LETTER pubs.acs.org/NanoLett
Novel Quantum Interference Effects in Transport through Molecular Radicals Justin P. Bergfield,*,† Gemma C. Solomon,‡ Charles A. Stafford,§ and Mark A. Ratner|| †
Departments of Chemistry and Physics, University of California, Irvine, California 92697, United States Nano-Science Center and Department of Chemistry, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen Ø, Denmark § Department of Physics, University of Arizona, 1118 East Fourth Street, Tucson, Arizona 85721, United States Department of Chemistry, Northwestern University, Evanston, Illinois 60208, United States
)
‡
bS Supporting Information ABSTRACT: We investigate electronic transport through molecular radicals and predict a correlation-induced transmission node arising from destructive interference between transport contributions from different charge states of the molecule. This quantum interference effect has no single-particle analog and cannot be described by effective single-particle theories. Large errors in the thermoelectric properties and nonlinear currentvoltage response of molecular radical junctions are introduced when the complementary wave and particle aspects of the electron are not properly treated. A method to accurately calculate the low-energy transport through a radical-based junction using an Anderson model is given. KEYWORDS: Quantum transport, molecular radicals, DFT, many-body, Mott-node
T
he vast majority of calculations of electronic transport through single molecules strongly bound to electrodes assume that the process occurs via the tunneling of phase coherent electron waves with the neutral ground electronic state of the system describing the entirety of the process. However, the very nature of charge transport, whereby individual electrons tunnel (virtually) onto and off of the molecule, suggests that this may not be a complete physical description. Here we investigate electronic transport in radical-based single-molecule junctions (SMJs) and predict a new type of transmission node arising from interferences between transport pathways involving different charge states of the molecule, requiring that the complementary wavelike and particle-like nature of the electron be treated simultaneously. Such “Mott nodes” are independent of junction topology and are predicted to occur in the experimentally relevant transport regime of all molecular radical-based junctions. As an example, we consider transport through a AupentadienyldithiolAu SMJ, shown schematically in Figure 1. The simulated transmission spectra and molecular charge imbalance for the π-electrons of this radical-based junction are shown as a function of the leads’ chemical potential μ in Figure 2. The transport was calculated using a many-body theory based on the molecular Dyson equation (MDE), which includes all states of the isolated radical exactly.1 These calculations utilize a basis of frozen nuclei since we focus on transport in the cotunelling regime, where structural relaxation is not expected to be significant. Image charge effects2 from the nearby metal electrodes are not included in the calculations because they have no significant r 2011 American Chemical Society
Figure 1. A schematic diagram of the AupentadienyldithiolAu junction we use as an example throughout this article.
effect on the Mott node. All simulations presented here are for junctions operating at room temperature where we assume phase coherent transport. The effect of dephasing is discussed in the Supporting Information. When μ is commensurate with the molecular midgap energy μ0,3 the transmission spectrum exhibits a node, a spectral feature indicative of interference.410 However, the origin of this node is not the specific molecular topology, since the pentadienyl radical is a linear molecule whose π-orbital electronic system lacks any topological nodes. Instead, this Mott node is a manifestation of complete destructive interference between the transmission amplitudes for electrons and holes, quantities involving different charge Received: March 28, 2011 Revised: May 24, 2011 Published: June 10, 2011 2759
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Figure 2. The transmission probability T(E) for both spin species (R and β) and the average molecular charge imbalance ÆQæ/e are shown near the HOMO and LUMO resonances11 of a pentadienyldithiol radical SMJ as a function of lead chemical potential μ. Both spins contribute equally to the transport since there is no broken spinsymmetry.12 The spectrum encompasses three charge states (1,0,þ1) of the molecule and exhibits both Coulomb blockade peaks and a midgap node when μ = μ0. This Mott-node stems from quantum interferences between transmission amplitudes involving different charge states of the molecule. Calculations were performed using MDE theory1 for the π-orbitals of a SMJ operating at room temperature with a symmetric lead-molecule coupling Γ = 0.5 eV. e is the electron’s charge.
