Landauer Formula Research Articles

Magnetoresistance originated from charge-spin conversion in ferromagnet

Transverse magnetoresistance in a ferromagnetic/nonmagnetic/ferromagnetic trilayer originated from charge-spin conversion by anomalous Hall effect is investigated theoretically. Solving the spin diffusion equation in bulk and using the spin-dependent Landauer formula at the ferromagnetic/nonmagnetic interface, an analytical formula of the transverse resistivity is obtained. The charge-spin conversion by the anomalous Hall effect contributes to the magnetoresistance in a manner proportional to the square of the spin anomalous Hall angle. The angular dependence of the magnetoresistance is basically identical to that of planar Hall effect, but has an additional term which depends on the relative angle of the magnetizations in two ferromagnets.

Enhanced Spin Figure of Merit in a Molecular Junction

We study the spin-dependent thermoelectric effect in a molecular junction where a semiconductor quantum dot in a magnetic field is trapped between two conducting polymer leads. The Green’s function method along with the Landauer formula based on the Holstein small polaron model is applied to study the thermo-spin effects in this system. The narrow bandwidth of the small polaron band in the polymer leads results in sharp carrier density of states, which offers the opportunity to enhance the spin Seebeck coefficient. At the same time, the spin conductance and electron thermal conductance can be tuned by tuning the affinity mismatch and chemical potential flexibly. Consequently, significant enhancement of spin figure of merit can be achieved. Detailed calculation of the influence of band mismatch, width of a small polaron band, chemical potential, temperature, and magnetic field on the thermo-spin effects is shown in this work. Our results show that the value of spin figure of merit is larger than 1 at a wid...

Tunneling Current of Electron in Armchair Graphene Nanoribbon Bipolar Transistor Model Using Transfer Matrix Method

The tunneling current of n-p-n bipolar junction transistor AGNR-based is modeled with semi-numerical method. The exponential solution from Schrödinger equation is used and solved analytically. The potential profile of n-p-n BJT divided into several segments in the numerical method. Then, the solved analytical result is used in the numerical method to compute the electron transmittance. Transfer Matrix Method (TMM) is the numerical method used to compute the electron transmittance. From the calculated transmittance the tunneling current can be computed by using Landauer formula with aid of Gauss-Legendre Quadrature (GLQ). Next, the tunneling current is computed with several change of variables which are base-emitter voltage (VBE), base-collector voltage (VBC), temperature and the AGNR’s width. The computed tunneling current shows that the larger value of applied voltage for both VBE and VBC results in larger value of tunneling current. At the lower temperature, the current is larger. The computed tunneling current shows that at wider width of AGNR, the current is also larger. This is due to the decreased band-gap energy (Eg) because of the wider width of AGNR.

Modelling of Drain Current in Tunnelling Field-Effect Transistor Based on Strained Armchair Graphene Nanoribbons

A tunnelling field-effect transistor (TFET) based armchair graphenenanoribbons (AGNRs) with variation of uniaxial strain has been modeled. Bandgap of strained AGNR estimated by an extended tight binding method is applied to obtain electrical characteristics of a TFET under the quantum capacitance limit device approximation. Furthermore, the electron transmittance is calculated by utilizing the WKB (Wentzel–Kramers–Brillouin) approach. The obtained transmittance is then used to calculate the drain current by employing the Landauer formula. The results show that strain parameter has significant effect on the current. In other words, the electrical characteristics of AGNR TFET can be tuned by the strain of AGNR.

Anisotropic magnetotransport in Dirac-Weyl magnetic junctions

We theoretically study the anisotropic magnetotransport in Dirac-Weyl magnetic junctions where a doped ferromagnetic Weyl semimetal is sandwiched between doped Dirac semimetals. We calculate the conductance using the Landauer formula and find that the system exhibits extraordinarily large anisotropic magnetoresistance (AMR). The AMR depends on the ratio of the Fermi energy and the strength of the exchange interaction. The origin of the AMR is the shift of the Fermi surface in the Weyl semimetal and the mechanism is completely different from the conventional AMR originating from the spin dependent scattering and the spin-orbit interaction.

