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Thermoelectric properties of ZnO nanowires: A first principle research
Physics Letters A 375 (2011) 2878–2881
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Thermoelectric properties of ZnO nanowires: A first principle research Chaoren Liu ∗ , Jingbo Li State Key Laboratory for Superlattices and Microstructures, Institute of Semiconductors, Chinese Academy of Sciences, P.O. Box 912, Beijing 100083, China
a r t i c l e
i n f o
Article history: Received 5 April 2011 Received in revised form 11 June 2011 Accepted 12 June 2011 Available online 14 June 2011 Communicated by R. Wu Keywords: Thermoelectric property ZnO nanowire First principle calculation Boltzmann transport equation
a b s t r a c t By means of ab-initio electronic structure calculation and one-dimensional Boltzmann transport equation solution, we investigate the size dependent thermoelectric (TE) properties of n-type ZnO nanowires (NWs) and surface passivation effects. As demonstrated by our calculations, largest figure of merit ZT achievable in thin NWs is larger than that in wide NWs, whereas being restrained by higher demand of n-type doping. Moreover, bare NWs are superior in TE application comparing with the passivated. To compete with conventional TE materials, lattice thermal conductivity of ZnO NWs should be at least 2 orders of magnitude lower than bulk value. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Thermoelectric (TE) materials are regarded as promising candidates in solving energy crisis. However, the complicated interaction between electronic conductivity σ , Seebeck coefficient S and thermal conductivity κ (sum of electron contribution κe and lattice contribution κl ) makes it difficult to enhance the figure of merit ZT = σ S 2 T /(κe + κl ), which has remained around 1.0 for many years. ZT should be increased to no less than 3.0 if we want to put TE materials into application. Lee et al. [1] and Heremans et al. [2] raised ZT by means of distorting density of states (DOS) nearby Fermi level through doping special atoms. Others realized high ZT in complex alloys [3], taking advantage of their low thermal conductivity. Recently, quantum confinement was proposed to be another effective approach [4,5]. On one hand, it is well known that low-dimensional materials have a DOS of sharp shape at band edge [6], which is favorable to high TE efficiency [7]. On the other hand, the large surface to volume ratio in low-dimensional systems intensifies scattering of phonon, decreasing thermal conductivity as a result. It was reported, for example, that the thermal conductivity of silicon NWs was 100 times smaller than bulk value [8,9]. In laboratory, the development of advanced material preparation methods makes it possible to grow NWs with different diameters and surface geometries in different directions, which facilitates the experimental investigation of TE properties in nanomaterials [8,10]. ZnO NWs have received great attention for many years, mostly in their electronic and optical properties [11,12]. To the best of my
*
Corresponding author. Tel.: +86 0 15811043227; fax: +86 10 82305056. E-mail addresses: [email protected] (C. Liu), [email protected] (J. Li).
0375-9601/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2011.06.024
knowledge, there are no reports related to TE properties in ZnO NWs both experimentally and theoretically. In this Letter, we utilize a combination of density functional theory (DFT) and Boltzmann transport equation simulation (BTES) to give a description of TE properties in both bare and passivated ¯ ) facet surfaces ZnO NWs, which are [0001] oriented with six (1010 because this is commonest in experiments. Due to the fact that asprepared ZnO is n-type and p-type doping is difficult in ZnO, we only study the former kind in this Letter. In the first place, we discuss their structural and electronic properties and find that, before passivation, surface Zn atoms relax inward relative to O atoms manifestly, in consequence prolonging NWs in growth direction while, after passivation, the relaxation becomes negligible, which demonstrates our successful dangling bonds saturation. The investigation of TE properties tells that ZT can be tuned up through carrier doping and bare NWs are superior to passivated ones in TE application. Furthermore, ZT can be improved by decreasing NWs dimension, along with an increasing requirement of free electron doping. 2. Calculation method Two bare NWs, with hexagonal cross sections and different diameters (3.69 Å and 9.78 Å), are constructed from LDA relaxed wurtzite bulk ZnO (a = 3.24 Å, c = 5.23 Å, u = 0.387) and labeled as B1 and B2. Then we saturate the dangling bonds on B1 and B2 with pseudo-hydrogen atoms, which have equal amount of nucleus and electron charge, and label the counterparts as P1 and P2 respectively. Each dangling bond on surface Zn atoms is terminated with one H1.5 atom, whose charge amount is 1.5e (e = 1.6 × 10−19 C), while each dangling bond on surface O atoms
C. Liu, J. Li / Physics Letters A 375 (2011) 2878–2881
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Table 1 LDA bandgap and effective mass ([0001] direction) of B1, B2, P1 and P2.