states of the molecule. When μ = μ0 these amplitudes cancel, giving rise to a node, whereas when μ < μ0 or μ > μ0 the transport is dominated by holes or electrons, respectively, and the destructive interference between the amplitudes is not complete. Fundamentally, a description of this type of interference requires a transport theory that correctly accounts both for the charge fluctuations of the system and quantum coherence, since the splitting of the transmission amplitude poles is a particle effect (Coulomb blockade) while their interference is a quantum wave effect. Electronic transport calculations involving molecules often utilize effective single-particle spin-unrestricted methods, such as unrestricted HartreeFock (UHF) or an implementation of density functional theory (DFT), where the molecular orbitals for each spin are solved independently, leading to states with broken spin-rotation symmetry. This allows for a better description of the molecular energy spectrum, particularly for the case of molecular radicals, where spin-restricted methods fail to describe a HOMOLUMO gap. However, transport depends not only on energy levels but also on the matrix elements between the initial and final states of the system. Because of the discrete nature of the charges being transported, these matrix elements involve different charge states of the molecule and require a many-body theory for their proper description. An alternative computational approach to the transport problem is to solve the few-body molecular Hamiltonian exactly and to treat electron hopping between molecule and electrodes as a perturbation. This approach was initially used to describe molecular junction transport in the sequential-tunneling regime.1315 Higher-order tunneling processes may be included in the density-matrix formalism16,17 and the closely related superoperator Green’s function approach,1820 where the expansion is typically truncated at second19 or fourth order.16,21 Here, we utilize a MDE theory1 based on ordinary nonequilibrium Green’s functions (NEGFs), which allows for the inclusion of tunneling
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Figure 3. The electronic transmission spectrum for the R and β spins of a AupentadienyldithiolAu radical junction calculated using spinunrestricted DFT. The midgap energy μ0 was chosen as the energy origin rather than the Fermi energy of gold in order to be consistent with the MDE calculations.
processes to all orders for each class of diagram. We emphasize, however, that any transport theory that properly accounts for the complementary wave-particle nature of the electron should exhibit Mott nodes. Currently, most modern transport calculations involve implementations of DFT, a theory that would give the exact electronic ground state if the exact exchange-correlation functional were known. A spin-unrestricted DFT calculation of the transmission through a pentadienyl radical SMJ is shown in Figure 3, where no transmission nodes are found. Despite DFT’s sophistication, existing implementations of the exchange-correlation functional do not properly describe the quantization of charge14,2224 and consequently the resulting transport does not exhibit a Mott node. The thermopower spectrum of a pentadienyl radical SMJ calculated using both MDE and DFT is shown in Figure 4. In the MDE simulation, the effect of tunneling through σ-orbitals was included by adding a phenomenological constant Tσ = 1.27 105 to the transmission probability, where Tσ = ANeβN and the values AN = 0.0006, βN = 0.77 and N = 5 are consistent with the thiol end-groups of the pentadienyl SMJ.2527 In the limit that Tσ = 0, the thermopower reaches a maximum value of 156 μV/K.9 The peak thermopowers found using MDE and DFT were 42 and 7 μV/K, respectively. Since the thermoelectric response is enhanced near transmission nodes9,10 and MDE theory predicts a midgap node while DFT does not, the peak thermopower predicted by the former is larger than the latter. This enhancement is a clear experimental signature of Mott nodes. In order to investigate the nature of the Mott node it is necessary to consider transport theory in some detail. The transmission function of a junction T(E) can be used to compactly express a number of important linear-response transport quantities9 such as the zero-bias conductance G ¼ e2 L
ð0Þ
ð1Þ
and the thermopower S¼ where L 2760
ðνÞ
1 ¼ h
Z
1 L eT L
ð1Þ ð0Þ
Df dEðE μÞ TðEÞ DE ν
ð2Þ
ð3Þ
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broadened and imbued with finite lifetimes. Using the molecular Dyson equation1 the full Green’s function of the SMJ may then be written in terms of Gmol as GðEÞ ¼ Gmol ðEÞ þ Gmol ðEÞΔΣðEÞGðEÞ
ð9Þ
Σ(0) C .
Figure 4. The thermopower S of a AupentadienyldithiolAu SMJ is shown near the midgap energy μ0 calculated using MDE and spinunrestricted DFT. The peak thermopowers Speak found using MDE and DFT were 42 and 7 μV/K, respectively. In the MDE simulation, the effect of tunneling through σ-orbitals was included by adding a phenomenological constant Tσ = 1.27 105 to the π-orbital transmission probability.2527 Neglecting tunneling through σ-orbitals, Speak = 156 μV/K near the midgap node. The atomic positions used in both the MDE and DFT calculations were identical. The thermopower is independent of the lead-molecule coupling Γ.
