An analytical Seebeck coefficient model for disordered organic semiconductors

Physics Letters A 381 (2017) 3441–3444

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Physics Letters A www.elsevier.com/locate/pla

An analytical Seebeck coefficient model for disordered organic semiconductors Xuewen Shi a,b,c , Nianduan Lu a,b,c , Guangwei Xu a,b,c , Jinchen Cao a,b,c , Zhiheng Han a,b,c , Guanhua Yang a,b,c , Ling Li a,b,c , Ming Liu a,b,c a b c

Key Laboratory of Microelectronic Devices & Integrated Technology, Institute of Microelectronics, Chinese Academy of Sciences, Beijing 100029, China University of Chinese Academy of Sciences, Beijing 100049, China Jiangsu National Synergetic Innovation Center for Advanced Materials (SICAM), Nanjing, 210009, China

a r t i c l e

i n f o

Article history: Received 22 March 2017 Received in revised form 23 August 2017 Accepted 5 September 2017 Available online 6 September 2017 Communicated by R. Wu Keywords: Seebeck coefficient Organic semiconductor Percolation theory

a b s t r a c t An analytical Seebeck coefficient model for disordered organic semiconductors based on hopping transport and percolation theory is proposed here. This model demonstrates the relationships between Seebeck coefficient and temperature, carrier concentration as well as disorder degree of materials. As compared with experimental data, the simulated results show a convincing coincidence with experimental results. Moreover, the effect from doping is addressed. The calculation results show that the Seebeck coefficient will decrease with increasing doping ratio, after passing a minimum value then a sharp increase of Seebeck coefficient appears. © 2017 Elsevier B.V. All rights reserved.

1. Introduction Recently, researchers have witnessed the rapid development of organic semiconductors, which provide a number of merits, for instance, lower cost, flexibility, easy to synthesis, capability of massive production, and less weight. These advantages are already presented in the form of the display, sensors and novel photovoltaic cells [1,2]. Another promising application of organic semiconductors is the thermoelectric devices owing to their intrinsically low thermal conductivity and potential low cost associated with ease of low temperature processing [3]. While some organic thermoelectric materials have been reported [4], the development of organic thermoelectric devices is still subjected to inadequate fundamental investigation involving the development of new materials and device architecture. One of the most important characteristics for all organic thermoelectric materials is energy conversion efficiency, which can be described as thermoelectric figure of merit Z T = S 2 σ T /k, where S is Seebeck coefficient, σ denotes electrical conductivity, k is thermal conductivity and T is absolute temperature. It is apparent that increasing S can significantly enhance the figure of merit compared with other parameters. However, due to lacking of thorough understanding of the relationships between Seebeck coefficient and material parameters, such as degree

E-mail addresses: [email protected] (N. Lu), [email protected] (L. Li). http://dx.doi.org/10.1016/j.physleta.2017.09.006 0375-9601/© 2017 Elsevier B.V. All rights reserved.

of disorder, doping level and conductivity, it has taken researchers numerous efforts to choose appropriate materials in laboratory by altering a number of material parameters within a large range to fulfill practical applications. In this paper, an analytical Seebeck coefficient model in disordered organic semiconductors was proposed. The model illustrates the relations between Seebeck coefficient and disorder degree of materials, carrier concentration as well as temperature via deducing basic current equation from Seebeck effect. The proposed model can successfully fit existing experimental results, which proves that it can be used to predict Seebeck coefficient for materials and helps researchers to synthesis specific materials. Furthermore, we also analyze the effect of doping on Seebeck coefficient. 2. Model theory There is a direct energy conversion from heat to electricity in thermoelectric materials. When semiconductors are placed in a temperature gradient environment, the carriers (electrons or holes) can diffuse from heat side to cool side. At the same time, the accumulated carriers in cool side will induce an electric field which will counteract the diffusion. The current is therefore the superposition of diffusion current (from temperature gradient) and drift current (from electric field) as

J = σE +q

d ( D n n) dx

,

(1)

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X. Shi et al. / Physics Letters A 381 (2017) 3441–3444

where σ is conductivity in semiconductors, D n is diffusion coefficient, n is carrier concentration, q is elementary charge, and F E = − 1q ddx with  F denoting Fermi level. Based on Einstein relation, Dμn = (Dσn ) = k BqT , where μ is the carrier mobility and k B is nq the Boltzmann constant, equation (1) can be rewritten as

 J =σ −

1 d F q dx

1 k B d( T σ ) dT

+

σ q

dT

dx

 .

(2)

In an open circuit, J = 0 and one can infer from equation (2) that the Seebeck coefficient is [5]

S=

1 d F q dT

=

1 k B d( T σ )

σ q

dT

(3)

.