E g (eV) m∗ (me ) a
B1
B2
P1
P2
Bulk
1.99 0.350
1.47 0.295
4.05 0.855
2.16 0.395
0.79 0.181a
See Ref. [20].
tronic structures, carrier concentrations, temperatures and so on, which makes it too complex to be obtained precisely. For lack of experimental dependence of electronic mobility on carrier concentration in ZnO NWs, we obtain the relaxation time in NWs by fitting the calculated electronic mobility in bulk ZnO to experimental values [15]. This approximation has been testified reasonable by Shi et al. [16] and Vo et al. [17] if we just care about the qualitative tendency rather than the accuracy of results. 3. Structural and electronic properties
Fig. 1. (Color online.) Views of as-constructed B1, B2, P1 and P2 in [0001] direction. Zn, O and pseudo-hydrogen atoms are represented as blue, red and gray balls.
with one H0.5, whose charge amount is 0.5e. The bond lengths of Zn–H1.5 and O–H0.5 are assumed to be the same as in ZnH2 and H2 O. We put the views of all NWs in Fig. 1 in [0001] direction. The initial structures are then relaxed with quasi-Newton algorithm until Hellmann–Feynman force acting on each atom is smaller than 0.01 eV/Å. We allow the lattice constant c to change freely since we find that ZnO [0001] NWs tend to stretch after relaxation. The periodic boundary condition is applied and NWs are separated by 10 Å vacuum layers, which proves sufficient to eliminate the interaction between different images. The atomistic relaxation and electronic structure calculation are performed with density functional theory (DFT) as implemented in VASP code [13]. Local density approximation (LDA) is utilized to treat exchange-correlation energy. Frozen-core projected augmented wave methods [14] is utilized to describe the ion–electron interactions. Zn3d electrons are explicitly treated as valence electrons. Monkhorst–Pack special k-point grid (1 × 1 × 8) is used to integrate in Brillouin zone. The electronic conductivity σ , Seebeck coefficient S and electronic thermal conductivity κe are obtained by means of processing electronic structures with the solution of one-dimensional Boltzmann transport equation in constant relaxation time approximation [7]:
+∞ ∂ f0 (ε ), σ =e dε − ∂ε 2
(1)
−∞
+∞ dε −
Tσ S = e −∞
+∞ dε −
T κ0 = −∞
(ε ) =
∂ f0 (ε )(ε − μ), ∂ε
∂ f0 (ε )(ε − μ)2 , ∂ε
νx (k)2 τ (k)δ ε − ε(k) ,
(2)
(3)
(4)
k
κe = κ0 − T σ S 2 ,
(5)
where e is charge of electron; μ is chemical potential; T is absolute temperature; f 0 is Fermi distribution function; νx (k) = (1/¯h)(∂ E (k)/∂ kx ); τ is relaxation time. τ varies with different elec-
We start by investigating the structural properties of bare and passivated ZnO NWs. Since the relaxation of NWs with different diameters follows similar trend, we take B2 and P2 as the example. Surface Zn atoms move inward relative to outmost O atoms, resulting in a distorted hexagonal cross section with three sides short (4.65 Å) and three sides long (5.93 Å). To release stress, NWs stretch in growth direction, with lattice constant c of B2, for example, increasing from bulk value 5.23 Å to 5.41 Å. The angle of surface O–Zn–O, in view of [0001] direction, increases to 140◦ from 120◦ while that of Zn–O–Zn decreases to 114◦ . A