In the elastic transport regime, the nonlinear current through a two-terminal junction can also be expressed in terms of the transmission function28 Z e ¥ I ¼ TðEÞ½f2 ðEÞ f1 ðEÞ ð4Þ h ¥ where fγ(E) = {1 þ exp[β(E μγ)]}1 is the FermiDirac distribution for lead γ at inverse temperature β = 1/kBT. From eqs 14, it is clear that any errors in T(E) lead directly to errors in observable transport quantities. The elastic transmission function for a two-terminal junction is given by29 TðEÞ ¼ TrfΓL ðEÞGðEÞΓR ðEÞG† ðEÞg
ð5Þ
where Γγ(E) is the tunneling-width matrix for lead γ and the trace includes spin. The retarded Green’s function G(E) of the junction may be written as ð1Þ
GðEÞ ¼ ½SE Hmol ΣðEÞ1
ð6Þ
þ is the molecular Hamiltonian (see where Supporting Information) which we formally separate into onebody and two-body terms. S is an overlap matrix, which in an orthonormal basis reduces to the identity matrix, and Hmol=H(1) mol
H(2) mol
ΣðEÞ ¼ ΣLT ðEÞ þ ΣRT ðEÞ þ ΣC ðEÞ
ð7Þ
is the self-energy, including the effect of both a finite leadmolecule coupling via ΣL,R T and many-body interactions via the Coulomb self-energy ΣC(E). The tunneling self-energy matrices are related to the tunneling-width matrices by ΣγT(E) [ΣγT(E)]† = iΓγ(E). It is advantageous to define a molecular Green’s function Gmol(E) = limΓγf0þG(E). In the sequential tunneling regime,1 where lead-molecule coherence can be neglected, the molecular Green’s function within MDE theory is given by ð1Þ
ð0Þ
Gmol ðEÞ ¼ ½SE Hmol ΣC ðEÞ1 H(1) mol
ð8Þ
and the Coulomb where all one-body terms are included in self-energy Σ(0) accounts for the effect of all two-body intramolecular many-body correlations exactly. In the presence of finite lead-molecule coupling Γγ 6¼ 0 and the molecular states are
Gmol is found by where ΔΣ = ΣT þ ΔΣC and ΔΣC = ΣC exact diagonalization of Hmol, including all charge states and excited states of the molecule, and the correction to the Coulomb self-energy ΔΣC is found using NEGF methods.1 We consider transport in the elastic cotunneling approximation1 where ΔΣC = 0. In the basis of exact eigenstates of the molecular Hamiltonian Gmol is given by1 Gmol ðEÞ ¼
½P ðνÞ þ P ðν0 ÞCðν, ν0 Þ E Eν0 þ Eν þ i0þ 0
∑ ν, ν
ð10Þ
where ν and ν0 label eigenstates of Hmol and P (ν) is the probability that state ν is occupied. In linear-response, P (ν) = eβ(EνμNν)/Z , where Z =∑νeβ(EνNνμ) is the grand canonical partition function. The many-body matrix element † ½Cðν, ν0 Þnσ, mσ0 ¼ Æνjdnσ jν0 æÆν0 jdmσ 0 jνæ
ð11Þ
where dnσ (d†nσ) annihilates (creates) an electron with spin σ on the nth atomic orbital of the molecule and ν and ν0 label molecular eigenstates with different charge occupancies. In effective single-particle theories, such as UHF and current implementations of DFT, many-body interactions are treated approximately, resulting in a static energy shift of the one-body ~ (1) ~ (1) ~ ~ν, where H(1) energies E mol þ Σ(E) f H mol and H mol|νæ = Eν|νæ. The molecular Green’s function in this case is given by ð1Þ ~ mol Gmol ðEÞ ¼ ½SE H þ i0þ 1
ð12Þ
Although a junction’s Green’s function is generally too complicated to work with analytically, the low-energy excitation spectrum most relevant experimentally of a molecular radical (i.e., the region between the HOMO and LUMO resonances11) is typically dominated by the unpaired electron. This allows us to map the molecular problem onto a simpler Anderson model to a very good approximation with two parameters: the effective lead~γ and charging energy U. In what follows, we molecule coupling Γ investigate the Anderson mapping of the pentadienyl radical SMJ using UHF and MDE and find that the Anderson mapping of this junction is extremely accurate, quantitatively reproducing the low-energy spectrum of the full junction. An “Anderson molecule” has the following Hamiltonian30 Hmol ¼ εv nv þ εV nV þ Unv nV
ð13Þ
where nσ = d†σdσ is the number operator for spin σ and U is the charging energy. The occupancy of an Anderson molecule can vary between zero and two, where a natural many-body eigenbasis is the set of Fock states {|0æ,|væ,|Væ,|vVæ} with corresponding eigenenergies {0, εv, εV, εv þ εV þ U}. The Anderson junction therefore has two electronic addition resonances for each spin species σ: one at εσ and another at εσh þ U. In mapping the full molecular spectrum onto an Anderson model the two free parameters Γ~γ and U are chosen to produce the correct resonance widths and HOMOLUMO gap, respectively. A similar mapping is discussed in ref 31. The solution of the Anderson model provides a paradigm to describe Coulomb blockade and the Kondo effect in quantum dots and SMJs.32 For temperatures less than the Kondo 2761
