectrodes band gaps. Semiconductor-semiconductor junctionsalso display different molecule-electrode interactions thansimilar metal-metal junctions. These effects are particularlynoticeable through the increased molecular-level shifting causedby adsorption to semiconductors and are very sensitive to howthe molecule bonds to the semiconductor surface. Furthermore,the presence of surface states, a consequence of the bondingconfiguration in these models, drastically changes the electronictransport. Surface states dominate the transmission in somecases, allowing the largest currents for molecular site levels inthe band gap. While discussed very generally here, the effectsof these surface states on transport need to be understood in

more detail. Extensions to junctions with one metal and onesemiconductor electrode, subject to the band lineup problem,are also envisioned.

Unlike the situation for metals, molecular energy levelshifting, introduced through (E), is non-negligible for semi-conductors. Essentially no differences in the transmisson func-tions were observed between the inclusion and exclusion ofNA(E) for metal-metal junctions. Semiconductors, however,evidenced both bending and shifting with the inclusion ofKD(E) (and its AB and AS limits). This bending of the high-transmission ridges (HTRs) makes them resonant with somemolecular energy levels while not with others, breaking aconservation of transmission seen in metal-metal junctions.While physically interesting in its own right, such an effect mayalso be advantageous for certain applications. Consider amolecule with a low-energy state connected between twosemiconductor electrodes (without surface states, for simplicity).If this molecular state is below the HTRs, relatively lowtransport will be observed through the junction. When we adda gating voltage, we move the molecular level relative to thesurfaces, bringing it into the range of the resonances, therebyobtaining high transport. Additional gating voltage displaces thelevel out of this range, reducing the current. From this, we canimagine molecular transistors. Semiconductor electrodes alsoinvite the extension of coherent control schemes to manipulateelectric current. The use of sub-band gap light and optimal

control theory to command transport and device functionalityholds exciting potential.

Figure 10. Transmissions (left column) and currents (right column)for junctions with two titanium dioxide electrodes. (a, b) Both electrodesconnected with R < 0; (c, d) mixed R > 0 and R < 0 connectivity; (e,

f) both atR

> 0. As in Figure 8, we observe strips of zero transmissionand a minimum bias threshold for current. The insets similarly showthe transmissions with AS(E) neglected, indicating the importance ofmolecular-level shifting due to semiconductor adsorption. The currentsare normalized to the maximum current [realized in (b, f)], and themodel TiO2 parameters are listed in Table 1.



11/12

In closing, we remind the reader that many of the parameters used to model transport were qualitatively chosen, and the modelsresults should only be qualitatively interpreted. Due to the evidenced band gap, we believe these models capture much of thefundamental physics and chemistry of adsorption to semiconductors, although inherent errors are introduced by describing three-dimensional electrodes with one-dimensional models. A comparison of results from these one-dimensional models with those fromelectronic structure calculations may reveal additional information about electronic transport through molecule-semiconductorjunctions.

Acknowledgment. We appreciate many enlightening discussions with Dr. Gemma C. Solomon and Prof. Vladimiro Mujica. Weare grateful to the NSF [Chem and MRSEC (Grant No. DMR-0520513) programs] and the MNRF program of the DoD for support.

M.G.R. acknowledges support from the DoE Computational Science Graduate Fellowship Program (Grant No. DE-FG02-97ER25308).

Appendix

Integration for KD(E). The eigenvalues (eq 18) and eigenvectors (eq 21) are used in eq 2 for calculating KD(E). Three changeof variable substitutions are needed, u ) R2 + 12 + 22 + 212 cos(2k), x ) (u)1/2, and y ) -(u)1/2. Furthermore, the sign functionin the eigenvalues splits the bounds of integration into two halves, (0, /2) and (/2, ). This split leads to an additional signsubtlety during the conversion of k to u; sin(2k) is positive for 0 < k < /2 and negative for /2 < k < . Finally

for [R2 + (1 - 2)2]1/2 e |E| e [R2 + (1 + 2)2]1/2. KD(E) ) 0 for all other E.

References and Notes

(1) ORegan, B.; Gratzel, M. Nature 1991, 353, 737740.(2) Nitzan, A. Chemical Dynamics in Condensed Phases: Relaxation,

Transfer, and Reactions in Condensed Molecular Systems; Oxford Univer-sity Press: New York, 2006.

(3) Nitzan, A. Annu. ReV. Phys. Chem. 2001, 52, 681750.(4) Datta, S. Quantum Tranport: Atom to Transistor; Cambridge

University Press: New York, 2005.(5) Introducing Molecular Electronics; Cuniberti, G., Richter, K., Fagas,

G., Eds.; Springer-Verlag: New York, 2005.(6) Di Ventra, M. Electrical Transport in Nanoscale Systems; Cam-

bridge University Press: New York, 2008.

