Charles A Stafford
 Professor, Physics
 Member of the Graduate Faculty
Contact
 (520) 6264260
 PhysicsAtmospheric Sciences, Rm. 000347
 Tucson, AZ 85721
 stafford@physics.arizona.edu
Awards
 Professor Leon and Pauline Blitzer Award for Excellence in the Teaching of Physics and Related Sciences
 The Blitzer Teaching Award Fund, Spring 2018
 Award for Excellence in Graduate Physics Teaching
 Department of Physics, University of Arizona, Spring 2011
Interests
Teaching
Physics
Research
Theoretical Condensed Matter Physics, Nonequilibrium Quantum Systems
Courses
202122 Courses

Dissertation
PHYS 920 (Fall 2021) 
Independent Study
PHYS 599 (Fall 2021) 
Statistical Mechanics
PHYS 528 (Fall 2021)
202021 Courses

Directed Research
PHYS 492 (Spring 2021) 
Independent Study
PHYS 599 (Spring 2021) 
Introductory Mechanics
PHYS 141 (Spring 2021) 
Senior Capstone
PHYS 498 (Spring 2021) 
Directed Research
PHYS 492 (Fall 2020) 
Independent Study
PHYS 599 (Fall 2020) 
Statistical Mechanics
PHYS 528 (Fall 2020)
201920 Courses

Directed Research
PHYS 492 (Spring 2020) 
Honors Intro Mechanics
PHYS 161H (Spring 2020) 
Independent Study
PHYS 599 (Spring 2020) 
Introductory Mechanics
PHYS 140 (Spring 2020) 
Introductory Mechanics
PHYS 141 (Spring 2020) 
Directed Research
PHYS 492 (Fall 2019) 
Independent Study
PHYS 599 (Fall 2019) 
Quantum Mechanics
PHYS 570A (Fall 2019)
201819 Courses

Honors Thesis
PHYS 498H (Spring 2019) 
Independent Study
PHYS 599 (Spring 2019) 
SolidState Physics
PHYS 460 (Spring 2019) 
Honors Thesis
PHYS 498H (Fall 2018) 
Independent Study
PHYS 599 (Fall 2018) 
Quantum Mechanics
PHYS 570A (Fall 2018)
201718 Courses

Dissertation
PHYS 920 (Spring 2018) 
Honors Intro Mechanics
PHYS 161H (Spring 2018) 
Introductory Mechanics
PHYS 141 (Spring 2018) 
Dissertation
PHYS 920 (Fall 2017) 
Independent Study
PHYS 499 (Fall 2017) 
Independent Study
PHYS 599 (Fall 2017) 
Statistical Mechanics
PHYS 528 (Fall 2017)
201617 Courses

Directed Research
PHYS 492 (Spring 2017) 
Dissertation
PHYS 920 (Spring 2017) 
Honors Intro Mechanics
PHYS 161H (Spring 2017) 
Senior Capstone
PHYS 498 (Spring 2017) 
Directed Research
PHYS 492 (Winter 2016) 
Independent Study
PHYS 499 (Winter 2016) 
Directed Research
PHYS 492 (Fall 2016) 
Dissertation
PHYS 920 (Fall 2016) 
Statistical Mechanics
PHYS 528 (Fall 2016)
201516 Courses

Directed Research
PHYS 392 (Summer I 2016) 
Honors Thesis
PHYS 498H (Summer I 2016) 
Directed Research
PHYS 392 (Spring 2016) 
Directed Research
PHYS 492 (Spring 2016) 
Dissertation
PHYS 920 (Spring 2016) 
Independent Study
PHYS 599 (Spring 2016) 
Quantum Theory
PHYS 371 (Spring 2016)
Scholarly Contributions
Journals/Publications
 Shastry, A., Xu, Y., & Stafford, C. A. (2019). The third law of thermodynamics in open quantum systems. The Journal of Chemical Physics, 151(6), 064115.
 Inui, S., Stafford, C. A., & Bergfield, J. P. (2018). Emergence of Fourier's law of heat transport in quantum electron systems. ACS Nano, 12(5), 4304–4311. doi:10.1021/acsnano.7b08816More infoThe microscopic origins of Fourier's venerable law of thermal transport inquantum electron systems has remained somewhat of a mystery, given thatprevious derivations were forced to invoke intrinsic scattering rates farexceeding those occurring in real systems. We propose an alternativehypothesis, namely, that Fourier's law emerges naturally if many quantum statesparticipate in the transport of heat across the system. We test this hypothesissystematically in a graphene flake junction, and show that the temperaturedistribution becomes nearly classical when the broadening of the individualquantum states of the flake exceeds their energetic separation. We develop athermal resistor network model to investigate the scaling of the sample andcontact thermal resistances, and show that the latter is consistent withclassical thermal transport theory in the limit of large level broadening.[Journal_ref: ]
 Stafford, C. A., & Shastry, A. (2017). Local entropy of a nonequilibrium fermion system. The Journal of Chemical Physics, 146(9), 092324.
 Shastry, A., & Stafford, C. A. (2016). Temperature and voltage measurement in quantum systems far from equilibrium. Phys. Rev. B, 94, 155433.
 Stafford, C. A. (2016). Local temperature of an interacting quantum system far from equilibrium. Phys. Rev. B, 93, 245403.
 Stafford, C., & Shastry, A. (2016). "Local entropy of a nonequilibrium fermion system". ArXiv eprints.
 Bergfield, J. P., Ratner, M. A., Stafford, C. A., & Di Ventra, M. (2015). Tunable quantum temperature oscillations in graphene nanostructures. PHYSICAL REVIEW B, 91.
 Gong, L., Buerki, J., Stafford, C. A., & Stein, D. L. (2015). Lifetimes of metal nanowires with broken axial symmetry. PHYSICAL REVIEW B, 91.
 Lusk, M. T., Stafford, C. A., Zimmerman, J. D., & Carr, L. D. (2015). Control of exciton transport using quantum interference. PHYSICAL REVIEW B, 92.
 Shastry, A., & Stafford, C. A. (2015). Cold spots in quantum systems far from equilibrium: Local entropies and temperatures near absolute zero. PHYSICAL REVIEW B, 92.
 Bergfield, J. P., & Stafford, C. A. (2014). Thermoelectric corrections to quantum voltage measurement. PHYSICAL REVIEW B, 90(23), 235438.More infoA generalization of Buttiker's voltage probe concept for nonzero temperatures is an open third terminal of a quantum thermoelectric circuit. An explicit analytic expression for the thermoelectric correction to an ideal quantum voltage measurement in linear response is derived and interpreted in terms of local Peltier cooling/heating within the nonequilibrium system. The thermoelectric correction is found to be large (up to +/ 24% of the peak voltage) in a prototypical ballistic quantum conductor (graphene nanoribbon). The effects of measurement nonideality are also investigated. Our findings have important implications for precision local electrical measurements.
 Meair, J., Bergfield, J. P., Stafford, C. A., & Jacquod, P. (2014). Local temperature of outofequilibrium quantum electron systems. PHYSICAL REVIEW B, 90(3), 035407.More infoWe show how the local temperature of outofequilibrium, quantum electron systems can be consistently defined with the help of an external voltage and temperature probe. We determine sufficient conditions under which the temperature measured by the probe (i) is independent of details of the systemprobe coupling, (ii) is equal to the temperature obtained from an independent currentnoise measurement, (iii) satisfies the transitivity condition expressed by the zeroth law of thermodynamics, and (iv) is consistent with Carnot's theorem. This local temperature therefore characterizes the system in the common sense of equilibrium thermodynamics, but remains well defined even in outofequilibrium situations with no local equilibrium.
 Barr, J. D., Stafford, C. A., & Bergfield, J. P. (2013). Effective field theory of interacting pi electrons. PHYSICAL REVIEW B, 86(11).More infoWe develop a pielectron effective field theory (piEFT) wherein the twobody Hamiltonian for a pielectron system is expressed in terms of three effective parameters: the piorbital quadrupole moment, the onsite repulsion, and a dielectric constant. As a first application of this piEFT, we develop a model of screening in molecular junctions based on image multipole moments, and use this to investigate the reduction of the HOMOLUMO gap of benzene. Beyond this, we also use piEFT to calculate the differential conductance spectrum of the prototypical benzenedithiolAu singlemolecule junction and the pielectron contribution to the van der Waals interaction between benzene and a metallic electrode.
 Bergfield, J. P., Story, S. M., Stafford, R. C., & Stafford, C. A. (2013). Probing Maxwell's Demon with a Nanoscale Thermometer. ACS NANO, 7(5), 44294440.More infoA precise definition for a quantum electron thermometer is given, as an electron reservoir coupled locally (e.g., by tunneling) to a sample, and brought into electrical and thermal equilibrium with it. A realistic model of a scanning thermal microscope with atomic resolution is then developed, including the effect of thermal coupling of the probe to the ambient environment. We show that the temperatures of individual atomic orbitals or bonds in a conjugated molecule with a temperature gradient across it exhibit quantum oscillations, whose origin can be traced to a realization of Maxwell's demon at the singlemolecule level. These oscillations may be understood in terms of the rules of covalence describing bonding in pielectron systems. Fourier's law of heat conduction is recovered as the resolution of the temperature probe Is reduced, indicating that the macroscopic law emerges as a consequence of coarse graining.
