Supplementary information
Thermal Management in MoS2 Based Integrated Device using NearField Radiation
Jiebin Peng,1 Gang Zhang,2,a) and Baowen Li3
1
Department of Physics, National University of Singapore, Singapore 117546, Singapore
2
Institute of High Performance Computing, A*STAR, Singapore 138632, Singapore
3
Department of Mechanical Engineering, University of Colorado, Boulder, Colorado
80309, USA
In principle, the heat energy transfer between two objects can be described by the
electromagnetic flux resulting from fluctuating current sources inside each object:
thermal fluctuations in the first object (high temperature) induce correlations between
electric currents, which are related to the real part of its conductivity; next, the fluctuating
currents in the first object induce electromagnetic fields in the second object (low
temperature); then, the heat energy transfer is realized by the Poynting flux around the
second object. Under steady state, the heat transfer from object-1 to object-2 is higher
than heat transfer along the opposite direction and results in a net heat flow from high
temperature object to low temperature object. The total radiative heat transfer between
two objects is given by:1
H 

0
d
1 ( , T1 )   2 ( , T2 ) ( f far  f near )
2
(S1)
 k T
Where (, T )   (e b  1) is the average energy of photons. In this Landau
like formula, the spectral transfer function ffar (fnear) describes the contribution of far field
(near field) radiation by considering the nonlocal optical effects. For two semi-infinite
planes, which are separated by vacuum, the radiative heat transfer can be calculated from
the Green’s functions as Fourier integrals along the transverse coordinates (x, y). In z
component (perpendicular with plane), we can mathematically find spectral transfer
function (fnear) by solving inhomogeneous ordinary differential equations.
It is necessary to justify the applicability of Eq. 3 (in the main text) in studying near
field radiation between ultra-thin 2D materials. Based on fluctuation electrodynamics, the
optical conductivity σ(x, y, z) of semi-infinite plane is used to study the near field
radiation. To study 2D material in x-y plane with optical conductivity of σ(x, y), the form
of optical conductivity can be changed to σ(x, y)δ(z). This describes the surface of semiinfinite plane by surface optical conductivity σ(x, y) and others are described by vacuum.
Thus the system is equivalent to 2D material. With this treatment, Ilic et al. have justified
the applicability of Eq. 3 in studying near field radiative heat transfer between graphene
and semi-infinite plane.2 In 1999, in an independent work, Pendry also demonstrated the
equation for p-polarized NFRHT between two flat surfaces is the same as Eq. 3.3
Actually, Eq. 3 has been used to study near field radiative heat transfer between two
graphene sheets.4 Therefore, it is clear that Eq.3 is applicable in studying near field
radiation between two 2D sheets.
From equation 3 in the letter, we need get the reflection coefficient r1(2) to calculate
the radiative heat transfer. In our calculation, the reflection coefficient r1(2) equals to (1ε)/ε.5 Here ε=1+γσ/(2ε0ω) is the dielectric function of different layers, σ is the optical
conductivity. We get that reflection coefficient depends on optical conductivity of
different materials, which can be divided into intra-band and inter-band parts. The intraband term can be described by the Drude model, which is the response for intra-band
processes at frequencies that are well below the band gap; and the inter-band term
describes excitation process with photon frequencies above the band gap. In our
calculation, in the limit of high carrier concentration, optical conductivity of Graphene is
presented as: 4-7
   Intra   Inter
e 2 2k b T 
i
 
In 2 cosh

2
  i  
2k b T 

e 2  
4  G ( )  G ( 2) 

d 
G ( )  i
4 
2
 0 ( ) 2  4 2

 Intra 
 Inter
G ( )  sinh(  k b T ) cosh(  k bT )  cosh(  k b T )
(S2)
Where μ is chemical potential of Graphene, 𝜏 ≈ 10−13 𝑠 is the relaxation time which
describes the scattering process of Graphene and is assumed that it is not varied with
temperature (at room temperature region).
Different from Graphene, MoS2 has in-equivalent K(K’) points in Brillouin zone due
to the lack of inversion symmetry. The calculation about optical property of MoS2 is
based on the below modified Hamiltonian.8
2
2 q

1 z
H   z  s
 t0 a0q    
(   z )
2
2
4m0
where    ( x ,  y ) is the Pauli matrices,
(S3)
   () for different K(K’) which
describes two independent valley in the first Brillouin zone, q  (q x , q y ) , a0 is lattice
constant, m0 is free electron mass, α and β are high order parameters which are connected
with electron or hole effective mass. The intrinsic optical conductivity of MoS2 also
contains two parts: intra-band part and inter-band part. For inter-band part of MoS2, the
calculation starts from the modified Hamiltonian of MoS2 and use the Kubo formula to
calculate the inter-band optical properties.9 On the other side, the intra-band optical
conductivity is calculated by the general formula in the limit of high carrier
concentration:5
 
int ra
d 2 p v v  df ( ( p ))
( )  2 
d

i   1
e2
(S4)
Where p is the wave vector, vα(β) is the group velocity of electron in conduction band, α
and β are the indices, f is the Fermi Dirac distribution function. In our numerical
calculation, the parameters are taken from first-principle calculation to describe
properties of MoS2.8,9 For intraband part, we take the relaxation time data from Cai et
al.,10 where τ equals to 1.889×10-14 s for electron doping case of MoS2. The adopted
parameters for calculation of optical conductivity of MoS2 are summarized in Table. S1.
Table S1. Numerical parameters for monolayer MoS2.8
Δ (ev)
λ (ev)
t0 (ev)
α
β
1.9
0.08
1.68
0.43
2.21
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