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temperature TK the Mott node will be filled in by spin-flip tunneling processes. Using parameters found from the MDE simulation of the pentadienyl radical SMJ TK ≈ 5 105K at the center of the HOMOLUMO gap (see Supporting Information), indicating that there will be little or no observable effect at room temperature for this junction. The MDE molecular Green’s function for spin σ may be derived from the molecular Hamiltonian eq 13 using eq 10 giving33 ½Gmol ðEÞσσ ¼
1 Ænσ æ Ænσ æ þ þ E εσ þ i0 E εσ U þ i0þ
ð14Þ
where the bar on the spin index σ indicates the opposite spin and the thermodynamic factors P (ν) and P (ν0 ) have been rewritten in terms of the average occupancy of spin σ h βðεσ μÞ βðεσ þ εσ þ U 2μÞ e þe ð15Þ Ænσ æ ¼ Z The effective parameters needed to map the many-body molecular problem onto an Anderson model are given by ~ γ ¼ TrfΓγ Cðν, ν0 Þg Γ
ð16Þ
and U ¼ εLUMO εHOMO
ð17Þ
where εLUMO = εHOMO = and is the ground state energy of the neutral molecule. Note that without any broken spin-symmetry (i.e., with εv = εV) each spin species’ molecular Green’s function has two poles, a consequence of the fact that there are two nonzero many-body matrix elements: Æσ|d†σ|0æ and ÆvV|d†σ|σhæ. In order to understand why the Mott node is not predicted by effective single-particle theories, we consider the molecular Green’s function in the UHF approximation, where eq 13 beUHF ~ ~ ~ (1) comes H mol = Hmol = ΣσEσnσ, with Eσ = εσ þ UÆnσ æ. In effective single-particle theories like UHF, particles interact with the static potential arising from the average potential made by the other particles. Using eq 12 with an orthonormal basis, the molecular Green’s function for spin σ in the UHF approximation is given by E(0) Nþ1
½GUHF mol ðEÞσσ
E(0) N ,
E(0) N
E(0) N1,
1 ¼ E εσ UÆnσ æ þ i0þ
Figure 5. The transmission spectrum of a pentadienyl radical SMJ calculated using MDE and the spectra of an Anderson model of the same junction calculated using MDE and UHF. Compared with the full MDE transmission spectrum, the Anderson MDE spectrum has a maximal absolute error of ∼105 in the midgap region. All transmission nodes in these spectra are Mott nodes; no topological paths exist in either the full pentadienyl molecule or in the Anderson molecule. Inset: The transmission function in the vicinity of the HOMO resonance showing that the UHF peak height is half the MDE peak height. In these calculations Γ = ~γ = 0.1225 eV and U = 5.85 eV. 0.5 eV, Γ
E(0) N
ð18Þ
which consists of only a single pole per spin species. The spacing ~σ = U|Ænv between the individual poles for up and down spins is ΔE nVæ|, which matches the spacing of the two poles in eq 14 when Ænæ = 1 and spin-rotation symmetry is maximally broken. UHF uses a spin-gap in the spectrum of an effective single-particle Hamiltonian to mimic a charge gap in the spectrum of the true interacting Hamiltonian. The transmission spectrum of a pentadienyl radical SMJ calculated using MDE and the spectra of an Anderson model of the same junction calculated using MDE and UHF are shown ~γ = as a function of chemical potential μ in Figure 5. Here Γ 0.1225 eV and U = 5.85 eV. All transmission nodes exhibited in these spectra are Mott nodes; no topological paths exist through the π-orbitals of either the full pentadienyl radical or the Anderson molecule that produce such nodes. From the figure, we see that the many-body treatment of the Anderson model accurately reproduces the low-energy spectrum of the full MDE calculation including the Mott node, although only two of the ten total resonances are included. In contrast, the UHF solution of
Figure 6. The current as a function of bias voltage for an Anderson junction using MDE and UHF. Each theory predicts steps in the current at the same voltages, but the heights of the two steps are different. For voltages between the two current steps, MDE predicts a plateau height of two-thirds while the spin-unrestricted approach gives one-half.