similar result has been reported by Xiang et al. [18] and Xu et al. [19]. After pseudo-hydrogen passivation, we find that the relaxation of surface atoms becomes unnoticeable, which demonstrates our success of surface state elimination. The bond length of Zn–H1.5 and O– H0.5 are relaxed to 1.64 Å and 1.05 Å, close to the values in ZnH2 (1.62 Å) and H2 O (0.94 Å). All ZnO NWs studied in this Letter prove to be semiconducting with direct bandgap at Γ point, which are shown in Table 1. We can see that bandgap increases when NW diameter decreases and all of them are larger than the bulk value. Passivation process enlarges the bandgap further. To understand these phenomena, we analyze the composition of bands and draw the charge density of valence band maximum (VBM) and conduction band minimum (CBM) in Fig. 2. VBM of bare NWs are contributed by Zn3d and O2p orbits, however of surface atoms, which indicates surface states, while CBM mainly consist of Zn4s orbits from inside atoms, which indicates bulk states. All of these are verified further in Fig. 2 (b1) and (c1). As indicated by Xiang [18], the O2plike surface state in B2 is only 80 meV above the VBM of bulk ZnO. In this case, when diminishing the diameter of bare NWs, quantum confinement effect will uplift CBM significantly because bulk state delocalizes greatly in radial direction. However, surface state responses to dimension constraint inertly due to its localized attribute. Hence the delocalized property of CBM as well as the low position of surface state result in a sensitive size-dependent bandgap of bare NW and a sharp increase of bandgap from bulk to NWs. In P1 and P2, Zn3d and O2p orbits in core part are the main component of VBM while CBM is primarily contributed by Zn4s and O3s orbits of inside atoms. From Fig. 2 (b2) and (c2), it is shown that the charge of both VBM and CBM locates mainly in inner part, which demonstrates their bulk state attributes. That is to say dangling bonds termination with pseudo-hydrogen successfully eliminates the surface states nearby VBM, meanwhile enlarging bandgap. Moreover, it is easy to notice that difference between the bandgap of B2 and B1 is 0.52 eV but the value for P2 and P1 is as large as 1.89 eV, which could be understood as follows. CBM might become more delocalized in radial direction after surface passiva-
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C. Liu, J. Li / Physics Letters A 375 (2011) 2878–2881
with free electrons, with carrier concentration obtained by dividing
Ef
Fig. 2. (Color online.) Band structures of B2 (a1) and P2 (a2) in [0001] direction; (b) and (c) are charge distribution of CBM and VBM with (b1), (c1) for B2 and (b2), (c2) for P2. Fermi levels are set at 0 eV for convenience.
tion and thus quantum confinement effect grows more significant in passivated NWs. 4. Thermoelectric properties The electronic conductivity σ , Seebeck coefficient S, electronic thermal conductivity κe and power factor P , having direct influence on figure of merit ZT, are functions of carrier concentration which are shown in Fig. 3. The systems are artificially doped
Fig. 3. (Color online.) Calculated dependence of electronic conductivity concentration.