Comparative investigation of electronic transport across three-dimensional nanojunctions

We carry out a layer-by-layer investigation to understand electron transport across metal-insulator-metal junctions. Interfacial structures of junctions were studied and characterized using first-principles density functional theory within the generalized gradient approximation. We found that as a function of the number of crystal layers the calculated transmission coefficients of multilayer silicene junctions decay much slower than for BN-based junctions We revisited the semiclassical Boltzmann theory of electronic transport and applied to multilayer silicene and BN-based junctions. The calculated resistance in the high-transmission regime is smaller than that provided by the Landauer formula. As the thickness of the barrier increases, results from the Boltzmann and the Landauer formulae converge. We provide a upper limit in the transmission coefficient below which, the Landauer method becomes valid. Quantitatively, when the transmission coefficient is lower than $ \sim 0.05 $ per channel, the error introduced by the Landauer formula for calculating the resistance is negligible. In addition, we found that the resistance of a junction is not entirely determined by the averaged transmission, but also by the distribution of the transmission over the first Brillouin zone.

Ballistic transport and quantum interference in InSb nanowire devices**Project supported by the National Key Basic Research and Development Project of the Ministry of Science and Technology of China (Grant No. 2016YFA0300601) and the National Natural Science Foundation of China (Grant Nos. 91221202, 91421303, 11374019, and 61321001).

An experimental realization of a ballistic superconductor proximitized semiconductor nanowire device is a necessary step towards engineering topological quantum electronics. Here, we report on ballistic transport in InSb nanowires grown by molecular-beam epitaxy contacted by superconductor electrodes. At an elevated temperature, clear conductance plateaus are observed at zero magnetic field and in agreement with calculations based on the Landauer formula. At lower temperature, we have observed characteristic Fabry–Pérot patterns which confirm the ballistic nature of charge transport. Furthermore, the magnetoconductance measurements in the ballistic regime reveal a periodic variation related to the Fabry–Pérot oscillations. The result can be reasonably explained by taking into account the impact of magnetic field on the phase of ballistic electron’s wave function, which is further verified by our simulation. Our results pave the way for better understanding of the quantum interference effects on the transport properties of InSb nanowires in the ballistic regime as well as developing of novel device for topological quantum computations.

Microscopic theory of a nonequilibrium open bosonic chain.

Quantum master equations form an important tool in the description of transport problems in open quantum systems. However, they suffer from the difficulty that the shape of the Lindblad dissipator depends sensibly on the system Hamiltonian. Consequently, most of the work done in this field has focused on phenomenological dissipators which act locally on different parts of the system. In this paper we show how to construct Lindblad dissipators to model a one-dimensional bosonic tight-binding chain connected to two baths at the first and last site, kept at different temperatures and chemical potentials. We show that even though the bath coupling is local, the effective Lindblad dissipator stemming from this interaction is inherently nonlocal, affecting all normal modes of the system. We then use this formalism to study the current of particles and energy through the system and find that they have the structure of Landauer's formula, with the bath spectral density playing the role of the transfer integral. Finally, we consider infinitesimal temperature and chemical potential gradients and show that the currents satisfy Onsager's reciprocal relations, which is a consequence of the fact that the microscopic quantum dynamics obeys detailed balance.

A three-terminal quantum dot heat engine based on ideal resonant tunneling

In this paper, the model of a three-terminal quantum dot heat engine consists of two quantum dots with a single energy level, a cavity and two electron reservoirs is established. According to Landauer formula the expressions for the heat current, the output power and the efficiency are derived analytically. The performance characteristic curves of the output power versus the efficiency are plotted by numerical calculation. Moreover, the optimal performance parameters are determined. Then we optimize the output power and the efficiency respectively, the influence of the width of energy level and the heat leak on performance of the three-terminal thermoelectric heat engine is discussed. Lastly, the variation of the corresponding efficiency at the maximum power output with the Carnot efficiency between two reservoirs is discussed and the corresponding efficiency is compared with Carnot efficiency and CA efficiency. It is shown that this three-terminal heat engine is irreversible due to the existence of the width of energy level and the heat leak. Thus the curve of the power output versus the efficiency is a loop-shaped one. And due to the heat leak, the characteristics of the efficiency and the width of energy level is a non-monotone curve. The efficiency at the maximum power will be greater than the CA efficiency when the heat leak and the width of energy level are not taken into account.

