Transport in electron−photon systems

doi:10.1007/s11467-023-1260-z

Front. Phys.

2023, Vol. 18

Issue (4) : 43602 https://doi.org/10.1007/s11467-023-1260-z

REVIEW ARTICLE

Transport in electron−photon systems

Jian-Sheng Wang¹(

), Jiebin Peng², Zu-Quan Zhang³, Yong-Mei Zhang⁴, Tao Zhu^5,⁶

¹. Department of Physics, National University of Singapore, Singapore 117551, Republic of Singapore
². Center for Phononics and Thermal Energy Science, China-EU Joint Center for Nanophononics, Shanghai Key Laboratory of Special Artificial Microstructure Materials and Technology, School of Physics Science and Engineering, Tongji University, Shanghai 200092, China
³. Department of Physics, Zhejiang Normal University, Jinhua 321004, China
⁴. College of Physics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
⁵. School of Electronic and Information Engineering, Tiangong University, Tianjin 300387, China
⁶. Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China

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Abstract

We review the description and modeling of transport phenomena among the electron systems coupled via scalar or vector photons. It consists of three parts. The first part is about scalar photons, i.e., Coulomb interactions. The second part is with transverse photons described by vector potentials. The third part is on ϕ = 0 or temporal gauge, which is a full theory of the electrodynamics. We use the nonequilibrium Green’s function (NEGF) formalism as a basic tool to study steady-state transport. Although with local equilibrium it is equivalent to the fluctuational electrodynamics (FE), the advantage of NEGF is that it can go beyond FE due to its generality. We have given a few examples in the review, such as transfer of heat between graphene sheets driven by potential bias, emission of light by a double quantum dot, and emission of energy, momentum, and angular momentum from a graphene nanoribbon. All of these calculations are based on a generalization of the Meir−Wingreen formula commonly used in electronic transport in mesoscopic systems, with materials properties represented by photon self-energy, coupled with the Keldysh equation and the solution to the Dyson equation.

Keywords quantum transport thermal radiation scalar and vector photons nonequilibrium Green's function

Corresponding Author(s): Jian-Sheng Wang

Issue Date: 17 March 2023

Cite this article:

Jian-Sheng Wang,Jiebin Peng,Zu-Quan Zhang, et al. Transport in electron−photon systems[J]. Front. Phys. , 2023, 18(4): 43602.

URL:

https://academic.hep.com.cn/fop/EN/10.1007/s11467-023-1260-z
https://academic.hep.com.cn/fop/EN/Y2023/V18/I4/43602

Fig.1 Heat current of the simple capacitor model, from Eqs. (10), (13), and (14). (a) Current

I L

vs. distance

d

T L = 1000

K and

T R = 300

K. (b) The current

I L

vs.

T L

while fixing

d = 1

nm,

T R = 300

K. Parameters are area

A = 1

nm², energy scale

E L = E R = U L = U R = 1

eV.

Fig.2 Diagrams for the heat current in the lowest order expansion. The solid lines denote the electron Green’s functions, and the dash lines for the photon Green’s functions. The boxes denote derivative of the bath self energy. The end of the arrow is associated with the first argument and beginning of the arrow with the second argument of the Green’s functions or self energies. The electron−photon vertex is associated with the matrix

M l

Fig.3 The self-consistent Born approximation diagrams. (a) The Hartree self-energy diagram, (b) the Fock self-energy diagram, (c) the polarization bubble

Π

. The double line with arrows signifies the full electron Green’s function

G

, the single dotted line denotes bare Coulomb

v

, while the double dotted line is the screened Coulomb

D

Fig.4 The retarded

Π L

of the left quantum dot. For parameter, see next Fig.5. Note that the real part is symmetry, while the imaginary part is anti-symmetric with respect to the frequency

ω

. Reproduced from Ref. [95].

Fig.5 Energy current calculation under SCBA and the blackbody limit in log−log plot. The temperature of dot

L

1000

K and dot

R

300

K. The chemical potential of dot

L

0

eV and dot

R

0.02

eV. Charge

Q = 1 e

, onsite energy

? L = ? R = 0

eV, and plate area is

A = 19.2 × 19.2

nm². The bath coupling is

Γ L = 1

eV,

Γ R = 0.5

eV, and

E 0, L = E 0, R =

50 eV. Reproduced from Ref. [95].

Fig.6 (a) Integrated spectral transfer function

f (ω)

as a function of frequency and (b)

g (q x)

as a function of wave vector in the driven direction, with different drift velocities: no drift (blue dash-dot line), total heat current density

I L / A = − 0.84

MW/m²;

v 1 = 5.0 × 105 m/s

(red dash line),

− 0.30

MW/m²; and

v 1 = 9.0 × 105 m/s

(black solid line),

+ 0.09

MW/m². The temperatures are

T L = 300

K and

T R = 320

K. The chemical potential of graphene

μ

is set as 0.1 eV. Gap distance

d

is set as 10 nm. The damping parameter is

η = 9

meV. Reproduced from Ref. [89].

Fig.7 Heat current density from left to right layer as a function of

T R

(temperature of the right layer) with drift velocity

v 1 = 9.0 × 105 m/s

(red solid line). The dotted line is a reference line for zero current density. The green circle indicates the point for “off temperature”. The chemical potential of graphene

μ

is set as 0.1 eV. Temperature of left layer of graphene

T L

is set as 300 K. Vacuum gap distance

d

is set as 10 nm. Reproduced from Ref. [89].

Fig.8 The first few terms of diagrammatic expansion of

l n Z

Fig.9 Distance dependence of the near-field heat flux ratio between two parallel single and multiple-layer graphene sheets. The temperatures of two sides are fixed at

T L =

1000 K and

T R =

300 K.

Fig.10 (a) Two-dot model connected to the left and right baths with strength

η

electron hops between two sites with hopping parameter

t

. (b) Energy levels of the two-dot model and relation to the bath distribution functions. When electron jumps from the excited state to the ground state, it releases energy

2 t = ℏ ω

Fig.11

N

objects of arbitrary shapes, each experiencing energy radiation, force and torque exerted to. There is one more special object called

∞

which is the space outside the sphere of radius

R

. The norm vector

n

is a unit vector pointing outwards from the enclosing surface.

Fig.12 One of the higher order contributions to the self-energy

Π

in a diagrammatic expansion known as Aslamazov-Larkin diagram. It consists of 6 electron Green’s functions (solid line with arrow) and 2 bare interactions

v

(dotted line). The electron line of a closed loop must belong to the same object, say,

α

and

β

for the two loops, respectively. Due to the existence of the

v

lines which introduce mutual interaction, it is impossible to write the self-energy as a sum over each object individually.

Fig.13 Illustration of bias-induced emissions of energy, momentum, and angular momentum from the conducting channel of a two-terminal transport device of a graphene strip. The length of the central channel is

l

. The semi-infinite left and right leads are extensions of the pristine graphene strip.

Fig.14 Results of (a) force component in the

x

direction, (b) power, and (c) torque component in the

z

direction as a function of the chemical potentials

μ L / t

and

μ R / t

for the two-terminal graphene strip system.

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