N = CBM dε N (ε ) f (ε ), where N (ε ) is density of states and f (ε ) is Fermi distribution function, by the volume of NWs in superlattice. Both σ and κe increase with carrier concentration because there are more carriers to transport electricity and heat, but S decreases when carrier density increases. Large cross section is favorable to σ and κe , which is also true in silicon NWs [16], but passivation process lowers them significantly. Because σ as well as κe are in inverse proportion to effective mass m∗ , we calculate the effective mass m∗ along c direction of all NWs and cite the experimental value of bulk ZnO in Table 1. By comparison, we can see that both quantum confinement and surface passivation enlarge m∗ , which increases from 0.181, 0.295 to 0.350 me for bulk, B2 and B1, from 0.295 to 0.395 me for B2, P2 and from 0.350 to 0.855 me for B1, P1. S is larger in smaller NWs and is enhanced further by passivation. This size dependence results from DOS narrowing phenomenon in quantum confinement. Power factor ( P = σ × S 2 ), important in TE application, is also shown in Fig. 3. It is presented that P reaches to a maximum at a specific carrier concentration for larger NWs but for smaller ones there are no maxima in the concentration range we are interested in. We can also find that P of larger NWs is larger in low carrier concentration area but, when carrier concentration increases to a certain point, smaller NWs hold a larger P instead. Having discussed σ , S and κe , then we move on to another important factor κl which is directly related to NWs dimension [21, 22]. Until now, there are no systematic researches on lattice thermal conductivity of ZnO NWs with different diameters and growth directions both theoretically and experimentally. In our calculation, we substitute the simulated κl of silicon NWs [22], which have the same sizes as ZnO NWs, for the values of ZnO NWs. ZT of B1, P1, B2 and P2 are plotted in Fig. 4, with κl to be 0.12 W m−1 K−1 for B1 and P1 and 0.30 W m−1 K−1 for B2 and P2. We can find that from B1 to P1 and from B2 to P2, ZT is repressed after surface passivation, mostly resulting from the drop of σ . It is obvious that ZT increases with carrier concentration at the first stage due to increase of σ and begins to drop at a specific carrier concentration due to decrease of S and increase of κe . A closer look will also find that the carrier concentration, where ZT reaches maximum, becomes larger from bare to passivated NWs and when NW size decreases. It has been claimed that quantum confinement lowers
σ , Seebeck coefficient S, thermal conductivity of electron κe and power factor P on free electron
C. Liu, J. Li / Physics Letters A 375 (2011) 2878–2881
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high ZT against doping difficulty while determining NW diameter. As indicated, κl need be reduced by approximately 2 orders of magnitude comparing to the bulk value in order to compete with conventional TE materials. Although most experimental ZnO NWs are much wider (> 10 nm) than ones investigated here, our calculation supplies a qualitative guide when optimizing the size, carrier doping and surface treatment of ZnO NW TE materials. Acknowledgements
Fig. 4. (Color online.) Dependence of ZT on free electron concentration calculated with κl = 0.12 W m−1 K−1 for B1 and P1, 0.30 W m−1 K−1 for B2 and P2.