Nonequilibrium quantum chains under multisite Lindblad baths.

We study a quantum XX chain coupled to two heat reservoirs that act on multiple sites and are kept at different temperatures and chemical potentials. The baths are described by Lindblad dissipators, which are constructed by direct coupling to the fermionic normal modes of the chain. Using a perturbative method, we are able to find analytical formulas for all steady-state properties of the system. We compute both the particle or magnetization current and the energy current, both of which are found to have the structure of Landauer's formula. We also obtain exact formulas for the Onsager coefficients. All properties are found to differ substantially from those of a single-site bath. In particular, we find a strong dependence on the intensity of the bath couplings. In the weak-coupling regime, we show that the Onsager reciprocal relations are satisfied.

(Invited) Large Mobility Modulation Due to Discrete Impurities in Nanowires

Semiconductor nanowires have been receiving great attention over the past decades, owing to their possible applications to chemical/biological sensors as well as future electronic/photonic devices [1,2]. Among other characteristics, the locality of ionized impurities plays a crucial role in optimizing the device performance; carrier transport is critically dependent on where the localized impurities are positioned inside nanoscale channels. In fact, this is the physical origin of the random dopant fluctuation found in threshold voltage, which is now of most serious concerns in integrating nanoscale Si-based devices [3]. Recently, we have theoretically carried out systematic investigations on the effects of the locality of multiple impurities on transport characteristics under quasi-one dimensional (quasi-1D) nanowire structures [4]. We have shown that both phase interference and phase randomization simultaneously play a dominant role in determining the impurity-limited resistance even under the fully coherent circumstances, in which no energy dissipating scattering is involved. In this presentation, we extend our approach further to study the polarity dependence of impurities on transport characteristics. We show that the impurity-limited mobility could be modulated over three orders of magnitude by changing the axial separation among impurities provided that the charge polarity of impurities is opposite to that of carriers. We employ a cylindrical nanowire with radius of 2 nm and ionized acceptors or donors are distributed at random in the Si channel. Typical potential profiles in the channel region consist of the long-range potential modulation spread over the channel and the localized short-range potential modulation. Here, we would like to point out that some confusion exists in many theoretical analyses, regarding the channel resistance caused by such potential profiles: The long-range potential modulation and the short-range scattering potential are mixed up. The former is caused mainly by the applied gate voltage and the work function difference between the substrate and gate, whereas the latter is attributed to the screened Coulomb potential of impurities and carriers. The channel resistance is dominated by the short-range scattering potential and the long-range potential simply modulates the carrier density in the channel region, rather than the channel resistance. Therefore, in order to calculate the impurity-limited resistance, it is essential to eliminate the long-range potential modulation from the channel potential. In the present calculations, the average impurity density in the channel is fixed at 2 x 1019 cm-3, so that the channel length varies according to the number of impurities doped in the channel, namely, L = 4 nm for a single impurity, 8 nm for two impurities, etc. The conductance is obtained from the Landauer formula in terms of the transmission probability calculated from the Lippmann-Schwinger equation with the short-range scattering potential. The impurity-limited resistance and mobility are then extracted by subtracting the quantum resistance from the inverse of the conductance. We show that the impurity-limited resistance (mobility) due to two ionized impurities located on the wire axis is greatly modulated as a function of the distance Δ between the impurities: The constructive phase interference is very strong at small Δ even at room temperature, whereas this phase interference diminishes at large Δ. Since no phase randomizing scattering is included in the present calculations, this randomization is attributed to the broadness of the energy spectrum of the incoming electrons from the reservoirs [4]. The most striking feature is that the impurity-limited resistance (mobility) is greatly suppressed (enhanced) at some particular value of Δ (around 1 nm) in the case of donors. We have also carried out more elaborate NEGF simulations with the screened Coulomb potential by placing the donor impurities at random inside the channel region. The above feature is still clearly visible, though the modulation is reduced owing to random impurity distributions. Nevertheless, this large mobility modulation is expected to affect significantly the device performance, especially, in the case of Junctionless-FETs. This study was supported in part by the MEXT under Grant- in-Aid for Scientific Research (B) (No. 15H03983). REFERENCES [1] V. Schmidt, et al., Adv. Mater. 21, 2681 (2009). [2] R. Rurali, Rev. Mod. Phys. 82, 427 (2010). [3] A. Asenov, IEEE ED-45, p.2505 (1998); N. Sano, et al., IEDM Tech. Dig. p.275 (2000). [4] N. Sano, J. Appl. Phys. 118, 244302 (2015); EUROSOI-ULIS 2016, p. 216 (2016).