 Bergfield, J. P., Story, S. M., Stafford, R. C., & Stafford, C. A. (2013). Probing maxwell's demon with a nanoscale thermometer. ACS Nano, 7(5), 44294440.More infoPMID: 23651014;Abstract: A precise definition for a quantum electron thermometer is given, as an electron reservoir coupled locally (e.g., by tunneling) to a sample, and brought into electrical and thermal equilibrium with it. A realistic model of a scanning thermal microscope with atomic resolution is then developed, including the effect of thermal coupling of the probe to the ambient environment. We show that the temperatures of individual atomic orbitals or bonds in a conjugated molecule with a temperature gradient across it exhibit quantum oscillations, whose origin can be traced to a realization of Maxwell's demon at the singlemolecule level. These oscillations may be understood in terms of the rules of covalence describing bonding in πelectron systems. Fourier's law of heat conduction is recovered as the resolution of the temperature probe is reduced, indicating that the macroscopic law emerges as a consequence of coarse graining. © 2013 American Chemical Society.
 Barr, J. D., Stafford, C. A., & Bergfield, J. P. (2012). Effective field theory of interacting π electrons. Physical Review B  Condensed Matter and Materials Physics, 86(11).More infoAbstract: We develop a πelectron effective field theory (πEFT) wherein the twobody Hamiltonian for a πelectron system is expressed in terms of three effective parameters: the πorbital quadrupole moment, the onsite repulsion, and a dielectric constant. As a first application of this πEFT, we develop a model of screening in molecular junctions based on image multipole moments, and use this to investigate the reduction of the HOMOLUMO gap of benzene. Beyond this, we also use πEFT to calculate the differential conductance spectrum of the prototypical benzenedithiolAu singlemolecule junction and the πelectron contribution to the van der Waals interaction between benzene and a metallic electrode. © 2012 American Physical Society.
 Barr, J., Bergfield, J., & Stafford, C. (2012). Effective Field Theory of InteractingpiElectrons. Physical Review B, 86, 115403.
 Bergfield, J. P., Barr, J. D., & Stafford, C. A. (2012). Transmission eigenvalue distributions in highly conductive molecular junctions. BEILSTEIN JOURNAL OF NANOTECHNOLOGY, 3, 4051.More infoBackground: The transport through a quantumscale device may be uniquely characterized by its transmission eigenvalues tau(n). Recently, highly conductive singlemolecule junctions (SMJ) with multiple transport channels (i.e., several tau(n) > 0) have been formed from benzene molecules between Pt electrodes. Transport through these multichannel SMJs is a probe of both the bonding properties at the leadmolecule interface and of the molecular symmetry.
 Bergfield, J. P., Barr, J. D., & Stafford, C. A. (2012). Transmission eigenvalue distributions in highly conductive molecular junctions. Beilstein Journal of Nanotechnology, 3(1), 4051.More infoPMID: 22428095;PMCID: PMC3304317;Abstract: Background: The transport through a quantumscale device may be uniquely characterized by its transmission eigenvalues τ n. Recently, highly conductive singlemolecule junctions (SMJ) with multiple transport channels (i.e., several τ n > 0) have been formed from benzene molecules between Pt electrodes. Transport through these multichannel SMJs is a probe of both the bonding properties at the leadmolecule interface and of the molecular symmetry. Results: We use a manybody theory that properly describes the complementary waveparticle nature of the electron to investigate transport in an ensemble of PtbenzenePt junctions. We utilize an effectivefield theory of interacting πelectrons to accurately model the electrostatic influence of the leads, and we develop an ab initio tunneling model to describe the details of the leadmolecule bonding over an ensemble of junction geometries. We also develop a simple decomposition of transmission eigenchannels into molecular resonances based on the isolated resonance approximation, which helps to illustrate the workings of our manybody theory, and facilitates unambiguous interpretation of transmission spectra. Conclusion: We confirm that PtbenzenePt junctions have two dominant transmission channels, with only a small contribution from a third channel with τ n
 Bergfield, J. P., Liu, Z., Burke, K., & Stafford, C. A. (2012). Bethe Ansatz Approach to the Kondo Effect within DensityFunctional Theory. PHYSICAL REVIEW LETTERS, 108(6).More infoTransport through an Anderson junction (two macroscopic electrodes coupled to an Anderson impurity) is dominated by a Kondo peak in the spectral function at zero temperature. We show that the singleparticle KohnSham potential of densityfunctional theory reproduces the linear transport, despite the lack of a Kondo peak in its spectral function. Using Bethe ansatz techniques, we calculate this potential for all coupling strengths, including the crossover from meanfield behavior to charge quantization caused by the derivative discontinuity. A simple and accurate interpolation formula is also given.
 Bergfield, J. P., Liu, Z., Burke, K., & Stafford, C. A. (2012). Bethe ansatz approach to the Kondo effect within densityfunctional theory. Physical Review Letters, 108(6).More infoPMID: 22401100;Abstract: Transport through an Anderson junction (two macroscopic electrodes coupled to an Anderson impurity) is dominated by a Kondo peak in the spectral function at zero temperature. We show that the singleparticle KohnSham potential of densityfunctional theory reproduces the linear transport, despite the lack of a Kondo peak in its spectral function. Using Bethe ansatz techniques, we calculate this potential for all coupling strengths, including the crossover from meanfield behavior to charge quantization caused by the derivative discontinuity. A simple and accurate interpolation formula is also given. © 2012 American Physical Society.
 Burki, J., Stafford, C., & Stein, D. (2012). A NanoTransistor Based on GateInduced Thermal Switching. International Journal of Theoretical and Applied Nanotechnology, 1, 3037.
 Gong, L., Buerki, J., Stafford, C. A., & Stein, D. L. (2012). Lifetimes of metal nanowires with broken axial symmetry. PHYSICAL REVIEW B, 91(3).More infoWe present a theoretical approach for understanding the stability of simple metal nanowires, in particular, monovalent metals such as the alkalis and noble metals. Their cross sections are of order 1 nm, so that small perturbations from external (usually thermal) noise can cause large geometrical deformations. The nanowire lifetime is defined as the time required for making a transition into a state with a different crosssectional geometry. This can be a simple overall change in radius, or a change in the crosssection shape, or both. We develop a stochastic field theoretical model to describe this noiseinduced transition process in which the initial and final states correspond to locally stable states on a potential surface derived by solving the Schrodinger equation for the electronic structure of the nanowire numerically. The numerical string method is implemented to determine the optimal transition path governing the lifetime. Using these results, we tabulate the lifetimes of sodium and gold nanowires for several different initial geometries.
 Liu, Z., Bergfield, J. P., Burke, K., & Stafford, C. A. (2012). Accuracy of density functionals for molecular electronics: The Anderson junction. PHYSICAL REVIEW B, 85(15).More infoThe exact groundstate exchangecorrelation functional of KohnSham density functional theory yields the exact transmission through an Anderson junction at zero bias and temperature. The exact impurity charge susceptibility is used to construct the exact exchangecorrelation potential. We analyze the successes and limitations of various types of approximations, including smooth and discontinuous functionals of the occupation, as well as symmetrybroken approaches.
 Liu, Z., Bergfield, J. P., Burke, K., & Stafford, C. A. (2012). Accuracy of density functionals for molecular electronics: The Anderson junction. Physical Review B  Condensed Matter and Materials Physics, 85(15).More infoAbstract: The exact groundstate exchangecorrelation functional of KohnSham density functional theory yields the exact transmission through an Anderson junction at zero bias and temperature. The exact impurity charge susceptibility is used to construct the exact exchangecorrelation potential. We analyze the successes and limitations of various types of approximations, including smooth and discontinuous functionals of the occupation, as well as symmetrybroken approaches. © 2012 American Physical Society.
 Solomon, G. C., Bergfield, J. P., Stafford, C. A., & Ratner, M. A. (2012). When "small" terms matter: Coupled interference features in the transport properties of crossconjugated molecules. BEILSTEIN JOURNAL OF NANOTECHNOLOGY, 2, 862871.More infoQuantum interference effects offer opportunities to tune the electronic and thermoelectric response of a quantumscale device over orders of magnitude. Here we focus on singlemolecule devices, in which interference features may be strongly affected by both chemical and electronic modifications to the system. Although not always desirable, such a susceptibility offers insight into the importance of "small" terms, such as throughspace coupling and manybody chargecharge correlations. Here we investigate the effect of these small terms using different Hamiltonian models with Huckel, gDFTB and manybody theory to calculate the transport through several singlemolecule junctions, finding that terms that are generally thought to only slightly perturb the transport instead produce significant qualitative changes in the transport properties. In particular, we show that coupling of multiple interference features in crossconjugated molecules by throughspace coupling will lead to splitting of the features, as can correlation effects. The degeneracy of multiple interference features in crossconjugated molecules appears to be significantly more sensitive to perturbations than those observed in equivalent cyclic systems and this needs to be considered if such supernodes are required for molecular thermoelectric devices.