valence band and uplifts conduction band, hence increasing the ionization energy and making effective doping more difficult [23]. Khanal [23] also reported that 9 × 1019 cm−3 was the limit of free electron concentration in ZnO NWs with a diameter under 60 nm. That is to say although achievable ZT of smaller NWs (B1 and P1) is higher than that of larger ones (B2 and P2), the higher carrier concentration requirement grows into a new challenge. Especially, ZT of P1 is higher than that of P2 only when carrier concentration is higher than 8.53 × 1019 cm−3 . Thereby, we should balance the dimension of NWs to a satisfactory value from the view of high ZT as well as low carrier concentration requirement. Maximal κl , making highest ZT obtainable larger than 1.0, for B2 and P2 are also calculated (0.316 and 0.157 W m−1 K−1 ), approximately 2 orders of magnitude lower than the bulk value which is 23 W m−1 K−1 [24]. 5. Conclusion In summary, the structural and electronic properties of both bare and passivated ZnO NWs are investigated. Surface Zn atoms of bare NWs relax inward relative to O atoms, as a result elongating the NWs in growth direction, and after surface passivation the relaxation becomes negligible. CBM is a delocalized state in both bare and passivated NWs while VBM of bare NWs is a surface state, successfully removed after pseudo-hydrogen saturation. Then, TE properties are discussed. It is found that σ and κe increase with cross section area but S decreases when the dimension of NWs increases. Furthermore, surface passivation reduces σ and κe while improves S. ZT increases with carrier concentration at the first stage and later begins to drop down. The carrier concentration, in which ZT reaches the peak, is low in large bare NWs. In TE application, bare NW is favored and the highest ZT obtainable is larger in smaller NWs, however requiring heavier n-type doping. In consideration of doping limit in ZnO NWs, we should weigh
J.L. gratefully acknowledges financial support from the “OneHundred Talent Plan” of the Chinese Academy of Sciences and National Science Fund for Distinguished Young Scholar (Grant No. 60925016). This work is supported by the National High Technology Research and Development program of China under Contract No. 2009AA034101. References [1] J.-H. Lee, J.Q. Wu, J.C. Grossman, Phys. Rev. Lett. 104 (2010) 016602. [2] J.P. Heremans, V. Jovovic, E.S. Toberer, A. Saramat, K. Kurosaki, A. Charoenphakdee, S. Yamanaka, G.J. Snyder, Science 321 (2008) 554. [3] E.A. Skrabek, D.S. Trimmer, CRC Handbook of Thermoelectrics, CRC, Boca Raton, 1995. [4] T.C. Harman, P.J. Taylor, M.P. Walsh, B.E. LaForge, Science 303 (2004) 818. [5] Y. Ma, Q. Hao, B. Poudel, Y.C. Lan, B. Yu, D.Z. Wang, G. Chen, Z.F. Ren, Nano Lett. 8 (2008) 2580. [6] M.S. Dresselhaus, G. Chen, M.Y. Tang, R.G. Yang, H. Lee, D.Z. Wang, Z.F. Ren, J.P. Fleurial, P. Gogna, Adv. Mater. 19 (2007) 1043. [7] G.D. Mahan, J.O. Sofo, Proc. Natl. Acad. Sci. USA 93 (1996) 7437. [8] A.I. Hochbaum, R.K. Chen, R.D. Delgado, W.J. Liang, E.C. Garnett, M. Najarian, A. Majumdar, P.D. Yang, Nature 451 (2008) 163. [9] N. Yang, G. Zhang, B. Li, Nano Lett. 8 (2008) 276. [10] C.H. Yu, L. Shi, Z. Yao, D.Y. Li, A. Majumdar, Nano Lett. 5 (2005) 1842. [11] C.H. Chen, S.J. Chang, S.P. Chang, M.J. Li, I.C. Chen, T.J. Hsueh, C.L. Hsu, Appl. Phys. Lett. 95 (2009) 223101. [12] J. Goldberger, D.J. Sirbuly, M. Law, P. Yang, J. Phys. Chem. B 109 (2005) 9. [13] G. Kresse, J. Hafner, Phys. Rev. B 47 (1993) R558; G. Kresse, J. Hafner, Phys. Rev. B 48 (1993) 3115. [14] P.E. Blöchl, Phys. Rev. B 50 (1994) 17953; G. Kresse, D. Joubert, Phys. Rev. B 59 (1999) 1758. [15] T. Makino, Y. Segawa, A. Tsukazaki, A. Ohtomo, M. Kawasaki, Phys. Stat. Sol. C 3 (2006) 956. [16] L.H. Shi, D.L. Yao, G. Zhang, B.W. Li, Appl. Phys. Lett. 95 (2009) 063102. [17] T.T.M. Vo, A.J. Williamson, V. Lordi, G. Galli, Nano Lett. 8 (2008) 1111. [18] H.J. Xiang, J.L. Yang, J.G. Hou, Q.S. Zhu, Appl. Phys. Lett. 89 (2006) 223111. [19] H. Xu, A.L. Rosa, T. Frauenheim, R.Q. Zhang, S.T. Lee, Appl. Phys. Lett. 91 (2007) 031914. [20] W.J. Fan, J.B. Xia, P.A. Agus, S.T. Tan, S.F. Yu, X.W. Sun, J. Appl. Phys. 99 (2006) 013702. [21] M. Wang, Z.Y. Guo, Phys. Lett. A 374 (2010) 4312. [22] Z. Wang, N. Mingo, Appl. Phys. Lett. 97 (2010) 101903. [23] D.R. Khanal, J.W.L. Yim, W. Walukiewicz, J. Wu, Nano Lett. 7 (2007) 1186. [24] T. Tsubota, M. Ohtaki, K. Eguchi, H. Arai, J. Mater. Chem. 7 (1997) 85.
