Supplementary information
Thermoelectric Properties of Si/SiB3 Sub-Micro Composite Prepared by Melt-Spinning Technique
I.
Theoretical model
We used the model proposed by Morelli,s1) which is based on the Debye-Callaway formalism.s2) In this
model, the lattice thermal conductivity is expressed by
  i T  i x x 4e x

C


dx
2

i
4 x
i
x
0


i
 T   x x e
 N x  e  1
1
1 i 3 
 ,
C
  C iT 3 
dx

C
T
2
i
4 x
i
x
0
 x x e
 T
3
i 3
e 1
0  i x C i x  e x  1 2 dx
N
R
2
 lat





(1)

where Ci is the lattice heat capacity, vi is the phonon group velocity, i is the Debye temperature, ()1
is the scattering rate for normal phonon processes, (c)-1 is the sum of all resistive scattering processes,
and (c)-1 = ()-1 + (c)-1. The superscript notation i represents the phonon branch: one longitudinal
(L) and two degenerate transverse (T) phonon branches. Ci and x are given by
k B4
C 
2 2  3v i
(2)

,
k BT
(3)
i
and
x
where  is the Planck constant, kB is the Boltzmann constant,  is the phonon frequency, and vi is the
phonon velocity of branch i. For a pure single crystal, the resistive scattering rate is assumed to be the
sum of the scattering rates due to phonon-phonon umklapp scattering  Ui 1 , point defect scattering


1
i
PDiso
due to the presence of isotopes, and scattering from the boundaries  Bi  :
 
i 1
R
1
 
  Ui
1

i
  PDiso

1
 
  Bi
1
.
(4)
The functional forms of these scattering rates and the required parameters for pure single-crystal Si
are given in Ref. s1 and are not reproduced here. In the present study, we added the scattering rates
due to phonon-electron, grain-boundary scattering, and impurity scattering in order to calculate the
thermal conductivity of the heavily B-doped polycrystalline Si:
      
i 1
R
i 1
U
            
1
i
PDiso
i 1
B
1
i
PDim
1
i
GB
i 1
pe
,
(5)
wheres3)


1
i
PDim

Vk B4  4 4 ,
x T
4 4 vi3
(6)
 
1
i
GB

vi
,
D
(7)
ands4)
 1




1  exp   m*vi2  E F  k BT   2 2 8m*vi2 k BT   2k BT  


E m vi k BT  
1

 2
,



 ln
4


4 d  1 m*v 2  k BT
 1 * 2


2 2
* 2


1

exp
m
v

E
k
T



8
m
v
k
T



k
T


i 


i
F
B
i B
B  
 2
2



 
 
i
pe
2
def
*3
(8)
where V is the volume per atom, D is the grain size, Edef is the electron-phonon deformation potential,
m* is density-of-states effective mass, d is the density, and EF is the Fermi energy.  is the phononscattering parameter which is expressed bys3)
  j
2
2

   j  
 M  M j 

   
f j 
 ,
M




 


(9)
where M is the averaged atomic mass, Mj is the atomic mass of impurity j, fi is the fractional
concentration of the impurity,  is the averaged cube root of the atomic volume, and j is the cube
root of the atomic volume of impurity j. Generally, is regarded as an adjustable parameter. We used
literature or slightly modified values for all parameters other than , which was determined by fitting
to the experimental value of Si99B1-Arc. The parameters used in the present study are summarized in
Table SI.
Table SI. Electron-modeling parameters.
Property
Sound velocity
Volume per atom
Parameter
vL
vT
θL
θT
V
Value
8340 (m/s)
5840 (m/s)
586 (K)
240 (K)
9.47 (Å3)
Grain size
D
Electron-phonon
deformation potential
Density-of-states
effective mass
Fermi energy
Edef
13 (μm)a
7 (μm)a
3 (μm)a
3 (eV)
m*
0.81·me (kg)
EF
0.28 (eV)
Debye temperature
Note
Ref. s1
Ref. s1
Ref. s1
Ref. s1
Evaluated from the lattice
parameter 4.231 Å, which is
estimated from the lattice
parameter of B-doped Si.
This study
Ref. s5
Ref. s6
Estimated from the carrier
density of 5 × 1020 cm-3 and the
effective mass.
Density
d
2318 (kg/m3)
Estimated from the lattice
parameter and averaged atomic
mass of Si99B1.
Parameter relevant to ε
6
Fitted to the experimental value
impurity scattering
of Si99B1-Arc-SPS.
a 13 μm for Si B -Arc-SPS, 7 μm for Si B -Arc-SPS, and 3 μm for Si B -MS-SPS.
99 1
92 8
92 8
II. Thermal stability during measurement
Because nonequilibrium techniques such as SPS and melt-spinning were used in this work, it is
important to show that no change was occurred during the transport property measurements. The
experimental data measured during both heating and cooling thermal cycles is shown in Fig. s1. As
can been seen, hysteresis was not observed. The data obtained during heating and cooling cycles show
an excellent agreement in the resistivity, Seebeck coefficient and total thermal conductivity. This
results suggest that these samples were stable under the transport property measurements conditions.
Fig. s1. The experimental data of Si92B8-MS-SPS measured during both heating and cooling thermal
cycles.
References
s1) D. T. Morelli, J. P. Heremans, and G. A. Slack, Phys. Rev. B, 66(2002), 195304.
s2) J. Callaway, Phys. Rev., 113(1959), 1046.
s3) B. Abeles, Phys. Rev., 131(1963), 1906.
s4) E. F. Steigmeier and A. Abeles, Phys. Rev., 136(1964), A1149.
s5) H. R. Shanks, P. D. Maycock, P. H. Sidles, and G. C. Danielson, Phys. Rev., 130(1963), 1743.
s6) M. Lundstrom, Fundamentals of Carrier Transport, 2nd ed. (Cambridge University Press,
Cambridge, United Kingdom, 2000).