A synthetic approach for enhanced thermoelectric properties of PEDOT:PSS bulk
composites
Kaya Wei, Troy Stedman, Zhen-Hua Ge, Lilia M. Woods*, and George S. Nolas*
Department of Physics, University of South Florida, Tampa, FL 33620, USA.
Supplementary Information
A. Thermal Averaging of Fluctuation Induced Electric Field
Any quantity 𝐻(ℰ 𝑇 ) that is dependent on the thermally induced field is averaged over the probability
2𝜀 𝑤𝐴 1/2 (−𝜀 𝑤𝐴ℰ 2 /2𝑘 𝑇)
𝐵
𝑇
) 𝑒 0
𝐵𝑇
𝑃(ℰ 𝑇 ) = ( 𝜋𝑘0
where 𝑎 =
using the following
4𝑎 1/2 ∞
2
⟨𝐻⟩ = (
) ∫ 𝐻(ℰ 𝑇 )𝑒 (−𝑎ℰ𝑇 /𝑘𝐵 𝑇) 𝑑ℰ 𝑇
𝜋𝑘𝐵 𝑇
0
𝜀0 𝑤𝐴
.
2
(A.1)
B. Charge and Heat Currents Calculations
Without loss of generality, the general expressions for the charge and heat currents can be written when
the external field ℰ and temperature gradient ∇𝑇 are directed along 𝑥̂ according to Fig. 1 When the
fluctuation-induced field points to the right, the Fermi levels of the left and right reservoirs are 𝜇𝐿 = 𝜇 −
𝑒𝑤(ℰ𝑇 + ℰ) and 𝜇𝑅 = 𝜇, respectively. Assuming ℰ 𝑇 > ℰ, when the fluctuation-induced field points to
the left, the Fermi levels of the left and right reservoirs are 𝜇𝐿 = 𝜇 and 𝜇𝑅 = 𝜇 − 𝑒𝑤(ℰ𝑇 −
ℰ), respectively.
1
Since both types of orientations are equally probable, one writes 𝐽 = 2 (𝑗(ℰ + ℰ 𝑇 ) + 𝑗(ℰ − ℰ 𝑇 )), 𝐽𝐿𝑄 =
1 𝑄
(𝑗 (ℰ
2 𝐿
1
+ ℰ 𝑇 ) + 𝑗𝐿𝑄 (ℰ − ℰ 𝑇 )) , 𝐽𝑅𝑄 = 2 (𝑗𝑅𝑄 (ℰ + ℰ 𝑇 ) + 𝑗𝑅𝑄 (ℰ − ℰ 𝑇 )). The currents are further expressed:
∞
𝐽 = ∫ 𝑑𝐸[𝑓(𝐸, 𝑇 + ∆𝑇) − 𝑓(𝐸 + 𝑒𝑤(ℰ 𝑇 + ℰ), 𝑇)]𝑀(ℰ 𝑇 + ℰ, 𝐸)
0
∞
(B.1a)
+ ∫ 𝑑𝐸[𝑓(𝐸 + 𝑒𝑤(ℰ 𝑇 − ℰ), 𝑇 + ∆𝑇) − 𝑓(𝐸, 𝑇)]𝑀(ℰ 𝑇 − ℰ, 𝐸)
0
𝐽𝐿𝑄
1 ∞
= ∫ 𝑑𝐸[𝑓(𝐸, 𝑇 + ∆𝑇) − 𝑓(𝐸 + 𝑒𝑤(ℰ 𝑇 + ℰ), 𝑇)][𝐸 − 𝜇 + 𝑒𝑤(ℰ 𝑇 + ℰ)]𝑀(ℰ 𝑇
𝑒 0
+ ℰ, 𝐸)
1 ∞
+ ∫ 𝑑𝐸[𝑓(𝐸 + 𝑒𝑤(ℰ 𝑇 − ℰ), 𝑇 + ∆𝑇) − 𝑓(𝐸, 𝑇)][𝐸 − 𝜇]𝑀(ℰ 𝑇
𝑒 0
− ℰ, 𝐸)
(B.1b)
1
1 ∞
𝑄
𝐽𝑅 = ∫ 𝑑𝐸[𝑓(𝐸, 𝑇 + ∆𝑇) − 𝑓(𝐸 + 𝑒𝑤(ℰ 𝑇 + ℰ), 𝑇)][𝐸 − 𝜇]𝑀(ℰ 𝑇 + ℰ, 𝐸)
𝑒 0
1 ∞
+ ∫ 𝑑𝐸[𝑓(𝐸 + 𝑒𝑤(ℰ 𝑇 − ℰ), 𝑇 + ∆𝑇) − 𝑓(𝐸, 𝑇)][𝐸 − 𝜇
𝑒 0
+ 𝑒𝑤(ℰ 𝑇 − ℰ)]𝑀(ℰ 𝑇 − ℰ, 𝐸)
(B.1c)
where
𝐸
𝑒𝑚
.(B.1d)
∫
𝐷(ℰ 𝑇 , 𝐸𝑥 )𝑑𝐸𝑥
(2𝜋)2 ℏ3 0
1
Note that the heat current traveling through the junction is 𝐽𝑄 = 2 (𝐽𝐿𝑄 + 𝐽𝑅𝑄 ). Applying linear response
𝑀(ℰ 𝑇 , 𝐸) =
theory to the currents, we obtain
𝐽 = ℒ11 ℰ + ℒ12 ∇𝑇
𝐽𝑄 = ℒ21 ℰ + ℒ22 ∇𝑇
(B.2a)
(B.2b)
where
𝜕𝐽
𝜕𝐽
, ℒ12 = 𝑤
𝜕ℰ ℰ=∆𝑇=0
𝜕𝑇ℰ=∆𝑇=0
𝜕𝐽𝑄