(7) J. Phys.: Condens. Matter 2008, 20. A special issue that providesan excellent collection of articles on The conductivity of single moleculesand supramolecular architectures.

(8) Magoga, M.; Joachim, C. Phys. ReV. B 1997, 56, 47224729.(9) Yaliraki, S. N.; Ratner, M. A. J. Chem. Phys. 1998, 109, 5036

5043.(10) Emberly, E. G.; Kirczenow, G. Phys. ReV. B 1998, 58, 10911

10920.(11) Hall, L. E.; Reimers, J. R.; Hush, N. S.; Silverbrook, K. J. Chem.

Phys. 2000, 112, 15101521.(12) Hipps, K. W. Science 2001, 294, 536537.(13) Hihath, J.; Arroyo, C. R.; Rubio-Bollinger, G.; Tao, N.; Agrat, N.

Nano Lett. 2008, 8, 16731678.(14) Kristensen, I. S.; Mowbray, D. J.; Thygesen, K. S.; Jacobsen, K. W.

J. Phys.: Condens. Matter 2008, 20, 374101.(15) Tian, W.; Datta, S.; Hong, S.; Reifenberger, R.; Henderson, J. I.;

Kubiak, C. P. J. Chem. Phys. 1998, 109, 28742882.(16) Metzger, R. M.; Xu, T.; Peterson, I. R. J. Phys. Chem. B 2001,105, 72807290.

KD(E) ) limNf

2n)1

N 4212 sin2 (2kn)(E - sign[cos(kn)]R2 + 12 + 22 + 212 cos(2kn))

(2N + 1)[(kn - R)2

+ 12

+ 22

+ 212 cos(2kn)]

) 821

2

0

dksin2 (2k)(E - sign[cos(k)]R2 + 12 + 22 + 212 cos(2k))

(k - R)2

+ 12

+ 22

+ 212 cos(2k)

) 821

20/2

dksin2 (2k)(E - R2 + 12 + 22 + 212 cos(2k))

(R2 + 12 + 22 + 212 cos(2k) - R)2 + 12 + 22 + 212 cos(2k)+

8212

/2

dk

sin2 (2k)(E + R2 + 12 + 22 + 212 cos(2k))

(-R2 + 12 + 22 + 212 cos(2k) - R)2 + 12 + 22 + 212 cos(2k)

)-2

22 R2+(1 + 2)2

R2+(1 - 2)2 du(E - u)[R2 + (1 + 2)2 - u][u - R2 - (1 + 2)2]

(u - R)2 + u - R2-

22

2 R2+(1 - 2)2R2+(1 + 2)2 du (E + u)(-1)[

R

2

+ (1 + 2)2

- u][u - R2

- (1 + 2)2

](-u - R)2 + u - R2

)2

22 R2+(1 - 2)2

R2+(1 + 2)2 dx(E - x)[R2 + (1 + 2)2 - x2][x2 - R2 - (1 + 2)2]

x - R-

2

22 -R2+(1 + 2)2

-R2+(1 - 2)2 dy

(E - y)[R2 + (1 + 2)2 - y2][y2 - R2 - (1 + 2)2]y - R

)2

22

[R2 + (1 + 2)2

- E2][E2 - R2 - (1 - 2)

2]

(E - R)2



12/12

(17) Ward, D. R.; Halas, N. J.; Ciszek, J. W.; Tour, J. M.; Wu, Y.;Nordlander, P.; Natelson, D. Nano Lett. 2008, 8, 919924.

(18) McCreery, R.; Dieringer, J.; Solak, A. O.; Snyder, B.; Nowak,A. M.; McGovern, W. R.; Duvall, S. J. Am. Chem. Soc. 2003, 125, 1074810758.

(19) Guisinger, N. P.; Greene, M. E.; Basu, R.; Baluch, A. S.; Hersam,M. C. Nano Lett. 2004, 4, 5559.

(20) Guisinger, N. P.; Yoder, N. L.; Hersam, M. C. Proc. Natl. Acad.Sci. U.S.A. 2005, 102, 88388843.

(21) Piva, P. G.; DiLabio, G. A.; Pitters, J. L.; Zikovsky, J.; Rezeq, M.;Dogel, S.; Hofer, W. A.; Wolkow, R. A. Nature 2005, 435, 658661.

(22) Yoder, N. L.; Guisinger, N. P.; Hersam, M. C.; Jorn, R.; Kaun,

C.-C.; Seideman, T. Phys. ReV. Lett. 2006, 97, 187601.(23) Rakshit, T.; Liang, G.-C.; Ghosh, A. W.; Datta, S. Nano Lett. 2004,

4, 18031807.(24) Rakshit, T.; Liang, G.-C.; Ghosh, A. W.; Hersam, M. C.; Datta, S.

