Angular Momentum Radiation

by a Benzene Molecule

Jian-Sheng Wang

1

Outline

• Experimental motivation

• Electron Green’s functions G

• Electron-photon interaction and photon Green’s

functions D

• NEGF “technologies”

• Application examples2

Experimental motivation

3

Radiation from thermal objects,

far-field effect

4

Stefan-Boltzmann law:

4S T

What if closer than wavelength ?

Near-field effect

• Rytov fluctuational electrodynamics (1953)

• Polder & van Hove (PvH) theory (1971)

• Phonon tunneling/phonon polaritons (Mahan

2011, Xiong at al 2014, Chiloyan et al 2015, …)

• Other mechanism?

5

Experiments

6

Ottens, et al, PRL 107,

014301 (2011).

Kim, et al, Nature 528, 387 (2015).

A recent experiment that does not

agree with any theory

7

Heat transport between a Au tip and surface is

measured, obtain much larger values than conventional

theory predicts. Nature Comm 2017, Kloppstech, et al.

0

0

1

,

f

f

t

t

t

B

BE

D

DB J

J 0 D E

Fluctuational electrodynamics

8

0

0

0

1

( )

t

t

t

B

BE

E

DB J

J E K

J 0

Rytov 1953: Polder & van Hove 1971:

random

variables

Electron Green’s functions G

9

Single electron quantum mechanics

10

/

, ( ) (0)

We define the (retarded) Green's function by

1, 0( ) ( ) , ( )

0, 0

then

( ) ( ) (0), 0

Hti

r iHt

r

di H t e

dt

tiG t t e t

t

t i G t t

Many-electron Hamiltonian and

Green’s functions

11

1

2

†

Fourier transform

1†

ˆ , ...

( , ') ( ') { ( ), ( ')} ( )

N

r r

jk j k

c

c

H c Hc c

c

iG t t t t c t c t G E E i H

†

0

( , ') ( ) ( ')jk j k

iG t t c t c t

Annihilation

operator c is a

column vector, H

is N by N matrix.

{A, B} =AB+BA

† †

† † † †

†

0

0

| 0 |1

j k k j jk

j k k j

j k k j

j j

c c c c

c c c c

c c c c

c

Equilibrium fluctuation-dissipation

theorems

12

( )

†

fermions:

1 1( ), ,

1

(1 )( ),

bosons:

1( ),

1

(1 )( )

r a

E

B

r a a r

r a

r a

G f G G fe k T

G f G G G G

D N D D Ne

D N D D

Perturbation theory, single electron

13

1 1 1 1

1 1

use the identity ( )

Let , ,

then

The last equation is known as Dyson or Lippmann-Schwinger

equation

r r

r r r r

H h V

A B B B A A

G A z H g B z h z E i

G g g VG

Electron-photon interaction and photon

Green’s functions D

14

Electrons & electrodynamics

15

†

' '

, ' '

22 2

0 0

0

int

int

( , )

†

ˆ exp ( )

1 1 +

2

, 0, , ,

use trapzoidal rule for line integral

l

l ll l l l

l l ll

e

l

l l l

iH c H c e d q

dVt

H H H

H I A x y z

q ec c

A l r

AA

†

' ' '

'

, ,

1h.c.

2

x

l l l

y

z

l l ll l l l

l

AA I q

A

A

iec H c

I

I R R

Gauge invariance

16

†

' '

, ' '

22 2

0 0

0

ˆ exp ( )

1 1 +

2

, or + ,

exp ( , )

Assume Coulomb gauge, 0, in the pro

l

l ll l l l

l l ll

l l l

iH c H c e d q

dVt

fA A f f

t

ec i f t c

A l r

AA

A A

r

A of

Commutation relation of the fields

17

2 220 0

2

0

2

0

1,

2 2

( ), ( ') ( '), ( ) ( )

( ), ( ') ( '), ( ) ( )j k jk k k

dV dV cc

ic

A i A

A A

r r r r r r

r r r r r r

Transverse delta function

18

3

3 2

3

3 2

1 2 3

( )2

32 1( ) , , 1, 2,3

3 4

( , , )

See C. Cohen-Tannoudji, et al, ``Photons & Atoms,'' page 42, Eq.(33)

j k i

jk jk

j k

jk jk

k kde

k

x xj k

r r

x x x

k rk

r

r

r

Heisenberg equations of motion,

iℏ𝑑 𝑂

𝑑𝑡= 𝑂,𝐻 , for electron and fields

19

' '

' '

2

2

0

22 †0

' '2 2, ' '

0

exp ( )

1, ( ),

1( ), exp

( ), 0

l

lll l l l

l l

l l

l

l

l ll l

l l l

dc iei H c d e c

dt

q cc

iec H c d

c t i

A

A l r

r r

AA Π r A l

j r j

Poynting scalar/vector

20

0

0

1

2

1

2

1

1

2

u

u

A

J

J

S E B

S j E E j

Photon Green’s function

21

0

0

( , ; ', ') ( ') [ ( , ), ( ', ')] , , 0, , ,

( , ; ', ') ( ', ') ( , )