 Bergfield, J. P., Barr, J. D., & Stafford, C. A. (2011). The number of transmission channels through a singlemolecule junction. ACS Nano, 5(4), 27072714.More infoPMID: 21417212;Abstract: We calculate transmission eigenvalue distributions for PtbenzenePt and PtbutadienePt junctions using realistic stateoftheart manybody techniques. An effective field theory of interacting πelectrons is used to include screening and van der Waals interactions with the metal electrodes. We find that the number of dominant transmission channels in a molecular junction is equal to the degeneracy of the molecular orbital closest to the metal Fermi level. © 2011 American Chemical Society.
 Bergfield, J. P., Solomon, G. C., Stafford, C. A., & Ratner, M. A. (2011). Novel quantum interference effects in transport through molecular radicals. Nano Letters, 11(7), 27592764.More infoPMID: 21661755;Abstract: We investigate electronic transport through molecular radicals and predict a correlationinduced transmission node arising from destructive interference between transport contributions from different charge states of the molecule. This quantum interference effect has no singleparticle analog and cannot be described by effective singleparticle 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 lowenergy transport through a radicalbased junction using an Anderson model is given. © 2011 American Chemical Society.
 Schulz, L., Willis, M., Nuccio, L., Shusharov, P., Fratini, S., Pratt, F. L., Gillin, W. P., Kreouzis, T., Heeney, M., Stingelin, N., Stafford, C. A., Beesley, D. J., Bernhard, C., Anthony, J. E., McKenzie, I., Lord, J. S., & Drew, A. J. (2011). Importance of intramolecular electron spin relaxation in small molecule semiconductors. Physical Review B  Condensed Matter and Materials Physics, 84(8).More infoAbstract: Electron spin relaxation rate (eSR) is investigated on several organic semiconductors of different morphologies and molecular structures, using avoided level crossing muon spectroscopy as a local spin probe. We find that two functionalized acenes (polycrystalline tri(isopropyl)silylpentacene and amorphous 5,6,11,12tetraphenyltetracene) exhibit eSRs with an Arrheniuslike temperature dependence, each with two characteristic energy scales similar to those expected from vibrations. Polycrystalline tris(8hydroxyquinolate)gallium shows a similar behavior. The observed eSR for these molecules is no greater than 0.85 MHz at 300 K. The variety of crystal structures and transport regimes that these molecules possess, as well as the local nature of the probe, strongly suggest an intramolecular phenomenon general to many organic semiconductors, in contrast to the commonly assumed spin relaxation models based on intermolecular chargecarrier transport. © 2011 American Physical Society.
 Solomon, G. C., Bergfield, J. P., Stafford, C. A., & Ratner, M. A. (2011). When "small" terms matter: Coupled interference features in the transport properties of crossconjugated molecules. Beilstein Journal of Nanotechnology, 2(1), 862871.More infoPMID: 22259770;PMCID: PMC3257512;Abstract: Quantum interference effects offer opportunities to tune the electronic and thermoelectric response of a quantumscale device over orders of magnitude. Here we focus on singlemolecule devices, in which interference features may be strongly affected by both chemical and electronic modifications to the system. Although not always desirable, such a susceptibility offers insight into the importance of "small" terms, such as throughspace coupling and manybody chargecharge correlations. Here we investigate the effect of these small terms using different Hamiltonian models with Hückel, gDFTB and manybody theory to calculate the transport through several singlemolecule junctions, finding that terms that are generally thought to only slightly perturb the transport instead produce significant qualitative changes in the transport properties. In particular, we show that coupling of multiple interference features in crossconjugated molecules by throughspace coupling will lead to splitting of the features, as can correlation effects. The degeneracy of multiple interference features in crossconjugated molecules appears to be significantly more sensitive to perturbations than those observed in equivalent cyclic systems and this needs to be considered if such supernodes are required for molecular thermoelectric devices. © 2011 Solomon et al.
 Stafford, C., Bergfield, J. P., Barr, J. D., & Stafford, C. A. (2011). The number of transmission channels through a singlemolecule junction. ACS nano, 5(4).More infoWe calculate transmission eigenvalue distributions for PtbenzenePt and PtbutadienePt junctions using realistic stateoftheart manybody techniques. An effective field theory of interacting πelectrons is used to include screening and van der Waals interactions with the metal electrodes. We find that the number of dominant transmission channels in a molecular junction is equal to the degeneracy of the molecular orbital closest to the metal Fermi level.
 Bergfield, J. P., Jacquod, P., & Stafford, C. A. (2010). Coherent destruction of Coulomb blockade peaks in molecular junctions. Physical Review B  Condensed Matter and Materials Physics, 82(20).More infoAbstract: Coherent electronic transport in singlemolecule junctions is investigated in the Coulomb blockade regime. Both the transmission phase and probability are calculated for junctions with various contact symmetries. A dramatic suppression of the Coulomb blockade peaks is predicted for junctions where multiple atomic orbitals of the molecule couple to a single electrode although the charging steps are unaffected. © 2010 The American Physical Society.
 Bergfield, J. P., Solis, M. A., & Stafford, C. A. (2010). Giant thermoelectric effect from transmission supernodes. ACS Nano, 4(9), 53145320.More infoPMID: 20735063;Abstract: We predict an enormous orderdependent quantum enhancement of thermoelectric effects in the vicinity of higherorder interferences in the transmission spectrum of a nanoscale junction. Singlemolecule junctions based on 3,3′biphenyl and polyphenyl ether (PPE) are investigated in detail. The nonequilibrium thermodynamic efficiency and power output of a thermoelectric heat engine based on a 1,3benzene junction are calculated using manybody theory and compared to the predictions of the figureofmerit ZT. © 2010 American Chemical Society.
 Stafford, C., Bergfield, J. P., Solis, M. A., & Stafford, C. A. (2010). Giant thermoelectric effect from transmission supernodes. ACS nano, 4(9).More infoWe predict an enormous orderdependent quantum enhancement of thermoelectric effects in the vicinity of higherorder interferences in the transmission spectrum of a nanoscale junction. Singlemolecule junctions based on 3,3'biphenyl and polyphenyl ether (PPE) are investigated in detail. The nonequilibrium thermodynamic efficiency and power output of a thermoelectric heat engine based on a 1,3benzene junction are calculated using manybody theory and compared to the predictions of the figureofmerit ZT.
 Bergfield, J. P., & Stafford, C. A. (2009). Manybody theory of electronic transport in singlemolecule heterojunctions. Physical Review B  Condensed Matter and Materials Physics, 79(24).More infoAbstract: A manybody theory of molecular junction transport based on nonequilibrium Green's functions is developed, which treats coherent quantum effects and Coulomb interactions on an equal footing. The central quantity of the manybody theory is the Coulomb selfenergy matrix ΣC of the junction. ΣC is evaluated exactly in the sequentialtunneling limit, and the correction due to finite tunneling width is evaluated selfconsistently using a conserving approximation based on diagrammatic perturbation theory on the Keldysh contour. Our approach reproduces the key features of both the Coulomb blockade and coherent transport regimes simultaneously in a single unified transport theory. As a first application of our theory, we have calculated the thermoelectric power and differential conductance spectrum of a benzenedithiolgold junction using a semiempirical π electron Hamiltonian that accurately describes the full spectrum of electronic excitations of the molecule up to 810 eV. © 2009 The American Physical Society.
 Bergfield, J. P., & Stafford, C. A. (2009). Thermoelectric signatures of coherent transport in singlemolecule heterojunctions. Nano Letters, 9(8), 30723076.More infoPMID: 19610653;Abstract: An exact expression for the heat current in an interacting nanostructure is derived and used to calculate the thermoelectric response of three representative singlemolecule junctions formed from isoprene, 1,3benzenedithiol, and [18]annulene. Dramatic enhancements of the thermopower S and Lorenz number L are predicted when the junction is tuned across a node in the transmission function, with universal maximum values S max ) (π/3 1/2)(k B/e) and L max ) (7π 2/5)(k B2/e 2). The effect of a finite minimum transmission probability due, e.g., to incoherent processes or additional nonresonant channels, is also considered. © 2009 American Chemical Society.
 Bürki, J., Stafford, C. A., & Stein, D. L. (2008). Order of phase transitions in barrier crossing. Physical Review E  Statistical, Nonlinear, and Soft Matter Physics, 77(6).More infoAbstract: A spatially extended classical system with metastable states subject to weak spatiotemporal noise can exhibit a transition in its activation behavior when one or more external parameters are varied. Depending on the potential, the transition can be first or second order, but there exists no systematic theory of the relation between the order of the transition and the shape of the potential barrier. In this paper, we address that question in detail for a general class of systems whose order parameter is describable by a classical field that can vary in both space and time, and whose zeronoise dynamics are governed by a smooth polynomial potential. We show that a quartic potential barrier can have only secondorder transitions, confirming an earlier conjecture. We then derive, through a combination of analytical and numerical arguments, both necessary and sufficient conditions to have a firstorder vs a secondorder transition in noiseinduced activation behavior, for a large class of systems with smooth polynomial potentials of arbitrary order. We find in particular that the order of the transition is especially sensitive to the potential behavior near the top of the barrier. © 2008 The American Physical Society.