the Anderson model fails to predict the midgap Mott node, even though the resonance energies are correct. There is also a peak height discrepancy, shown in the inset of the same figure, where the UHF peak heights are only 50% of those found using MDE. With the analytic form of the molecular Green’s functions we can investigate the origin of the Mott node in the Anderson junction directly, since whenever Gmol(μ0) = 0 then T(μ0) = 0. The molecular Green’s function eq 14 has two poles, corresponding to a hole-excitation and an electron-excitation. When μ = μ0, the numerator of each pole is the same but the denominators have opposite signs, giving rise to a Mott node. In contrast, the UHF molecular Green’s function eq 17 neglects the different charge states of the molecule and possesses only a single pole per spin species and, consequently, neglects the particle-hole interference which produces the Mott node. The reduced number of poles in the UHF Green’s function is also the reason for the 50% transmission peak height reduction, since only half as many poles 2762
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Nano Letters contribute on resonance as compared to the many-body MDE Green’s function. The nonlinear current through a pentadienyl radical SMJ, described by the corresponding Anderson model, is shown as a function of bias voltage in Figure 6, using both the MDE and UHF Green’s functions. The Coulomb staircase was calculated ~/p to highlight using eq 4 and has been normalized in units of eΓ the relative step height found in each theory. A current step occurs at a voltage ΔV where the leads’ chemical potentials correspond to molecular addition energies, opening an additional transport channel of the junction. In the simulations, the equilibrium chemical potential μ0 = 5 eV, the charging energy U = 5.85 eV and ε = 3 eV such that the first step occurs at ΔV = 2 V and the second 2U away at ΔV = 13.7 V. As the figure shows, each theory predicts steps in the current at the same voltages, although the heights of the two steps are different. For voltages between the two current steps, only two charge states of the junction are energetically accessible, and the many-body theory predicts a plateau height of two-thirds while the spin-unrestricted approach gives one-half. The inability of mean-field theories to correctly describe the relative step heights in the currentvoltage curve of a SMJ in the Coulomb blockade regime has been put forward as a “smoking gun” for the importance of many-body correlations.14 In UHF theory, Gmol has only a single pole per spin species, meaning that both step heights of the Coulomb staircase of the Anderson model are identical (one-half in normalized units) since when ΔV = 2 V the spin-up-only channel opens and when ΔV = 13.7 V the spin-down-only channel opens. In contrast, the MDE Gmol has two poles per spin species so when ΔV = 2 V a charge channel opens transporting electrons of both spins. But when ΔV = 13.7 V the step is reduced since, according to the Pauli-exclusion principle, only one spin-channel is available.34 The first current step height is therefore twice as high as the second giving two-thirds and one-third in normalized units, respectively. The incorrect pole structure of the UHF Green’s function is a direct consequence of the spin-unrestricted mean-field theory used in deriving it. As we’ve seen, approximations inherent to such theories give rise to a qualitatively incorrect transmission function and lead to significant errors in the thermoelectric properties and nonlinear currentvoltage curves of radical-based molecular junctions. However, by mapping the full radical problem onto the much simpler Anderson model and evaluating that model using a many-body theory, the transport can be accurately described in the low-energy regime most relevant experimentally. In conclusion, we predict the existence of Mott nodes, correlation-induced spectral features arising from interferences between transport pathways involving different charge states of the molecule. These nodes do not stem from the topology of a junction; even linear molecules and molecules without any internal structure, such as the Anderson molecule, exhibit them provided there is on average an unpaired electron on the molecule. Effective single-particle theories like UHF and DFT are known to accurately describe the wavelike nature of the electron in SMJ transport. However, they fail to describe the Coulomb blockade regime3538 where the electric charge on the junction is quantized. Fundamentally, Mott nodes require that the Coulomb blockade and coherent tunneling transport regimes be treated simultaneously, since they are a consequence of quantum coherence between different molecular charge states. In addition to being an interesting interference phenomenon, such nodes can also serve as a litmus test for any many-body transport theory, since their
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prediction necessitates that the particle-like and wavelike nature of the electron be treated on an equal footing.