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Thermoelectric properties of ZnO nanowires: A first principle research Chaoren Liu ∗ , Jingbo Li State Key Laboratory for Superlattices and Microstructures, Institute of Semiconductors, Chinese Academy of Sciences, P.O. Box 912, Beijing 100083, China
a r t i c l e
i n f o
Article history: Received 5 April 2011 Received in revised form 11 June 2011 Accepted 12 June 2011 Available online 14 June 2011 Communicated by R. Wu Keywords: Thermoelectric property ZnO nanowire First principle calculation Boltzmann transport equation
a b s t r a c t By means of ab-initio electronic structure calculation and one-dimensional Boltzmann transport equation solution, we investigate the size dependent thermoelectric (TE) properties of n-type ZnO nanowires (NWs) and surface passivation effects. As demonstrated by our calculations, largest figure of merit ZT achievable in thin NWs is larger than that in wide NWs, whereas being restrained by higher demand of n-type doping. Moreover, bare NWs are superior in TE application comparing with the passivated. To compete with conventional TE materials, lattice thermal conductivity of ZnO NWs should be at least 2 orders of magnitude lower than bulk value. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Thermoelectric (TE) materials are regarded as promising candidates in solving energy crisis. However, the complicated interaction between electronic conductivity σ , Seebeck coefficient S and thermal conductivity κ (sum of electron contribution κe and lattice contribution κl ) makes it difficult to enhance the figure of merit ZT = σ S 2 T /(κe + κl ), which has remained around 1.0 for many years. ZT should be increased to no less than 3.0 if we want to put TE materials into application. Lee et al. [1] and Heremans et al. [2] raised ZT by means of distorting density of states (DOS) nearby Fermi level through doping special atoms. Others realized high ZT in complex alloys [3], taking advantage of their low thermal conductivity. Recently, quantum confinement was proposed to be another effective approach [4,5]. On one hand, it is well known that low-dimensional materials have a DOS of sharp shape at band edge [6], which is favorable to high TE efficiency [7]. On the other hand, the large surface to volume ratio in low-dimensional systems intensifies scattering of phonon, decreasing thermal conductivity as a result. It was reported, for example, that the thermal conductivity of silicon NWs was 100 times smaller than bulk value [8,9]. In laboratory, the development of advanced material preparation methods makes it possible to grow NWs with different diameters and surface geometries in different directions, which facilitates the experimental investigation of TE properties in nanomaterials [8,10]. ZnO NWs have received great attention for many years, mostly in their electronic and optical properties [11,12]. To the best of my
*
Corresponding author. Tel.: +86 0 15811043227; fax: +86 10 82305056. E-mail addresses: [email protected] (C. Liu), [email protected] (J. Li).
0375-9601/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2011.06.024
knowledge, there are no reports related to TE properties in ZnO NWs both experimentally and theoretically. In this Letter, we utilize a combination of density functional theory (DFT) and Boltzmann transport equation simulation (BTES) to give a description of TE properties in both bare and passivated ¯ ) facet surfaces ZnO NWs, which are [0001] oriented with six (1010 because this is commonest in experiments. Due to the fact that asprepared ZnO is n-type and p-type doping is difficult in ZnO, we only study the former kind in this Letter. In the first place, we discuss their structural and electronic properties and find that, before passivation, surface Zn atoms relax inward relative to O atoms manifestly, in consequence prolonging NWs in growth direction while, after passivation, the relaxation becomes negligible, which demonstrates our successful dangling bonds saturation. The investigation of TE properties tells that ZT can be tuned up through carrier doping and bare NWs are superior to passivated ones in TE application. Furthermore, ZT can be improved by decreasing NWs dimension, along with an increasing requirement of free electron doping. 2. Calculation method Two bare NWs, with hexagonal cross sections and different diameters (3.69 Å and 9.78 Å), are constructed from LDA relaxed wurtzite bulk ZnO (a = 3.24 Å, c = 5.23 Å, u = 0.387) and labeled as B1 and B2. Then we saturate the dangling bonds on B1 and B2 with pseudo-hydrogen atoms, which have equal amount of nucleus and electron charge, and label the counterparts as P1 and P2 respectively. Each dangling bond on surface Zn atoms is terminated with one H1.5 atom, whose charge amount is 1.5e (e = 1.6 × 10−19 C), while each dangling bond on surface O atoms
C. Liu, J. Li / Physics Letters A 375 (2011) 2878–2881
2879
Table 1 LDA bandgap and effective mass ([0001] direction) of B1, B2, P1 and P2.