𝜕𝐽𝑄
ℒ21 =
, ℒ22 = 𝑤
𝜕ℰ ℰ=∆𝑇=0
𝜕𝑇 ℰ=∆𝑇=0
Further simplifications can be obtained by defining the following integrals:
ℒ11 =
(B.3a)
.(B.3b)
∞
𝐿1 (ℰ 𝑇 , 𝑇) = ∫ 𝑑𝐸(𝑓(𝐸, 𝑇) − 𝑓(𝐸 + 𝑒𝑤ℰ 𝑇 , 𝑇))𝑀(ℰ 𝑇 , 𝐸)
(B.3c)
0
∞
𝐿2 (ℰ 𝑇 , 𝑇) = ∫ 𝑑𝐸(𝑓(𝐸, 𝑇) + 𝑓(𝐸 + 𝑒𝑤ℰ 𝑇 , 𝑇))𝑀(ℰ 𝑇 , 𝐸)
(B.3d)
0
1 ∞
𝑞
𝐿1 (ℰ 𝑇 , 𝑇) = ∫ 𝑑𝐸(𝑓(𝐸, 𝑇) − 𝑓(𝐸 + 𝑒𝑤ℰ 𝑇 , 𝑇))[𝐸 − 𝜇]𝑀(ℰ 𝑇 , 𝐸)
𝑒 0
∞
1
𝑞
𝐿2 (ℰ 𝑇 , 𝑇) = ∫ 𝑑𝐸(𝑓(𝐸, 𝑇) + 𝑓(𝐸 + 𝑒𝑤ℰ 𝑇 , 𝑇))[𝐸 − 𝜇]𝑀(ℰ 𝑇 , 𝐸)
𝑒 0
Therefore, Eqs. (B.3a,b) can equivalently be given as
ℒ11 = 2
𝑞
𝜕𝐿1
,
𝜕ℰ 𝑇
𝜕𝐿1
𝜕𝐿1
+ 𝑉𝑇
+ 𝑤𝐿1 ,
𝜕ℰ 𝑇
𝜕ℰ 𝑇
Thermally averaging Eqs. (B.2a,b) gives
ℒ21 = 2
𝜕𝐿2
𝜕𝑇
𝑞
𝜕𝐿
𝑉𝑇 𝜕𝐿2
= 𝑤( 2+
)
𝜕𝑇
2 𝜕𝑇
ℒ12 = 𝑤
ℒ22
⟨𝐽⟩ = ⟨ℒ11 ⟩ℰ + ⟨ℒ12 ⟩∇𝑇
⟨𝐽𝑄 ⟩ = ⟨ℒ21 ⟩ℰ + ⟨ℒ22 ⟩∇𝑇
so that the transport properties may be identified as
𝜎 = ⟨ℒ11 ⟩, 𝑆 = −
⟨ℒ12 ⟩
⟨ℒ21 ⟩⟨ℒ12 ⟩
, 𝜅𝐸 = ⟨ℒ22 ⟩ −
⟨ℒ11 ⟩
⟨ℒ11 ⟩
(B.3e)
(B.3f)
(B.3g)
(B.3h)
(B.4a)
(B.4b)
(B.4c)
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Figure S1 shows the XRD patterns of the 30 wt% Bi0.5Sb1.5Te3 together with PEDOT:PSS and
Bi0.5Sb1.5Te3 and PEODT:PSS for comparison. The XRD results on the 30 wt%, 60 wt%, and 90
wt% specimens are similar to that of the 30 wt% specimen, i.e. only the pattern due to Bi0.5Sb1.5Te3
is observed, the XRD pattern of PEDOT:PSS being more of a “background” as expected for
organic materials.
Figure S1. XRD patterns of (a) Bi0.5Sb1.5Te3 powder, (b) 30 wt% Bi0.5Sb1.5Te3 with PEDOT:PSS
and (c) PEODT:PSS bulk. The XRD of the 30 wt% specimen is representative of the XRD
spectra for all three polymer/inorganic specimens.
Figure S2 shows SEM images. The regions with light gray color in Figures S2 (b), (c) and (d)
represent the location of the inorganic material. This area increases with increasing Bi0.5Sb1.5Te3
content, as expected; larger inorganic regions are observed when the level of the inorganic content
is higher than 30 wt%, as shown in Figure S2 and described in the manuscript.
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Figure S2. SEM images of (a) PEODT:PSS bulk, (b) 30 wt% Bi0.5Sb1.5Te3 with PEDOT:PSS, (c)
60 wt% Bi0.5Sb1.5Te3 with PEODT:PSS, (d) 90 wt% Bi0.5Sb1.5Te3 with PEODT:PSS.
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