 Dietz, B., Friedrich, T., Metz, J., MiskiOglu, M., Richter, A., Schäfer, F., & Stafford, C. A. (2007). Rabi oscillations at exceptional points in microwave billiards. Physical Review E  Statistical, Nonlinear, and Soft Matter Physics, 75(2).More infoAbstract: We experimentally investigated the decay behavior with time t of resonances near and at exceptional points, where two complex eigenvalues and also the associated eigenfunctions coalesce. The measurements were performed with a dissipative microwave billiard, whose shape depends on two parameters. The t2 dependence predicted at the exceptional point on the basis of a twostate matrix model could be verified. Outside the exceptional point the predicted Rabi oscillations, also called quantum echoes in this context, were detected. © 2007 The American Physical Society.
 Mares, A. I., Urban, D. F., Bürki, J., Grabert, H., Stafford, C. A., & Ruitenbeek, J. V. (2007). Electronic and atomic shell structure in aluminium nanowires. Nanotechnology, 18(26).More infoPMID: 21730404;Abstract: We report experiments on aluminium nanowires in ultrahigh vacuum at room temperature that reveal a periodic spectrum of exceptionally stable structures. Two 'magic' series of stable structures are observed: at low conductance, the formation of stable nanowires is governed by electronic shell effects whereas for larger contacts atomic packing dominates. The crossover between the two regimes is found to be smooth. A detailed comparison of the experimental results to a theoretical stability analysis indicates that, while the main features of the observed electronshell structure are similar to those of alkali and noble metals, a sequence of extremely stable wires plays a unique role in aluminium. This series appears isolated in conductance histograms and can be attributed to 'superdeformed' nonaxisymmetric nanowires. © IOP Publishing Ltd.
 Urban, D. F., Stafford, C. A., & Grabert, H. (2007). Scaling theory of the Peierls charge density wave in metal nanowires. Physical Review B  Condensed Matter and Materials Physics, 75(20).More infoAbstract: The Peierls instability in multichannel metal nanowires is investigated. Hyperscaling relations are established for the finitesize, temperature, and wavevector scaling of the electronic free energy. It is shown that the softening of surface modes at wave vector q=2 kF,ν leads to critical fluctuations of the wire's radius at zero temperature, where kF,ν is the Fermi wave vector of the highest occupied channel. This Peierls charge density wave emerges as the system size becomes comparable to the channel correlation length. Although the Peierls instability is weak in metal nanowires, in the sense that the correlation length is exponentially long, we predict that nanowires fabricated by current techniques can be driven into the chargedensitywave regime under strain. © 2007 The American Physical Society.
 Bürki, J., Stafford, C. A., & Stein, D. L. (2006). Comment on "nonlinear currentvoltage curves of gold quantum point contacts" [Appl. Phys. Lett. 87, 103104 (2005)]. Applied Physics Letters, 88(16).
 Urban, D. F., Bürki, J., Stafford, C. A., & Grabert, H. (2006). Stability and symmetry breaking in metal nanowires: The nanoscale freeelectron model. Physical Review B  Condensed Matter and Materials Physics, 74(24).More infoAbstract: A general linear stability analysis of simple metal nanowires is presented using a continuum approach that correctly accounts for materialspecific surface properties and electronic quantumsize effects. The competition between surface tension and electronshell effects leads to a complex landscape of stable structures as a function of diameter, cross section, and temperature. By considering arbitrary symmetrybreaking deformations, it is shown that the cylinder is the only generically stable structure. Nevertheless, a plethora of structures with broken axial symmetry is found at low conductance values, including wires with quadrupolar, hexapolar, and octupolar cross sections. These nonintegrable shapes are compared to previous results on elliptical cross sections, and their materialdependent relative stability is discussed. © 2006 The American Physical Society.
 Bürki, J., & Stafford, C. A. (2005). On the stability and structural dynamics of metal nanowires. Applied Physics A: Materials Science and Processing, 81(8), 15191525.More infoAbstract: This article presents a brief review of the nanoscale freeelectron model, which provides a continuum description of metal nanostructures. It is argued that surface and quantumsize effects are the two dominant factors in the energetics of metal nanowires, and that much of the phenomenology of nanowire stability and structural dynamics can be understood based on the interplay of these two competing factors. A linear stability analysis reveals that metal nanocylinders with certain magic conductance values G=1,3,6,12,17,23,34,42,51, 67,78,96,... times the conductance quantum are exceptionally stable. A nonlinear dynamical simulation of nanowire structural evolution reveals a universal equilibrium shape consisting of a magic cylinder suspended between unduloidal contacts. The lifetimes of these metastable structures are also computed. © SpringerVerlag 2005.
 Bürki, J., Stafford, C. A., & Stein, D. L. (2005). Theory of metastability in simple metal nanowires. Physical Review Letters, 95(9).More infoAbstract: Thermally induced conductance jumps of metal nanowires are modeled using stochastic GinzburgLandau field theories. Changes in radius are predicted to occur via the nucleation of surface kinks at the wire ends, consistent with recent electron microscopy studies. The activation rate displays nontrivial dependence on nanowire length, and undergoes first or secondorderlike transitions as a function of length. The activation barriers of the most stable structures are predicted to be universal, i.e., independent of the radius of the wire, and proportional to the square root of the surface tension. The reduction of the activation barrier under strain is also determined. © 2005 The American Physical Society.
 Zhang, C. ., Bürki, J., & Stafford, C. A. (2005). Stability of metal nanowires at ultrahigh current densities. Physical Review B  Condensed Matter and Materials Physics, 71(23).More infoAbstract: We develop a generalized grand canonical potential for the ballistic nonequilibrium electron distribution in a metal nanowire with a finite applied bias voltage. Coulomb interactions are treated in the selfconsistent Hartree approximation, in order to ensure gauge invariance. Using this formalism, we investigate the stability and cohesive properties of metallic nanocylinders at ultrahigh current densities. A linear stability analysis shows that metal nanowires with certain magic conductance values can support current densities up to 1011Acm2, which would vaporize a macroscopic piece of metal. This finding is consistent with experimental studies of gold nanowires. Interestingly, our analysis also reveals the existence of reentrant stability zonesgeometries that are stable only under an applied bias. © 2005 The American Physical Society.
 Bürki, J., Stafford, C. A., & Stein, D. L. (2004). Fluctuational instabilities of alkali and noble metal nanowires. Proceedings of SPIE  The International Society for Optical Engineering, 5471, 367379.More infoAbstract: We introduce a continuum approach to studying the lifetimes of monovalent metal nanowires. By modelling the thermal fluctuations of cylindrical nanowires through the use of stochastic GinzburgLandau classical field theories, we construct a selfconsistent approach to the fluctuationinduced "necking" of nanowires. Our theory provides quantitative estimates of the lifetimes for alkali metal nanowires in the conductance range 10 < G/G0 < 100 (where G0 = 2e2/h is the conductance quantum), and allows us to account for qualitative differences in the conductance histograms of alkali vs. noble metal nanowires.
 Urban, D. F., Bürki, J., Yanson, A. I., Yanson, I. K., Stafford, C. A., Ruitenbeek, J. V., & Grabert, H. (2004). Electronic shell effects and the stability of alkali nanowires. Solid State Communications, 131(910 SPEC. ISS.), 609614.More infoAbstract: Experimental conductance histograms for Na nanowires are analyzed in detail and compared to recent theoretical results on the stability of cylindrical and elliptical nanowires, using the freeelectron model. We find a onetoone correspondence between the peaks in the histograms and the most stable nanowire geometries, indicating that several of the commonly observed nanowires have elliptical cross sections. © 2004 Elsevier Ltd. All rights reserved.
 Urban, D. F., Bürki, J., Zhang, C. ., Stafford, C. A., & Grabert, H. (2004). JahnTeller distortions and the supershell effect in metal nanowires. Physical Review Letters, 93(18), 18640311864034.More infoPMID: 15525187;Abstract: The stability analysis of elliptical metal nanowires, using a jellium model, was presented. It was shown that a JahnTeller deformation breaking cylindrical symmetry can be energetically favorable, leading to stable nanowires with elliptic cross sections. It was found that the derived sequence of stable cylindrical and elliptical geometries explains the experimentally observed shell and superstructures for alkali metals. It was suggested that for gold, elliptical nanowires are even more likely to form since their eccentricity is smaller than for alkali metals.
 Bürki, J., Goldstein, R. E., & Stafford, C. A. (2003). Quantum Necking in Stressed Metallic Nanowires. Physical Review Letters, 91(25), 25450112545014.More infoPMID: 14754119;Abstract: The behavior of stressed metallic nanowires with radii comparable to the Fermi wavelength was studied. From a semiclassical energy functional that includes electronshell effects a partial differential equation for nanowire shape evolution was derived. A movement and interaction of kinks connecting locally stable radii was found. The results show that the metallic nanowire subject to tensile stress necks down smoothly as it elongates.