Phys. ReV. B 2005, 72, 125305.(25) Yu, L. H.; Gergel-Hackett, N.; Zangmeister, C. D.; Hacker, C. A.;

Richter, C. A.; Kushmerick, J. G. J. Phys.: Condens. Matter 2008, 20,374114.

(26) Hovis, J. S.; Liu, H.; Hamers, R. J. Surf. Sci. 1998, 402-404, 17.(27) Wolkow, R. A. Annu. ReV. Phys. Chem. 1999, 50, 413441.(28) Kirczenow, G.; Piva, P. G.; Wolkow, R. A. Phys. ReV. B 2005, 72,

245306.(29) Leftwich, T. R.; Madachik, M. R.; Teplyakov, A. V. J. Am. Chem.

Soc. 2008, 130, 1621616223.(30) Reuter, M. G.; Sukharev, M.; Seideman, T. Phys. ReV. Lett. 2008,

101, 208303.(31) Landauer, R. IBM J. Res. DeV. 1957, 1, 223231.

(32) Landauer, R. Philos. Mag. 1970, 21, 863867.(33) Imry, Y. In Directions in Condensed Matter Physics; Grinstein, G.,Mazenko, G., Eds.; World Scientific: River Edge, NJ, 1986; pp 101 -163.

(34) Cahay, M.; McLennan, M.; Datta, S. Phys. ReV. B 1988, 37, 1012510136.

(35) Jauho, A.-P.; Wingreen, N. S.; Meir, Y. Phys. ReV. B 1994, 50,55285544.

(36) Newns, D. M. Phys. ReV. 1969, 178, 11231135.(37) Anderson, P. W. Phys. ReV. 1961, 124, 4153.

(38) Koutecky, J. AdV. Chem. Phys. 1965, 9, 85168.(39) Foo, E. N.; Davison, S. G. Surf. Sci. 1976, 55, 274284.(40) Petersson, .; Ratner, M.; Karlsson, H. O. J. Phys. Chem. B 2000,

104, 84988502.(41) Mujica, V.; Ratner, M. A. Chem. Phys. 2006, 326, 197203.(42) Haydock, R.; Heine, V.; Kelly, M. J. J. Phys. C 1972, 5, 28452858.(43) Cini, M. Topics and Methods in Condensed Matter Theory; Springer-

Verlag: New York, 2007.(44) In full generality, Gelec(E) is a matrix, requiring two indices for a

specific element. Presently, we only need the diagonal elements; therefore,the second index is omitted.

(45) Xue, Y.; Datta, S.; Ratner, M. A. J. Chem. Phys. 2001, 115, 42924299.(46) Meir, Y.; Wingreen, N. S. Phys. ReV. Lett. 1992, 68, 25122515.(47) Foo, E.-N.; Wong, H.-S. Phys. ReV. B 1974, 9, 18571860.(48) Muscat, J. P.; Davison, S. G.; Liu, W. K. J. Phys. Chem. 1983, 87,

29772981.(49) Tamm, I. Phys. Z. Sowjetunion 1933, 1, 733746.(50) Shockley, W. Phys. ReV. 1939, 56, 317323.(51) Dy, K. S.; Wu, S.-Y.; Spratlin, T. Phys. ReV. B 1979, 20, 4237

4243.(52) Crain, J. N.; Pierce, D. T. Science 2005, 307, 703706.(53) We note that sin[kn(2N + 1)] in the Newns-Anderson case is not

equivalent to sin[2knN] + sin[2kn(N + 1)] here; however, they share thesame roots.

(54) The quantization condition for the KD model, eq 20, can berewritten as sin[2kn(N + 1)] + sin(2knN) ) 0, where 1/2. It can beverified that kn ) n/(2N + 1) when ) 1. When f 0, we get kn f

n/2N, and when f , kn f n/(2N + 2). We then infer that the kn areroughly evenly spaced (independent of ) and that the spacing is ap-proximately /(2N + 1).

(55) The reduction of [E2-R2-(1-2)2]1/2 to |E| in this combined limitcreates the sign disparity observed in NA(E).

(56) Mujica, V.; Kemp, M.; Ratner, M. A. J. Chem. Phys. 1994, 101,69566864.

JP811492U


Date post:	06-Apr-2018
Category:	Documents
Upload:	dfmso0
View:	219 times
Download:	0 times

Matthew G. Reuter, Thorsten Hansen, Tamar Seideman and Mark A. Ratner- Molecular Transport Junctions...

Documents