Tuning phonon transmission and thermal conductance by roughness at rectangular and triangular Si/Ge interface

Using the atomistic Green’s functions method in combination with the Landauer formula, we show that the phonon transmission and thermal conductance of periodic rectangular-shaped and triangular-shaped Si/Ge interfaces are both enhanced and tunable by roughness. For triangular-shaped interface, there is maximum phonon transmission and conductance with increasing roughness height, and the conductance can be tuned maximally by 22.3% compared with the flat interface. The maximum conductance of rectangular-shaped interface is enhanced by about 11.1%. The competing mechanisms between the broadening frequency transport window of rough interface and the increasing diffusing phonon scattering at the interface with higher roughness introduce the maximum transmission and conductance. Similar result is also obtained in non-periodic interfaces. The presented results provide insights into the thermal design of interfaces in nanoscale devices.

Temporal evolution of resonant transmission under telegraph noise

The environment of a quantum dot, which is connected to two leads, is modeled by telegraph noise, i.e. random Markovian jumps of the (spinless) electron energy on the dot between two levels. The temporal evolutions of the charge on the dot and of the currents in the leads are studied using a recently developed single-particle basis approach, which is particularly convenient for the averaging over the histories of the noise. In the steady state limit we recover the Landauer formula. At a very fast jump rate between the two levels, the noise does not affect the transport. As the jump rate decreases, the effective average transmission crosses over from the transmission through a single (average) level to an incoherent sum of the transmissions through the two levels. 13 pages, 6 figuresThe transient temporal evolution towards the steady state is dominated by the displacement current at short times, and by the Landauer current at long times. It contains oscillating terms, which decay to zero faster than for the case without noise. When the average chemical potential on the leads equals the dot's "original" energy, without the noise, the oscillations disappear completely and the transient evolution becomes independent of the noise.

Current-Induced Instability of a Perpendicular Ferromagnet in Spin Hall Geometry

We develop a theoretical formula of spin Hall torque in the presence of two ferromagnets. While the direction of the conventional spin Hall torque always points to the in-plane direction, the present system enables to manipulate the torque direction acting on one magnetization by changing the direction of another magnetization. Based on the diffusion equation of the spin accumulation and the Landauer formula, we derive analytical formula of the spin Hall torque. The present model provides a solution to switch a perpendicular ferromagnet deterministically at zero field using the spin Hall effect.

Effects of band hybridization on electronic properties in tuning armchair graphene nanoribbons

We explore a model of armchair graphene nanoribbons tuned by functionalizing the edge states. Edge modifications are modeled by changing the electronic energy of the edge states in specific periodic patterns. The model can be considered to mimic a controlled doping process with different elements. The band structure, density of states, conductance, and local density of states are calculated, using the tight binding approach, Green’s function methodology, and the Landauer formula. The results show interesting behaviors, which are considerably different from the properties of the perfect nanoribbons. The hybridization of conducting bands with non-conducting bands, which appear perfectly flat in the perfect ribbon, opens up and modifies gaps in conductance near the Fermi level. One particular pattern of edge functionalization causes a strong, symmetric, and systematic band gap change about the Fermi level, modifying the electronic characteristics in the energy dispersion, density of states, local density of states, and conductance.

A DC Thermal Model of Carbon Nanotube Field Effect Transistors for CAD Applications

In this paper a DC thermal model of Carbon Nanotube Field Effect Transistors (CNTFETs) is proposed. In particular we improve the semi-empirical model, already proposed by us, considering the temperature variation in the Landauer formula for the drain-source current. To confirm the validity of the proposed thermal model, the simulations are performed in different thermal conditions, obtaining I-V characteristics perfectly coincident with those of the Stanford-Source Virtual Carbon Nanotube Field-Effect Transistor model (VS-CNFET), but with a run time lower. This result makes the proposed model particularly suitable to be implemented in CAD applications.