’ ASSOCIATED CONTENT
bS
Supporting Information. Includes a discussion of the model Hamiltonian used in the MDE calculations, the Kondo temperature and the effect of dephasing on the Mott-node. This material is available free of charge via the Internet at http://pubs.acs.org.
’ AUTHOR INFORMATION Corresponding Author
*E-mail: jbergfi
[email protected].
’ ACKNOWLEDGMENT The authors wish to thank Carmen Herrmann for useful discussions. The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/ 2007-2013)/ERC Grant 258806. This work was supported by the Nonequilibrium Energy Research Center (NERC) which is an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under Award Number DE-SC0000989. ’ REFERENCES (1) Bergfield, J. P.; Stafford, C. A. Phys. Rev. B 2009, 79, 245125. (2) Bergfield, J. P.; Barr, J. D.; Stafford, C. A. ACS Nano 2011, 5, 2707–2714. (3) The energy μ0 = (εHOMO þ εLUMO)/2, where εHOMO and εLUMO are the HOMO and LUMO resonance energies,11 respectively. (4) Cardamone, D. M.; Stafford, C. A.; Mazumdar, S. Nano Lett. 2006, 6, 2422–2426. (5) Solomon, G. C.; Andrews, D. Q.; Goldsmith, R. H.; Hansen, T.; Wasielewski, M. R.; Van Duyne, R. P.; Ratner, M. A. J. Am. Chem. Soc. 2008, 130, 17301–17308. (6) Joachim, C.; Gimzewski, J. K.; Aviram, A. Nature 2000, 408, 541–548. (7) Markussen, T.; Stadler, R.; Thygesen, K. S. Nano Lett. 2010, 10, 4260–4265. (8) Bergfield, J. P.; Solis, M. A.; Stafford, C. A. ACS Nano 2010, 4, 5314–5320. (9) Bergfield, J. P.; Stafford, C. A. Nano Lett. 2009, 9, 3072–3076. (10) Bergfield, J. P.; Jacquod, P.; Stafford, C. A. Phys. Rev. B 2010, 82, 205405. (11) We define the “HOMO resonance” as the addition spectrum peak corresponding to the N 1 f N electronic transition of the molecule, where N is the charge of the neutral molecule. Similarly, we define the “ LUMO resonance” as the addition spectrum peak corresponding to the N f N þ 1 molecular charge transition. (12) Physically this system is spin symmetric (there are no external magnetic fields) and ÆSBæ = 0, where SB is the spin operator. (13) Hettler, M. H.; Wenzel, W.; Wegewijs, M. R.; Schoeller, H. Phys. Rev. Lett. 2003, 90, 076805. (14) Muralidharan, B.; Ghosh, A. W.; Datta, S. Phys. Rev. B 2006, 73, 155410. (15) Begemann, G.; Darau, D.; Donarini, A.; Grifoni, M. Phys. Rev. B 2008, 77, 201406. (16) K€ onig, J.; Schoeller, H.; Sch€on, G. Phys. Rev. Lett. 1997, 78, 4482–4485. (17) Schoeller, H.; K€onig, J. Phys. Rev. Lett. 2000, 84, 3686–3689. (18) Harbola, U.; Mukamel, S. Phys. Rep. 2008, 465, 191–222. 2763
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