Generally, conduction mechanism in disordered organic semiconductors is hopping theory [6,7], although transport process can also be described by using band-like model in single crystal pentacene [8]. In this work we adopt the hopping mechanism. Since generalized Einstein relation and Fermi–Dirac distribution function have been used in the model, the materials discussed should be non-degenerate semiconductors [9]. Following Vissenberg and Matters, we assume the density of (localized) states (DOS) follows exponential form [10],



Nt

g ( ) =

exp

kB T 0



Fig. 1. Temperature dependence of Seebeck coefficient in organic semiconductors. Line represents theory model and symbols is experimental data from Ref. [8].

 (−∞ <  ≤ 0),

kB T 0

(4)

where N t denotes the number of localized states per volume,  is energy, and T 0 is parameter indicating the width of DOS. The proportion of localized states occupied by carriers can be written as [10]

δ=

1

 g ( ) f ( ,  F )d( )

Nt



 exp

F kB T 0

     T T Γ 1− Γ 1+ . T0

(5)

T0

In equation (5), it is assumed that carriers occupy the sites with energy far below 0. The probability that carriers hop between sites depends on distance among sites and energy distribution of localized states. By using percolation theory and variable-range hopping (VRH) mechanism, Vissenberg et al. had given the expression of conductivity [10]:



σ (δ, T ) = σ0

 TT0

π Nt δ( TT0 )3

(6)

,

T )Γ (1 + TT0 ) T0

(2α )3 B c Γ (1 −

where α is a parameter describing overlap of electronic wave function, B c is constant denoting percolation criterion with the value of 2.8. Note that product of δ and N t is carrier concentration, thus it can be replaced by symbol n in equation (6). Substituting equation (6) into equation (3), one can obtain the expression of S:

S=

kB q

+

 1−

T0 T



n sin( πT T ) 0



0

ln

1

nT 04 sin( πT T )



(2α )3 B c T 4  

nπ cos

πT T0

−

4T 0 T

 n sin

πT T0

π T /T

 .

(7)

Approximation Γ (1 − TT )Γ (1 + TT )  sin(π T /0T ) is used when 0 0 0 calculating S. Finally, one can obtain the Seebeck coefficient, which is the function of temperature T , disorder parameter T 0 , and carrier concentration n.

Fig. 2. Relation between carrier concentration and Seebeck coefficient. The Seebeck coefficients for different materials are measured in FET. Lines and symbols are theoretical and experimental results, respectively. Experimental results are from Ref. [9] and [10].

3. Results and discussion Fig. 1 shows the comparison between the experimental data and theoretical calculation for temperature dependence of Seebeck coefficient. Symbols are experimental data from Adrian von Mühlenen [11], where Seebeck coefficient was measured in pentacene thin-film transistors and line is theoretical Seebeck coefficient calculated from equation (7). The fitting parameters are α = 8.4 × 107 cm−1 , n = 1 × 1021 cm−3 , Nt = 1021 cm−3 and T 0 = 440 K. The positive Seebeck coefficient values suggest that the dominant carrier is hole in pentacene material. From Fig. 1, one can also see that S decreases slowly with increase of temperature. Physically, it has been pointed out, higher Seebeck coefficient are often associated with very high density of states near the Fermi level [12], if the carrier concentration is constant (corresponding to the gate voltage in Ref. [11]), that is Fermi level keeps constant, the higher temperature results in the lowering of the transport energy and decrease of the Seebeck coefficient [13]. The dependence of Seebeck coefficient on carrier concentration is illustrated in Fig. 2. Symbols are the measured Seebeck coefficients from organic FETs based on rubrene [14], poly(2,5-bis(3alkylthiophen-2-yl)thieno(3,2-b)thiophene) (PBTTT) and indacenodithiophene-co-benzothiadiazole (IDTBT) [15] at room temperature, respectively. Lines are fitting results from equation (7). For

X. Shi et al. / Physics Letters A 381 (2017) 3441–3444

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Fig. 3. The dependence of S on T 0 in disordered organic semiconductors with temperature as a parameter. (For interpretation of the colors in this figure, the reader is referred to the web version of this article.)

rubrene, the model parameters are α = 3.5 × 107 cm−1 , T 0 = 350 K. For IDTBT, the parameters are α = 2.42 × 107 cm−1 , T 0 = 350 K. And as for PBTTT, the parameters are α = 2.3 × 107 cm−1 , T 0 = 670 K. Other parameters are the same as those in Fig. 1. It can be seen clearly that the model shows good agreement with experimental results. The physical origin of Seebeck coefficient decreases with increasing carrier concentration can also been explained: Increasing carrier density will lead to an increase of electrical conductivity, and the electron contribution of thermal conductivity will increase according to the Wiedemann Franz law [16]. Enhancing the electron component of thermal conductivity tends to decrease the entropy difference and decrease of Seebeck coefficient. In addition, Seebeck coefficient shows a linearity relation with log(n) both in the calculation of equation (7) and experiment data. The relationship between Seebeck Coefficient and T 0 at different carrier concentration with parameters α = 8.4 × 107 cm−1 , σ0 = 1010 S/m, T = 250 K, B c = 2.8 is demonstrated in Fig. 3. One can see that that S increases with disorder parameter T 0 and decreases with carrier concentration. For a larger disorder, it has a negative effect on mobility [17], thus the electrical conductivity decreases with increasing disorder. Otherwise, the conductivity is inversely proportional to Seebeck coefficient, the degeneration of conductivity introduced by disorder will result in the increase of Seebeck coefficient. Besides, Mendels et al. also demonstrates that increase of disorder in disordered organic semiconductors will raise transport energy and thus increase Seebeck coefficient [13]. Doping of organic semiconductors is an important technology which dramatically increases the density of the charge carriers and conductivity [18]. It should be mentioned that the proposed model also gives a prediction about Seebeck coefficient versus doping ratio at different dopant energy E d . Based on the proposed model, we made a further development by including additional effect from doping. It is well known, the characteristic and transport property of semiconductors can be changed by doping. From point of view of DOS, doping broadens the DOS [19,20]. Therefore doping effect in disordered organic semiconductors can be accounted by adding an additional DOS function in equation (4), which is expressed in equation (8) [21]