Free photon Green's function in frequency domain, | ' |

10 0 0

4

0

0

0

r

rxx xy xz

yx yy yz

zx zy zz

iD t t t t A t A t x y z

iD t t A t A t

R

R

d d dd

d d d

d d d

r r r r

r r r r

r r

,

2

0

2 33

0

1 ˆ ˆ( )4

1 ˆ ˆ( 3 )4

i Rc

i R i Rc c

d e U RRc R

i e eU RR

ci R i R

c c

[A, B]=AB-BA

NEGF “technologies”

22

A brief history of NEGF

• Schwinger 1961

• Kadanoff and Baym 1962

• Keldysh 1965

• Caroli, Combescot, Nozieres, and Saint-James

1971

• Meir and Wingreen 1992

23

Evolution operator on contour

24

2

1

2 1 2 1

3 2 2 1 3 1 3 2 1

1

1 2 2 1 1 2

0 0

( , ) exp ,

( , ) ( , ) ( , ),

( , ) ( , ) ,

( ) ( , ) ( , )

c

iU T H d

U U U

U U

O U t OU t

Contour-ordered Green’s function

25

0 '

( , ') ( ) ( ')

Tr ( ) C

AB C

iH d

C

iG T A B

it T A B e

t0

τ’

τ

Contour order: the operators

earlier on the contour are to the

right. See, e.g., H. Haug & A.-

P. Jauho.

Relation to other Green’s functions

26

'

( , ), or ,

( , ') ( , ') or

,

,

( ')

t

t

r t

t t

G GG G t t G

G G

G G G G

G G G G

G G G

t t G G

t0

τ’

τ

Transformation/Keldysh rotation

27

'

' '

' '

( , ') ( , ')

or

1 0 1 11, ,

0 1 1 12

1,

2

0

jj jj

t

t

T

z

r K

T T

z K a

t t t t

t t t t

r K

a

A A A t t

A AA A A

A A

R RR I

A AA R AR R AR

A A

A A A A A A A A

A A A A A A A A

G GG

G

Convolution, Langreth rule

28

2 3 1 2 2 3 1

11

( , ) ( , ) ( , )

0 0 0

or , , +

( )

,

(

n n n

r K r K r K

a a a

r r r a a a K r K K a

r r r r r r r r

K K r r K r K a K a a

AB D d d d A B D

C AB C AB

C C A A B B

C A B

C A B C A B C A B A B

G g g G G g g G G g

G g g G g G g G

G

1 ) (1 )r r a a r aG g G G G

Keldysh equation

29

1

(1 ) (1 )

But if is for noninteracting free particle, we have

(1 ) 0

so

r r a a r a

r r r r

r a

G G g G G G

g

G g G g g

G G G

Random phase approximation

30

int

†

int

( , )

0

0 0

ˆ

, 0, , ,

1( , ') ( ) ( ') RPA

Tr ( , ') ( ', )

e

l

c

H H H H

H I A c M c A x y z

T I Ii

i M G M G

D d d D

†

' ' '

'

, ,

1h.c.

2

x

l l l

y

z

l l ll l l l

l

AA I q

A

A

iec H c

I

I R R

Poynting vector

31

0

0 0 '

123 213 112

1classical Poynting vector is

Green's function expression

12Re ( , ', )

2 '

, , , , take the component , , or (1,2,3).

1, 1, 0,

i

ijk klm jm

jklm l

dS D

x

i j k l m x y z

r r

S E B

r r

Meir-Wingreen formula, photon bath at

infinity

32

, ,

†

0 0

0

, , '2 3, '0

Tr4

,

ˆ ˆ( , ) , | |

ˆ is unit matrix, / | |

ˆ ˆ( ) Tr ( ) Im ( )4

r a

r r r r r a r

r

r

l l

l l

dJ D D

D D D

D d d D D D

i c U RR

U R

T d U RRc

r r

r rx

y

z

r

Maxwell stress tensor and angular

momentum transfer

33

0

0

3

2 3

00

1

ˆ ˆ

ˆ ˆ2 4

T EE BB uU

dLd R R T R

dt

dd R R

c

Pure scalar photon

34

int

†

2

230

†

int

system

Total Hamiltonian:

electron:

scalar photon: ,2

Interaction: ( )

Green's functions: ( , ; ', ') ( , ) (

e

e

j j j

j

c

H H H H

H c Hc

H d cc

H e c c

iD T

r

r

r r r r

†

', ')

( ; ') ( ) ( ')jk c j k

iG T c c

Meir-Wingreen to Caroli formula

35

, ,

†

1 1 2 1 2

0

Tr4

,

, ( 1)