 Zhang, C. ., Kassubek, F., & Stafford, C. A. (2003). Surface fluctuations and the stability of metal nanowires. Physical Review B  Condensed Matter and Materials Physics, 68(16), 16541411654148.More infoAbstract: The surface dynamics and thermodynamics of metal nanowires are investigated in a continuum model. Competition between surface tension and electronshell effects leads to a rich stability diagram, with fingers of stability extending to extremely high temperatures for certain magic conductance values. The linearized dynamics of the nanowire's surface are investigated, including both acoustic surface phonons and surface selfdiffusion of atoms. On the stability boundary, the surface exhibits critical fluctuations, and the nanowire becomes inhomogeneous. Some stability fingers coalesce at higher temperatures, or exhibit overhangs, leading to reentrant behavior. The nonlinear surface dynamics of unstable nanowires are also investigated in a singlemode approximation. We find evidence that some unstable nanowires do not break, but rather neck down to the next stable radius.
 Affleck, I., Simon, P., Eckle, H., Johannesson, H., & Stafford, C. A. (2002). Comment on "Kondo resonance in a mesoscopic ring coupled to a quantum dot: Exact results for the AharonovBohmCasher effects" [1] (multiple letters). Physical Review Letters, 88(13), [d]13970111397021.
 Stafford, C. A. (2002). Metal nanowires: Quantum transport, cohesion, and stability. Physica Status Solidi (B) Basic Research, 230(2), 481489.More infoAbstract: Metal nanowires exhibit a number of interesting properties: their electrical conductance is quantized, their shotnoise is suppressed by the Pauli principle, and they are remarkably strong and stable. We show that many of these properties can be understood quantitatively using a nanoscale generalization of the freeelectron model. Possible technological applications of nanowires are also discussed.
 Eckle, H. ., Johannesson, H., & Stafford, C. A. (2001). Kondo resonance in a mesoscopic ring coupled to a quantum dot: Exact results for the AharonovBohmCasher effects. Physical Review Letters, 87(1), 016602/1016602/4.More infoAbstract: A onedimensional mesoscopic ring of free electrons coupled by a tunnel junction to a quantum dot at a Kondo resonance is described by an exactly solvable model for certain privileged values of the AharonovBohmCasher fluxes piercing the ring. This allows for an exact nonperturbative analysis of the relevant physics.
 Kassubek, F., Stafford, C. A., Grabert, H., & Goldstein, R. E. (2001). Quantum suppression of the Rayleigh instability in nanowires. Nonlinearity, 14(1), 167177.More infoAbstract: A linear stability analysis of metallic nanowires is performed in the freeelectron model using quantum chaos techniques. It is found that the classical instability of a long wire under surface tension can be completely suppressed by electronic shell effects, leading to stable cylindrical configurations whose electrical conductance is a magic number 1, 3, 5, 6, . . . times the quantum of conductance. Our results are quantitatively consistent with recent experiments with alkali metal nanowires.
 Eckle, H. ., Johannesson, H., & Stafford, C. A. (2000). AharonovBohm/Casher effect in a Kondo ring. Physica B: Condensed Matter, 284288(PART II), 18721873.More infoAbstract: The influence of a magnetic impurity or ultrasmall quantum dot on the spin and charge persistent currents of a mesoscopic ring is investigated. The system consists of electrons in a onedimensional ring threaded by spindependent AharonovBohm/Casher fluxes, and coupled via an antiferromagnetic exchange interaction to a localized electron. The problem is mapped onto a Kondo model for the evenparity channel plus free electrons in the oddparity channel. The twisted boundary conditions representing the fluxes couple states of opposite parity unless the twist angles φ + α satisfy φ + α = f απ, where f α are integers, with spin index α = ↑, ↓,. For these special values of φ + α, the model is solvable by the Bethe ansatz. © 2000 Elsevier Science B.V. All rights reserved.
 Eckle, H. ., Johannesson, H., & Stafford, C. A. (2000). Kondo Impurity in a Mesoscopic Ring: Charge Persistent Current. Journal of Low Temperature Physics, 118(56), 475483.More infoAbstract: We study the influence of a magnetic impurity or ultrasmall quantum dot on the charge persistent current of a mesoscopic ring. The system consists of electrons in a onedimensional ring threaded by spindependent AharonovBohm/Casher fluxes, coupled via an antiferromagnetic exchange interaction to a localized electron. By passing to a basis of electron states with definite parities, the problem is mapped onto a Kondo model for the evenparity channel plus free electrons in the oddparity channel. The twisted boundary conditions representing the fluxes couple states of opposite parity unless the twist angles satisfy φα = fαπ, where fα are integers, with spin index α = ↑, ↓. For these special values of φα, the model is solved exactly by a Bethe ansatz. Special cases are investigated in detail. In particular we show that the charge stiffness in the case φ↑ = φ↓ is insensitive to the presence of the magnetic impurity/quantum dot.
 Kassubek, F., Stafford, C. A., & Grabert, H. (2000). On universality in metallic nanocohesion. Physica B: Condensed Matter, 280(14), 438439.More infoAbstract: A semiclassical trace formula for the mesoscopic oscillations of the cohesive force in a metallic nanowire of constant cross section is derived. It is shown that the force oscillations are universal, in the sense that their r.m.s. amplitude is independent of the area of the cross section, only for cross sections with a continuous onedimensional symmetry. For a wire of rectangular cross section, the r.m.s. amplitude of the force oscillations is shown to be proportional to the aspect ratio of the cross section.
 Bürki, J., Stafford, C. A., Zotos, X., & Baeriswyl, D. (1999). Cohesion and conductance of disordered metallic point contacts. Physical Review B  Condensed Matter and Materials Physics, 60(7), 50005008.More infoAbstract: The cohesion and conductance of a point contact in a twodimensional metallic nanowire are investigated in an independentelectron model with hardwall boundary conditions. All properties of the nanowire are related to the Green function of the electronic scattering problem, which is solved exactly via a modified recursive Green function algorithm. Our results confirm the validity of a previous approach based on the WKB approximation for a long constriction, but find an enhancement of cohesion for shorter constrictions. Surprisingly, the cohesion persists even after the last conductance channel has been closed. For disordered nanowires, a statistical analysis yields welldefined peaks in the conductance histograms even when individual conductance traces do not show welldefined plateaus. The shifts of the peaks below integer multiples of 2e2/h, as well as the peak heights and widths, are found to be in excellent agreement with predictions based on random matrix theory, and are similar to those observed experimentally. Thus abrupt changes in the wire geometry are not necessary for reproducing the observed conductance histograms. The effect of disorder on cohesion is found to be quite strong and very sensitive to the particular configuration of impurities at the center of the constriction. ©1999 The American Physical Society.
 Kassubek, F., Stafford, C. A., & Grabert, H. (1999). Force, charge, and conductance of an ideal metallic nanowire. Physical Review B  Condensed Matter and Materials Physics, 59(11), 75607574.More infoAbstract: The conducting and mechanical properties of a metallic nanowire formed at the junction between two macroscopic metallic electrodes are investigated. Both two and threedimensional wires with a widenarrowwide geometry are modeled in the freeelectron approximation with hardwall boundary conditions. Tunneling and quantumsize effects are treated exactly using the scattering matrix formalism. Oscillations of order EF/λF in the tensile force are found when the wire is stretched to the breaking point, which are synchronized with quantized jumps in the conductance. The force and conductance are shown to be essentially independent of the width of the wide sections (electrodes). The exact results are compared with an adiabatic approximation; the latter is found to overestimate the effects of tunneling, but still gives qualitatively reasonable results for nanowires of length L≫λF, even for this abrupt geometry. In addition to the force and conductance, the net charge of the nanowire is calculated and the effects of screening are included within linear response theory. Mesoscopic charge fluctuations of order e are predicted that are strongly correlated with the mesoscopic force fluctuations. The local density of states at the Fermi energy exhibits riontrivial behavior that is correlated with fine structure in the force and conductance, showing the importance of treating the whole wire as a mesoscopic system rather than treating only the narrow part. © 1999 The American Physical Society.
 Stafford, C. A., Kassubek, F., Bürki, J., & Grabert, H. (1999). Universality in Metallic Nanocohesion: A Quantum Chaos Approach. Physical Review Letters, 83(23), 48364839.More infoAbstract: Convergent semiclassical trace formulas for the density of states and the cohesive force of a narrow constriction in an electron gas, whose classical motion is either chaotic or integrable, are derived. It is shown that mode quantization in a metallic point contact or nanowire leads to universal oscillations in its cohesive force: the amplitude of the oscillations depends only on a dimensionless quantum parameter describing the crossover from chaotic to integrable motion, and is of order 1 nN, in agreement with recent experiments.
 Kotlyar, R., Stafford, C. A., & Sarma, S. D. (1998). Addition spectrum, persistent current, and spin polarization in coupled quantum dot arrays: Coherence, correlation, and disorder. Physical Review B  Condensed Matter and Materials Physics, 58(7), 39894013.More infoAbstract: The groundstate persistent current and electron addition spectrum in twodimensional quantum dot arrays and onedimensional quantum dot rings, pierced by an external magnetic flux, are investigated using the extended Hubbard model. The collective multidot problem is shown to map exactly into the strongfield noninteracting finitesize Hofstadter butterfly problem at the spin polarization transition. The finitesize Hofstadter problem is discussed, and an analytical solution for limiting values of flux is obtained. In weak fields we prediet interesting flux periodic oscillations in the spin component along the quantization axis with a periodicity given by vh/e (ν≤1). The sensitivity of the calculated persistent current to interaction and disorder is shown to reflect the intricacies of various MottHubbard quantum phase transitions in twodimensional systems: the persistent current is suppressed in the antiferromagnetic Mottinsulating phase governed by intradot Coulomb interactions; the persistent current is maximized at the spin density wavecharge density wave transition driven by the nearestneighbor interdot interaction; the Mottinsulating phase persistent current is enhanced by the longrange interdot interactions to its noninteracting value; the strong suppression of the noninteracting current in the presence of random disorder is seen only at large disorder strengths; at halffilling even a relatively weak intradol Coulomb interaction enhances the disordered noninteracting system persistent current; in general, the suppression of the persistent current by disorder is less significant in the presence of the longrange interdot Coulomb interaction.