E g (eV) m∗ (me ) a
B1
B2
P1
P2
Bulk
1.99 0.350
1.47 0.295
4.05 0.855
2.16 0.395
0.79 0.181a
See Ref. [20].
tronic structures, carrier concentrations, temperatures and so on, which makes it too complex to be obtained precisely. For lack of experimental dependence of electronic mobility on carrier concentration in ZnO NWs, we obtain the relaxation time in NWs by fitting the calculated electronic mobility in bulk ZnO to experimental values [15]. This approximation has been testified reasonable by Shi et al. [16] and Vo et al. [17] if we just care about the qualitative tendency rather than the accuracy of results. 3. Structural and electronic properties
Fig. 1. (Color online.) Views of as-constructed B1, B2, P1 and P2 in [0001] direction. Zn, O and pseudo-hydrogen atoms are represented as blue, red and gray balls.
with one H0.5, whose charge amount is 0.5e. The bond lengths of Zn–H1.5 and O–H0.5 are assumed to be the same as in ZnH2 and H2 O. We put the views of all NWs in Fig. 1 in [0001] direction. The initial structures are then relaxed with quasi-Newton algorithm until Hellmann–Feynman force acting on each atom is smaller than 0.01 eV/Å. We allow the lattice constant c to change freely since we find that ZnO [0001] NWs tend to stretch after relaxation. The periodic boundary condition is applied and NWs are separated by 10 Å vacuum layers, which proves sufficient to eliminate the interaction between different images. The atomistic relaxation and electronic structure calculation are performed with density functional theory (DFT) as implemented in VASP code [13]. Local density approximation (LDA) is utilized to treat exchange-correlation energy. Frozen-core projected augmented wave methods [14] is utilized to describe the ion–electron interactions. Zn3d electrons are explicitly treated as valence electrons. Monkhorst–Pack special k-point grid (1 × 1 × 8) is used to integrate in Brillouin zone. The electronic conductivity σ , Seebeck coefficient S and electronic thermal conductivity κe are obtained by means of processing electronic structures with the solution of one-dimensional Boltzmann transport equation in constant relaxation time approximation [7]:
+∞ ∂ f0 (ε ), σ =e dε − ∂ε 2
(1)
−∞
+∞ dε −
Tσ S = e −∞
+∞ dε −
T κ0 = −∞
(ε ) =
∂ f0 (ε )(ε − μ), ∂ε
∂ f0 (ε )(ε − μ)2 , ∂ε
νx (k)2 τ (k)δ ε − ε(k) ,
(2)
(3)
(4)
k
κe = κ0 − T σ S 2 ,
(5)
where e is charge of electron; μ is chemical potential; T is absolute temperature; f 0 is Fermi distribution function; νx (k) = (1/¯h)(∂ E (k)/∂ kx ); τ is relaxation time. τ varies with different elec-
We start by investigating the structural properties of bare and passivated ZnO NWs. Since the relaxation of NWs with different diameters follows similar trend, we take B2 and P2 as the example. Surface Zn atoms move inward relative to outmost O atoms, resulting in a distorted hexagonal cross section with three sides short (4.65 Å) and three sides long (5.93 Å). To release stress, NWs stretch in growth direction, with lattice constant c of B2, for example, increasing from bulk value 5.23 Å to 5.41 Å. The angle of surface O–Zn–O, in view of [0001] direction, increases to 140◦ from 120◦ while that of Zn–O–Zn decreases to 114◦ . A similar result has been reported by Xiang et al. [18] and Xu et al. [19]. After pseudo-hydrogen passivation, we find that the relaxation of surface atoms becomes unnoticeable, which demonstrates our success of surface state elimination. The bond length of Zn–H1.5 and O– H0.5 are relaxed to 1.64 Å and 1.05 Å, close to the values in ZnH2 (1.62 Å) and H2 O (0.94 Å). All ZnO NWs studied in this Letter prove to be semiconducting with direct bandgap at Γ point, which are shown in Table 1. We can see that bandgap increases when NW diameter decreases and all of them are larger than the bulk value. Passivation process enlarges the bandgap further. To understand these phenomena, we analyze the composition of bands and draw the charge density of valence band maximum (VBM) and conduction band minimum (CBM) in Fig. 2. VBM of bare NWs are contributed by Zn3d and O2p orbits, however of surface atoms, which indicates