g doping ( ) =

Nt kB T 0

(−∞ <  ≤ 0),

 exp

 kB T 0

 +

Nd kB T 1

 exp

 − Ed



kB T 1 (8)

Fig. 4. Seebeck coefficient as a function of doping ratio for different E d . Inset: electrical conductivity as a function of doping ratio from Ref. [18]. (For interpretation of the colors in this figure, the reader is referred to the web version of this article.)

where T 1 represents length of dopant distribution, N d is the concentration of doping sates, and E d denotes the position of dopant in energy band diagram. Carrier concentration in doping semiconductors can be calculated by Fermi–Dirac distribution f ( ,  F )



g ( )

n=

1 + exp[

(  − F ) kB T

]

d .

(9)

As the preceding method, carriers transport in disordered organic semiconductors obeys hopping mechanism, and conductivity can be determined using percolation theory. As Li et al. has mentioned about doped organic semiconductors [21], the conductivity is function of sc , which is

σdoping = σ0 exp(−sc ),

(10)

where σ0 is the prefactor, sc is the exponent of conductance. sc is determined in the following section: B c is constant with value 2.8, and it relate to sc with equation

Bc =

κ+p N t exp(η) + N d exp(Υ )

(11)

,

κ = π N t2 ϕ 3 exp(2η) + π N d2 ξ 3 , p = π4 Nd Nt exp(η +Υ )(ϕ −1 +  +k T s − E T k B T sc ξ −1 )−3 , η =  F + , Υ = F kB T c d , ϕ = 4α0T , ξ = 4Tα1T . sc can kB T0 B 1 where

be obtained from numerical calculation and by using equation (10) one can obtain the conductivity. Substituting these expressions into equation (3), one can finally obtain the Seebeck coefficient N versus doping ratio N +dN : t

d

S doping =

1 d F q dT

=

1

k B d( T σdopoing )

σdopoing q

dT

(12)

In the proposed model, we assume  F = −3k B T 0 with E d = 1k B T 1 , 1.2k B T 1 and 2k B T 1 respectively. Fig. 4 illustrates the relation between S and doping ratio. It is found that, first, Seebeck coefficient decreases slowly, which reflects the intrinsic characteristics. Then an obvious decline of Seebeck coefficient occurs with the increase of doping ratio, which shows that doping effect cannot be neglected. Most important, a sharp increase of Seebeck coefficient occurs when doping ratio reaches a larger value depending on different E d . Shen et al have reported that the doping dependence electrical conductivity (as shown in inset of Fig. 4) can be

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X. Shi et al. / Physics Letters A 381 (2017) 3441–3444

attributed to DOS broadening from doping and manifold filling [18]. Since the Seebeck coefficient has an inverse relationship with conductivity, thus one can attribute the change of Seebeck coefficient to the same reason.

Academy of Sciences under Grant XDB12030400, and by the National 973 Program (Grant Nos. 2013CBA01604, 2013CB933504, and 2013CB933504). References

4. Conclusion In this work, we have proposed an analytical Seebeck coefficient model which gives reliable simulation results with respect to carrier concentration and temperature compared with experimental data. In addition, we analyzed the dependence of Seebeck coefficient on T 0 showing that materials with higher disorder will contribute to higher Seebeck coefficient. Moreover, effect of doping on Seebeck coefficient was covered as well. Introduction of dopant DOS will alter the distribution of carrier concentration. In low doping region, effect of doping is negligible, however, in high doping region, doping gradually affects the DOS and results in a minimum S together with a sharp increase after that. Acknowledgement This work was supported in part by the Opening Project of Key Laboratory of Microelectronic Devices and Integrated Technology, Institute of MicroElectronics Chinese Academy of Sciences, by National key research and development program (Grant Nos. 2016YFA0201802, 2017YFB0701703), by the National Natural Science Foundation of China (Grant Nos. 61574166, 61306117, 61221004, and 61376112), by the Beijing Training Project for the Leading Talents in S&T under Grant No. Z151100000315008, by the Strategic Priority Research Program through the Chinese

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