 Kotlyar, R., Stafford, C. A., & Sarma, S. D. (1998). Correlated charge polarization in a chain of coupled quantum dots. Physical Review B  Condensed Matter and Materials Physics, 58(4), R1746R1749.More infoAbstract: Coherent charge transfer in a linear array of tunnelcoupled quantum dots, electrostatically coupled to external gates, is investigated using the Bethe ansatz for a symmetrically biased Hubbard chain. Charge polarization in this correlated system is shown to proceed via two distinct processes: formation of bound states in the metallic phase, and chargetransfer processes corresponding to a superposition of antibound states at opposite ends of the chain in the Mottinsulating phase. The polarizability in the insulating phase of the chain exhibits a universal scaling behavior, while the polarization charge in the metallic phase of the model is shown to be quantized in units of e/2.
 Mila, F., Stafford, C. A., & Capponi, S. (1998). Persistent currents in a Möbius ladder: A test of interchain coherence of interacting electrons. Physical Review B  Condensed Matter and Materials Physics, 57(3), 14571460.More infoAbstract: Persistent currents in a Moebius ladder are shown to be very sensitive to the effects of intrachain interactions on the hopping of electrons between chains. Their periodicity as a function of flux is doubled for strong enough repulsive interactions because electrons cannot hop coherently between the chains and have to travel along the full edge of the Moebius ladder, thus encircling the flux twice. The interplay of topology and interactions is shown to lead to interesting finitesize effects on the odd harmonics of the persistent current.
 Stafford, C. A. (1998). Quantum theory of metallic nanocohesion. Physica E: LowDimensional Systems and Nanostructures, 1(14), 310312.More infoAbstract: The conducting and mechanical properties of ultrasmall metallic structures are calculated using the electronic scattering matrix, evaluated in the free electron approximation. Force oscillations of the order εF/λF are predicted when a metallic quantum wire is stretched to the breaking point, which are synchronized with quantized jumps in the conductance. Coherent backscattering from impurities is shown to lead to fine structure (a "quantum fingerprint") in the force oscillations. © 1997 Elsevier Science B.V. All rights reserved.
 Stafford, C. A., Kotlyar, R., & Sarma, S. D. (1998). Coherent resonant tunneling through an artificial molecule. Physical Review B  Condensed Matter and Materials Physics, 58(11), 70917102.More infoAbstract: Coherent resonant tunneling through an array of quantum dots in an inhomogeneous magnetic field is investigated using an extended Hubbard model. Both the multiterminal conductance of an array of quantum dots and the persistent current of a quantumdot molecule embedded in an AharanovBohm ring are calculated. The conductance and persistent current are calculated analytically for the case of a double quantum dot and numerically for larger arrays using a multiterminal BreitWignertype formula, which allows for the explicit inclusion of inelastic processes. Cotunneling corrections to the persistent current are also investigated, and it is shown that the sign of the persistent current on resonance may be used to determine the spin quantum numbers of the ground state and lowlying excited states of an artificial molecule. An inhomogeneous magnetic field is found to strongly suppress transport due to pinning of the spindensitywave ground state of the system, and giant magnetoresistance is predicted to result from the ferromagnetic transition induced by a uniform external magnetic field.
 Stafford, C. A., & Sarma, S. D. (1997). Coherent magnetotransport through an artificial molecule. Physics Letters, Section A: General, Atomic and Solid State Physics, 230(12), 7378.More infoAbstract: Coherent resonant tunneling through a double quantum dot in an inhomogeneous magnetic field is investigated using a generalized Hubbard model. The conductance is calculated analytically for the case of a single spin1/2 orbital per dot and numerically for dots containing multiple orbitals using a multiterminal BreitWigner type formula, which allows for the explicit inclusion of inelastic processes. Giant spindependent manybody corrections to the transport are predicted to be a clear signature of the formation of a molecularlike state in the system. © 1997 Elsevier Science B.V.
 Stafford, C. A., & Wang, D. F. (1997). Interactioninduced enhancement and oscillations of the persistent current. Physical Review B  Condensed Matter and Materials Physics, 56(8), R4383R4386.More infoAbstract: The persistent current I in integrable models of multichannel rings with both short and longranged interactions is investigated. I is found to oscillate in sign and increase in magnitude with increasing interaction strength due to interactioninduced correlations in the currents contributed by different channels. For sufficiently strong interactions, the contributions of all channels are found to add constructively, leading to a giant enhancement of I. Numerical results confirm that this paritylocking effect is robust with respect to intersubband scattering.
 Stafford, C. A., & Wang, D. F. (1997). Paritylocking effect in a stronglycorrelated ring. Zeitschrift fur Physik BCondensed Matter, 103(2), 323325.More infoAbstract: Orbital magnetism in an integrable model of a multichannel ring with longranged electronelectron interactions is investigated. In a noninteracting multichannel system, the response to an external magnetic flux is the sum of many diamagnetic and paramagnetic contributions, but we find that for sufficiently strong correlations, the contributions of all channels add constructively, leading to a parity (diamagnetic or paramagnetic) which depends only on the total number of electrons. Numerical results confirm that this paritylocking effect is robust with respect to subband mixing due to disorder. © SpringerVerlag 1997.
 Stafford, C. A., Baeriswyl, D., & Burki, J. (1997). Jellium model of metallic nanocohesion. Physical Review Letters, 79(15), 28632866.More infoAbstract: A unified treatment of the cohesive and conducting properties of metallic nanostructures in terms of the electronic scattering matrix is developed. A simple picture of metallic nanocohesion in which conductance channels act as delocalized chemical bonds is derived in the jellium approximation. Universal force oscillations of order εF/λF are predicted when a metallic quantum wire is stretched to the breaking point, which are synchronized with quantized jumps in the conductance.
 Wingreen, N. S., & Stafford, C. A. (1997). Quantumdot cascade laser: Proposal for an ultralowthreshold semiconductor laser. IEEE Journal of Quantum Electronics, 33(7), 11701173.More infoAbstract: We propose a quantumdot version of the quantumwell cascade laser of Faist et al. The elimination of single phonon decays by the threedimensional confinement implies a several orderofmagnitude reduction in the threshold current. The requirements on dot size (1020 nm) and on dot density and uniformity [one coupled pair of dots per (200 nm 3) with 5% nonuniformity] are dose to current technology.
 Büttiker, M., & Stafford, C. A. (1996). Charge transfer induced persistent current and capacitance oscillations. Physical Review Letters, 76(3), 495498.More infoAbstract: The transfer of charge between different regions of a phasecoherent mesoscopic sample is investigated. Charge transfer from a side branch quantum dot into a ring changes the persistent current through a sequence of plateaus of diamagnetic and paramagnetic states. In contrast, a quantum dot embedded in a ring exhibits sharp resonances in the persistent current, whose sign is independent of the number of electrons in the dot if the total number of electrons in the system is even. It is shown that such a mesoscopic system can be polarized appreciably not only by the application of an external voltage but also via an AharonovBohm flux.
 Stafford, C. A. (1996). Nonlinear conductance in resonant tunneling. Physical Review Letters, 77(13), 27702773.More infoAbstract: A multiterminal conductance formula describing resonant tunneling through an interacting mesoscopic system is derived and used to investigate the nonlinear conductance of a quantum dot. An explicit gaugeinvariant expression for the IV characteristic which depends sensitively on the full capacitance matrix is obtained. A voltage probe is found to have a dramatic effect on the nonlinear conductance.
 Stafford, C. A., & Wingreen, N. S. (1996). Resonant photonassisted tunneling through a double quantum dot: An electron pump from spatial rabi oscillations. Physical Review Letters, 76(11), 19161919.More infoAbstract: The time average of the fully nonlinear current through a double quantum dot, subject to an arbitrary combination of ac and dc voltages, is calculated exactly using the Keldysh nonequilibrium Green function technique. When driven on resonance, the system functions as an efficient electron pump due to Rabi oscillation between the dots. The pumping current is maximum when the coupling to the leads equals the Rabi frequency.