Impurity-limited resistance and phase interference of localized impurities under quasi-one dimensional nano-structures

The impurity-limited resistance and the effect of the phase interference among localized multiple impurities in the quasi-one dimensional (quasi-1D) nanowire structures are systematically investigated under the framework of the scattering theory. We derive theoretical expressions of the impurity-limited resistance in the nanowire under the linear response regime from the Landauer formula and from the Boltzmann transport equation (BTE) with the relaxation time approximation. We show that the formula from the BTE exactly coincides with that from the Landauer approach with the weak-scattering limit when the energy spectrum of the in-coming electrons from the reservoirs is narrow and, thus, point out a possibility that the distinction of the impurity-limited resistances derived from the Landauer formula and that of the BTE could be made clear. The derived formulas are applied to the quasi-1D nanowires doped with multiple localized impurities with short-range scattering potential and the validity of various approximations on the resistance are discussed. It is shown that impurity scattering becomes so strong under the nanowire structures that the weak-scattering limit breaks down in most cases. Thus, both phase interference and phase randomization simultaneously play a crucial role in determining the impurity-limited resistance even under the fully coherent framework. When the impurity separation along the wire axis direction is small, the constructive phase interference dominates and the resistance is much greater than the average resistance. As the separation becomes larger, however, it approaches the series resistance of the single-impurity resistance due to the phase randomization. Furthermore, under the uniform configuration of impurities, the space-average resistance of multiple impurities at room temperature is very close to the series resistance of the single-impurity resistance, and thus, each impurity could be regarded as an independent scattering center. The physical origin of this “self-averaging” under the fully coherent environments is attributed to the broadness of the energy spectrum of the in-coming electrons from the reservoirs.

Optimal Performance Analysis of a Three-Terminal Thermoelectric Refrigerator with Ideal Tunneling Quantum Dots**Supported by the National Natural Science Foundation of China under Grant No 11365015.

The model of a three-terminal thermoelectric refrigerator with ideal tunneling quantum dots is established. It consists of a cavity connected to two quantum dots embedded between two electron reservoirs at different temperatures and chemical potentials. According to the Landauer formula the expressions for the heat current, the cooling rate and the coefficient of performance (COP) are derived analytically. The performance characteristic curves of the cooling rate versus the coefficient of performance are plotted with numerical calculation. The optimal regions of the cooling rate and the COP are determined. Moreover, we optimize the cooling rate and the COP with respect to the position of energy level of the right quantum dot, respectively. The influence of the width of energy level and the temperature ratio on performance of the three-terminal thermoelectric refrigerator is analyzed. Lastly, when the width of energy level is small enough, the optimal performance of the refrigerator is discussed in detail.

Single-electron approach for time-dependent electron transport

We developed a new approach to electron transport in mesoscopic systems by using a particular single-particle basis. Although this basis generates redundant many-particle amplitudes, it greatly simplifies the treatment. By using our method for transport of non-interacting electrons, we generalize the Landauer formula for transient currents and time-dependent potentials. The result has a very simple form and clear physical interpretation. As an example, we apply it to resonant tunneling through a quantum dot where the tunneling barriers are oscillating in time. We obtain an analytical expression for the time-dependent (ac) resonant current. However, in the adiabatic limit this expression displays the dc current for zero bias (electron pumping).

Nonlinear transport in ionic liquid gated strontium titanate nanowires

Measurements of the current-voltage (I–V) characteristics of ionic liquid gated nanometer scale channels of strontium titanate have been carried out. At low gate voltages, the I–V characteristics exhibit a large voltage threshold for conduction and a nonlinear power law behavior at all temperatures measured. The source-drain current of these nanowires scales as a power law of the difference between the source-drain voltage and the threshold voltage. The scaling behavior of the I–V characteristic is reminiscent of collective electronic transport through an array of quantum dots. At large gate voltages, the narrow channel acts as a quasi-1D wire whose conductance follows Landauer's formula for multichannel transport.