( ) , ( ) Tr2

, 1,2

r a

r r r a r

r a r a

r a

r a

dJ D D

D D D

D d d D D D

N N

dJ T N N T D D

i

2( , ') ( , ') ( ', )jk jk kji e G G

Assuming local equilibrium

Random phase approximation

(RPA)

Application examples

36

Analytic Result: heat transfer between

ends of 1D chains

37

22

2

BB22

0

2 24 4

BB 1 2 2

8

, 1.750, ,4

1

( ) is the blackbody value,

d e cI j

c

d

e ej T T i

Analytic Result: far field total radiative

power of a two-site model

38

2 2 2

01 2 1 22 3

0

0 ( ) ( )

2 ' '3

1 1, , ' , 1, 2

1 1i i i ii it t

e v tI f f f f

c

atv f f i

e e

tT1, 1 T2, 2

use weak coupling

limit, 0

Two graphene sheets

39

1000 K

300 K

Ratio of heat flux to blackbody

value for graphene as a function of

distance d, JzBB = 56244 W/m2.

From J.-H. Jiang and J.-S. Wang,

PRB 96, 155437 (2017).

Current-carrying graphene sheets

40

Current-induced heat transfer

41

300 305 310 315 320 325 330 335 340-1.5x10

6

-1.0x106

-5.0x105

0.0

5.0x105

1.0x106

1.5x106

Curr

ent de

nsity (

W/m

2)

T2 (K)

v1=1.010

6 m/s

-1 0 10.0

5.0x105

1.0x106

-0.1 0.0 0.10.0

2.0x105

4.0x105

1 10 10010

3

104

105

106

107

108

109

1 10 10010

1

102

103

104

105

106

(d)(c)

(b)

Curr

ent density (W

/m2)

v1 (10

6 m/s)

(a)

Curr

ent density (W

/m2)

1R

-1L

(eV)

Curr

ent density (W

/m2)

d (nm)

Curr

ent density (W

/m2)

d (nm)

: Double-layer graphene.

T1=300K, varying T2 at

distance d = 10 nm,

chemical potential at 0.1 eV.

From Peng & Wang,

arXiv:1805.09493.

: T1=T2=300 K. (a) and (c) infinite

system (fluctuational electrodynamics),

(b) and (d) 4×4 cell finite system with

four leads (NEGF).

Carbon nanotubes

42

Heat transfer from 400K to

300K objects. (a), (b) zigzag

carbon nanotubes. (c), (d)

nano-triangles. d: gap

distance, M: nanotube

circumference, L: triangle

length. : dielectric constant.

From G. Tang, H.H. Yap, J.

Ren, and J.-S. Wang, Phys.

Rev. Appl. 11, 031004

(2019).

Light emission by a biased benzene

molecule

Parameters:

Bond length 𝑎0 =1.41 Angstrom

Nearest neighbor hopping parameter t = 2.54 eV

Bias voltage 𝜇tip = 3.0eV, 𝜇sub = −3.0eV

Tip coupling and Substrate coupling:

Γtip = Γsub = diag{Γ, Γ, Γ, Γ, Γ, Γ}, Γ = 0.5eV

The photon image is taken in the plane at 𝑧= 0.10nm, 0.15nm, 0.20nm. 43

Benzene radiation power & electric current

under bias

Parameters:

Bond length 𝑎0 =1.41 Angstrom

Nearest neighbor hopping parameter

t = 2.54 eV

𝜇sub = 0.0eV, 𝜇tip is set to be the

voltage bias.

Tip coupling and Substrate coupling:

Γtip = Γsub = diag{Γ, Γ, Γ, Γ, Γ, Γ}, Γ

= 0.2 eV

Sphere surface radius R = 0.1 mm

Unpublished work from Zuquan

Zhang.

44

Ie-Ie

Angular momentum emission

45

Symmetric leads, no

net angular

momentum emission

Unsymmetric

connection,

have emission

𝑥

𝑦

𝑧

16

3

2

5

4

𝐼𝑉

𝑥

𝑦

𝑧

𝜃

𝜙1

6

3

2

5

4

R

𝐼 𝑉

(a) (b)

Γ

Γ

Angular momentum emission

46

Hopping t = 2.5

eV, coupling Γ

=0.4 eV, size a =

0.14 nm, 𝑣0 =𝑎𝑡

ℏ.

0

1

2-2

-1

ε

t

2t

-t

-2t

Summary

• Fully quantum-mechanical, microscopic theory

for near-field and far field radiation is

proposed.

• 1D two-dot model, current-carrying graphene,

angular momentum emission, etc., are reported

47

Acknowledgements

• Students: Jiebin, Han Hoe, Jia-Hui, Zuquan

• Collaborators, Jingtao Lü, Gaomin Tang, JieRen, …

• Supported by MOE tier 2 and FRC grants

48