 Liu, D. Z., Hu, B. Y., Stafford, C. A., & Sarma, S. D. (1994). Dynamic magnetoconductance fluctuations and oscillations in mesoscopic wires and rings. Physical Review B, 50(8), 57995802.More infoAbstract: Using a finitefrequency recursive Green'sfunction technique, we calculate the dynamic magnetoconductance fluctuations and oscillations in disordered mesoscopic normalmetal systems, incorporating interparticle Coulomb interactions within a selfconsistent potential method. In a disorderd metal wire, we observe ergodic behavior in the dynamic conductance fluctuations. At low ω, the real part of the conductance fluctuations is essentially given by the dc universal conductance fluctuations while the imaginary part increases linearly from zero, but for ω greater than the Thouless energy and temperature, the fluctuations decrease as ω1/2. Similar frequencydependent behavior is found for the AharonovBohm oscillations in a metal ring. However, the Al'tshulerAronovSpivak oscillations, which predominate at high temperatures or in rings with many channels, are strongly suppressed at high frequencies, leading to interesting crossover effects in the ω dependence of the magnetoconductance oscillations. © 1994 The American Physical Society.
 Stafford, C. A., & Sarma, S. D. (1994). Collective coulomb blockade in an array of quantum dots: A motthubbard approach. Physical Review Letters, 72(22), 35903593.More infoAbstract: We investigate the electron addition spectrum in a class of Hubbardlike models which describe arrays of coupled quantum dots. Interdot tunneling leads to a sequence of two phase transitions separating a region of collective Coulomb blockade from a region where the Coulomb blockade of individual dots is maintained and a region where the Coulomb blockade is destroyed altogether. Observable experimental consequences of our theory are discussed.
 Stafford, C. A. (1993). Unusual lowtemperature thermopower in the onedimensional Hubbard model. Physical Review B, 48(11), 84308433.More infoAbstract: The lowtemperature thermoelectric power of the repulsiveinteraction onedimensional Hubbard model is calculated using an asymptotic Bethe ansatz for holons and spinons. The competition between the entropy carried by the holons and that carried by the backflow of the spinons gives rise to an unusual temperature and doping dependence of the thermopower which is qualitatively similar to that observed in the normal state of highTc superconductors. © 1993 The American Physical Society.
 Stafford, C. A., & Millis, A. J. (1993). Scaling theory of the MottHubbard metalinsulator transition in one dimension. Physical Review B, 48(3), 14091425.More infoAbstract: We use the Bethe ansatz equations to calculate the charge stiffness Dc=(L/2)d2E0/dc2c=0 of the onedimensional repulsiveinteraction Hubbard model for electron densities close to the Mott insulating value of one electron per site (n=1), where E0 is the groundstate energy, L is the circumference of the system (assumed to have periodic boundary conditions), and (Latin small letter h with strokec/e)c is the magnetic flux enclosed. We obtain an exact result for the asymptotic form of Dc(L) as L at n=1, which defines and yields an analytic expression for the correlation length in the Mott insulating phase of the model as a function of the onsite repulsion U. In the vicinity of the zerotemperature critical point U=0, n=1, we show that the charge stiffness has the hyperscaling form Dc(n,L,U)=Y+(/L), where =1n and Y+ is a universal scaling function which we calculate. The physical significance of in the metallic phase of the model is that it defines the characteristic size of the chargecarrying solitons, or holons. We construct an explicit mapping for arbitrary U and 1 of the holons onto weakly interacting spinless fermions, and use this mapping to obtain an asymptotically exact expression for the lowtemperature thermopower near the metalinsulator transition, which is a generalization to arbitrary U of a result previously obtained using a weakcoupling approximation, and implies holelike transport for 0
 Stafford, C. A., Millis, A. J., & Shastry, B. S. (1991). Finitesize effects on the optical conductivity of a halffilled Hubbard ring. Physical Review B, 43(16), 1366013663.More infoAbstract: We use the Betheansatz equations to calculate the total and zerofrequency spectral weight in the optical conductivity of the halffilled onedimensional Hubbard model as a function of the lattice size L and the onsite repulsion U. The zerofrequency spectral weight D scales as L1/2exp(L/) as L. Near U=0, varies as the inverse of the LiebWu charge gap. In the strongly correlated regime (Ut), 1=ln(U/t)1.48. $D is negative when L is a multiple of 4, corresponding to a negative inductance. We give a physical explanation of our results in terms of a simple model of ring exchange. The finitesize corrections to the total spectral weight scale as L2. We discuss the implications of our results for exact diagonalization calculations of the optical conductivity. © 1991 The American Physical Society.
 Schaefer, W., SchmittRink, S., & Stafford, C. (1990). Nonlinear optical response of twodimensional magnetoexcitons. Proceedings of SPIE  The International Society for Optical Engineering, 1280, 2434.More infoAbstract: We study the coherent nonlinear optical response of twodimensional excitons in a perpendicular magnetic field. The transition from twodimensional behaviour in the lowfield limit to quasizerodimensional behaviour in high fields is investigated and pronounced manybody effects which are highly field dependent are found. Differential pumpprobe spectra are calculated for various excitation conditions. It is shown that cyclotron motion can be time resolved with ultrashort optical pulses.
 Schafer, W., SchmittRink, S., & Stafford, C. (1990). Nonlinear optical response of 2D magnetoexcitons. XVII International Conference on Quantum Electronics. Digest of, 9899.More infoAbstract: Initial results of a numerical study of the coherent nonlinear optical response of 2D excitons in a strong perpendicular magnetic field are presented. By solving the semiconductor density matrix equations in a Landau basis (with up to 1700 orbitals), the response of the system to pump and probe pulses of various durations, excitation wavelengths, and intensities has been calculated. This includes both the nonlinear probe absorption as well as the twopulse selfdiffraction (fourwave mixing) signal generated parametrically. The results show that (interLandau level) Coulomb interactions generally persist up to the highest magnetic fields, giving rise to pronounced manybody effects in the nonlinear optical response. Timeresolved nonlinear absorption spectra for a cyclotron frequency equal to 16Ry E0 and the corresponding timeintegrated twopulse selfdiffracted signal for R = 3E0 are shown and discussed.
 Stafford, C., SchmittRink, S., & Schaefer, W. (1990). Nonlinear optical response of twodimensional magnetoexcitons. Physical Review B, 41(14), 1000010011.More infoAbstract: We study the nonlinear optical response of twodimensional excitons in a perpendicular magnetic field. The transition from twodimensional behavior in low fields to quasizerodimensional behavior in high fields is investigated, and pronounced interaction effects that are highly field dependent are found. Differential pumpprobe spectra are calculated for both continuouswave and impulsive excitation. It is shown that cyclotron motion can be time resolved with ultrashort optical pulses. © 1990 The American Physical Society.
 Stark, J. B., Knox, W. H., Chemla, D. S., Schäfer, W., SchmittRink, S., & Stafford, C. (1990). Femtosecond dynamics of excitons under extreme magnetic confinement. Physical Review Letters, 65(24), 30333036.More infoAbstract: The resonant optical nonlinearities of GaAs/AlGaAs quantum wells are measured with femtosecond time resolution, as the quasitwodimensional states are further confined into quasizero dimensions by a perpendicular magnetic field. We apply fields up to 12 T to demonstrate that excitonexciton Coulomb interactions are strongly modified by quasizerodimensional confinement, in agreement with manybody theory. These measurements demonstrate for the first time the remarkable result that at high magnetic fields an ensemble of 1s electronhole pairs behaves like a gas of noninteracting particles.
Presentations
 Stafford, C. A. (2019, August). Topological Effects in Quantum Thermodynamics. Workshop on Quantum Transport in Nanoscale Molecular Systems. Telluride, Colorado: TSRC.
 Stafford, C. A. (2019, November). Topological Effects in Quantum Thermodynamics. Chemical Physics Seminar. University of Arizona: Department of Chemistry.
 Stafford, C. A. (2019, October). Topological Effects in Quantum Thermodynamics. Physics Colloquium. Tucson, Arizona: Physics Department.
 Stafford, C. A., Jimenez, M., Shastry, A., & Xu, Y. (2019, March). Topological work in nonequilibrium quantum thermodynamics. APS March Meeting. Boston, MA: APS.
 Shastry, A., Rosales, M., & Stafford, C. A. (2018, March). Singleatom thermometer. APS March Meeting. Los Angeles, CA: American Physical Society.
 Stafford, C. A. (2018, March). Demons, Paradoxes, and the Laws of Quantum Thermodynamics. Blitzer Award Lecture. Tucson, AZ: University of Arizona.
 Stafford, C. A., & Shastry, A. (2018, March). Exact local entropy of a nonequilibrium quantum system. APS March Meeting. Los Angeles, CA: American Physical Society.
 Stafford, C. A., Jimenez, M., & Shastry, A. (2018, March). Maximum cooling of a quantum thermocouple. APS March Meeting. Los Angeles, CA: American Physical Society.
 Xu, Y., Evers, F., & Stafford, C. A. (2018, March). Topological effects in thermodynamics. APS March Meeting. Los Angeles, CA: American Physical Society.
 Inui, S., Shastry, A., & Stafford, C. A. (2017, March). Scanning thermal microscopy using electrical measurements. APS March Meeting. New Orleans, LA: American Physical Society.
 Jimenez, M., Shastry, A., & Stafford, C. A. (2017, October). Cooling limit of a quantum thermocouple. APS Four Corners Section meeting. Fort Collins, CO: American Physical Society.
 Stafford, C. A. (2017, August). Local thermodynamics of a quantum system far from equilibrium. Workshop on Quantum Transport in Nanoscale Molecular Systems, Telluride, CO. Telluride, CO: Telluride Science Research Center.
 Stafford, C. A., & Shastry, A. (2017, March). Local entropy of a nonequilibrium fermion system. APS March Meeting. New Orleans, LA: American Physical Societ.
 Xu, Y., Evers, F., & Stafford, C. A. (2017, October). Topological term in the First Law of Thermodynamics. APS Four Corner Section meeting. Fort Collins, CO: American Physical Society.
 Bergfield, J., Ratner, M., Stafford, C. A., & Di Vantra, M. (2016, March). Thermoelectric Corrections to Quantum Measurement. March Meeting. Baltimore, MD: American Physical Society.
 Shastry, A., & Stafford, C. A. (2016, March). Local temperatures and voltages in quantum systems far from equilibrium. March Meeting. Baltimore, MD: American Physical Society.
 Stafford, C. A. (2016, August). Local thermoelectric probes of nonequilibrium quantum systems. Theoretical Condensed Matter Physics PI Meeting. Gaithersburg, MD: Department of Energy.
 Stafford, C. A. (2016, June). Local thermoelectric measurements of nonequilibrium quantum systems. Theory Colloquium. University of Konstanz, Germany: Physics Department.
 Stafford, C. A. (2016, June). Theory of local thermoelectric measurements. Theory Colloquium. University of Regensburg, Germany: Physics Department.
 Stafford, C. A. (2016, March). Local thermoelectric probes of nonequilibrium quantum systems. March Meeting of the American Physical Society. Baltimore, MD: American Physical Society.
 Stafford, C. A. (2016, May). Local thermodynamic measurements in nonequilibrium quantum systems. Chemical Physics Seminar. University of Arizona: Deparment of Chemistry.
 Xu, Y., & Stafford, C. A. (2016, October). Persistent Heat Current? A Thermodynamic Paradox. 4 Corners Section Meeting. Las Cruces, New Mexico: American Physical Society.
 Stafford, C. A. (2015, July 21). Local thermodynamics of a nonequilibrium quantum system. Workshop on Quantum Transport in Nanoscale Molecular Systems. Telluride, CO: Telluride Science Research Center.
 Stafford, C. A. (2015, March). Attaining local temperatures close to absolute zero in a nonequilibrium quantum. American Physical Society March Meeting. San Antonio, Texas: American Physical Society.
 Stafford, C. A. (2015, March). Local temperature of an interacting quantum system far from equilibrium. American Physical Society March Meeting. San Antonio, Texas: American Physical Society.
 Stafford, C. A. (2015, March). Tunable Quantum Temperature Oscillations in Graphene Nanostructures. American Physical Society March Meeting. San Antonio, Texas: American Physical Society.
 Stafford, C. A. (2015, March). Tuning nanoscale thermoelectricity with electronelectron interactions. American Physical Society March Meeting. San Antonio, Texas: American Physical Society.
 Stafford, C. A. (2015, May 12). Converting Heat to Electricity: The Demon is in the Details. TEDx Tucson Salon, It’s Electric. Tucson Convention Center: TEDx Tucson Salon.More infoDr. Stafford's talk discusses his ideas of using nanotechnology to convert waste heat into electricity. In particular, he shows how we might take advantage of quantum oddities in the way heat is transferred across specific atomic structures, so that we could produce costeffective, nonpolluting electricity.
 Stafford, C. A. (2015, September 17). Local thermodynamic observables of nonequilibrium quantum systems. International Conference on SingleMolecule Electronics. Regensburg, Germany: Deutsche Forschungsgemeinschaft.
 Stafford, C. A. (2014, March). On local temperatures near absolute zero in nonequilibrium quantum systems. 2014 March Meeting. Denver, CO: American Physical Society.
 Stafford, C. A. (2014, March). Thermoelectric corrections to quantum voltage measurement. 2014 March Meeting. Denver, CO.
 Stafford, C. A. (2014, March). Tunable Quantum Temperature Oscillations in Graphene and Carbon Nanoribbons. 2014 March Meeting. Denver, CO.
 Stafford, C. A. (2014, October). Using NonEquilibrium Green's Functions to Study Nanoscale Thermoelectricity. APS Four Corners Section Meeting. Orem, Utah: American Physical Society.
 Stafford, C. A. (2013, April). Probing Maxwell’s Demon with a Nanoscale Thermometer. Physics Colloquium. Tucson, AZ: University of Arizona.More infoDate: 04/19
 Stafford, C. A. (2013, November). Local Temperature of OutofEquilibrium Quantum Electron Systems. Chemical Physics Seminar. Tucson, AZ: University of Arizona.More infoDate: 11/04
Poster Presentations
 Stafford, C. A. (2018, August). ManyBody Theory of Energy Transport and Conversion at the Nanoscale. Department of Energy Theoretical Condensed Matter Physics Principal Investigators’ Meeting. Gaithersburg, Maryland: U.S. Department of Energy.
 Shastry, A., & Stafford, C. A. (2015, December). Local temperatures and voltages in quantum systems far from equilibrium. Big Ideas in Quantum Materials. La Jolla, CA: Gordon and Betty Moore foundation.
 Shastry, A., & Stafford, C. A. (2014, October). Local temperature and voltage measurement of a quantum system far from equilibrium. International Workshop on Correlations, criticality and coherence in quantum systems. Evora, Portugal.
 Stafford, C. A. (2014, August). Local Thermodynamics of a Quantum System far from Equilibrium. US Department of Energy Theoretical Condensed Matter PI Meeting. Gaithersburg, Maryland: US Department of Energy.
 Stafford, C. A. (2013, August). Probing Maxwell’s Demon with a Nanoscale Thermometer. Workshop on QuantumTransport in Nanoscale Molecular Systems. Telluride, CO.More infoDate: 07/08
 Stafford, C. A. (2013, August). Probing Maxwell’s Demon with a Nanoscale Thermometer. Workshop on the StatisticalPhysics of Disordered Systems. New York, NY: Courant Institute.More infoDate: 08/23
 Stafford, C. A. (2013, March). Probing Maxwell’s Demon with a Nanoscale Thermometer. Chemical Physics Seminar. Tucson, AZ: University of Arizona.More infoDate: 03/04
 Stafford, C. A. (2013, October). Probing Maxwell’s Demon with a Nanoscale Thermometer. CondensedMatter Physics Seminar. Evanston, IL: Northwestern University.More infoDate: 10/03
 Stafford, C. A. (2013, October). Probing Maxwell’s Demon with a Nanoscale Thermometer. Four Corners Section MeetingAmerican Physical Society.More infoDate: 10/19
 Stafford, C. A. (2013, September). Probing Maxwell’s Demon with a Nanoscale Thermometer. Physics Colloquium. Golden, CO: Colorado School of Mines.More infoDate: 09/03
 Stafford, C. A. (2012, August). ManyBody Theory of Energy Transport and Conversion at the Nanoscale. Departmentof Energy Theoretical Condensed Matter Physics PI’s Meeting. Rockville, MD.More infoDate: 08/22
 Stafford, C. A. (2012, February). On the derivative discontinuity in molecular junctions. American Physical Society March Meeting. Boston, MA.More infoDate: 02/27
 Stafford, C. A. (2012, March). Carbon nanostructures for themoelectricity and thermal management: Modeling electric and thermal transport. Raytheon Industry Day. Tucson, AZ: University of Arizona.More infoDate: 03/27
 Stafford, C. A. (2012, March). QuantumEnhanced Thermoelectric Effects in Polycyclic Molecular Junctions. 2012 APS March Meeting. Boston, MA.
 Stafford, C. A. (2012, May). Charging effects and thermoelectricity in molecular junctions. Workshop on QuantumTransport in Molecular Nanostructures. Dublin, Ireland.More infoDate: 05/23
 Stafford, C. A. (2012, September). Probing Maxwell’s Demon with a Nanoscale Thermometer. Copenhagen University.More infoDate: 09/25
 Stafford, C. A. (2012, September). Thermoelectricity and Charging Effects in Molecular Junctions. Condensed Matter Seminar. Madrid, Spain: Universidad Aut´onoma de Madrid.More infoDate: 09/07
 Stafford, C. A. (2012, September). Thermoelectricity and Charging Effects in Molecular Junctions. Institut fur Nanotechnologie. Karlsruhe, Germany: Karlsruher Institut fur Technologie.More infoDate: 09/13
 Stafford, C. A. (2012, September). Thermoelectricity and Charging Effects in Molecular Junctions. Physics Department. Copenhagen, Denmark: Technical University of Denmark.More infoDate: 09/18
Others
 Barr, J., & Stafford, C. (2013, Fall). On Transmission Node Structure in Interacting Systems.More infoarXiv:1303.3618
 Bergfield, J., Ratner, M., Stafford, C., & Di, V. M. (2013, Fall). Tunable Quantum Temperature Oscillations in Graphene and Carbon Nanoribbons.More infoarXiv:1305.6602
 Burki, J., Stafford, C., & Stein, D. (2013, Fall). Nanoscale variable resistor/electromechanical transistor.More infoU.S. Patent No. 8,492,231
 Meair, J., Bergfield, J., Stafford, C., & Jacquod, P. (2013, Fall). Local Temperature of OutofEquilibrium Quantum Electron Systems.More infoarXiv:1306.6345