surface states, while CBM mainly consist of Zn4s orbits from inside atoms, which indicates bulk states. All of these are verified further in Fig. 2 (b1) and (c1). As indicated by Xiang [18], the O2plike surface state in B2 is only 80 meV above the VBM of bulk ZnO. In this case, when diminishing the diameter of bare NWs, quantum confinement effect will uplift CBM significantly because bulk state delocalizes greatly in radial direction. However, surface state responses to dimension constraint inertly due to its localized attribute. Hence the delocalized property of CBM as well as the low position of surface state result in a sensitive size-dependent bandgap of bare NW and a sharp increase of bandgap from bulk to NWs. In P1 and P2, Zn3d and O2p orbits in core part are the main component of VBM while CBM is primarily contributed by Zn4s and O3s orbits of inside atoms. From Fig. 2 (b2) and (c2), it is shown that the charge of both VBM and CBM locates mainly in inner part, which demonstrates their bulk state attributes. That is to say dangling bonds termination with pseudo-hydrogen successfully eliminates the surface states nearby VBM, meanwhile enlarging bandgap. Moreover, it is easy to notice that difference between the bandgap of B2 and B1 is 0.52 eV but the value for P2 and P1 is as large as 1.89 eV, which could be understood as follows. CBM might become more delocalized in radial direction after surface passiva-
2880
C. Liu, J. Li / Physics Letters A 375 (2011) 2878–2881
with free electrons, with carrier concentration obtained by dividing
Ef
Fig. 2. (Color online.) Band structures of B2 (a1) and P2 (a2) in [0001] direction; (b) and (c) are charge distribution of CBM and VBM with (b1), (c1) for B2 and (b2), (c2) for P2. Fermi levels are set at 0 eV for convenience.
tion and thus quantum confinement effect grows more significant in passivated NWs. 4. Thermoelectric properties The electronic conductivity σ , Seebeck coefficient S, electronic thermal conductivity κe and power factor P , having direct influence on figure of merit ZT, are functions of carrier concentration which are shown in Fig. 3. The systems are artificially doped
Fig. 3. (Color online.) Calculated dependence of electronic conductivity concentration.
N = CBM dε N (ε ) f (ε ), where N (ε ) is density of states and f (ε ) is Fermi distribution function, by the volume of NWs in superlattice. Both σ and κe increase with carrier concentration because there are more carriers to transport electricity and heat, but S decreases when carrier density increases. Large cross section is favorable to σ and κe , which is also true in silicon NWs [16], but passivation process lowers them significantly. Because σ as well as κe are in inverse proportion to effective mass m∗ , we calculate the effective mass m∗ along c direction of all NWs and cite the experimental value of bulk ZnO in Table 1. By comparison, we can see that both quantum confinement and surface passivation enlarge m∗ , which increases from 0.181, 0.295 to 0.350 me for bulk, B2 and B1, from 0.295 to 0.395 me for B2, P2 and from 0.350 to 0.855 me for B1, P1. S is larger in smaller NWs and is enhanced further by passivation. This size dependence results from DOS narrowing phenomenon in quantum confinement. Power factor ( P = σ × S 2 ), important in TE application, is also shown in Fig. 3. It is presented that P reaches to a maximum at a specific carrier concentration for larger NWs but for smaller ones there are no maxima in the concentration range we are interested in. We can also find that P of larger NWs is larger in low carrier concentration area but, when carrier concentration increases to a certain point, smaller NWs hold a larger P instead. Having discussed σ , S and κe , then we move on to another important factor κl which is directly related to NWs dimension [21, 22]. Until now, there are no systematic researches on lattice thermal conductivity of ZnO NWs with different diameters and growth directions both theoretically and experimentally. In our calculation, we substitute the simulated κl of silicon NWs [22], which have the same sizes as ZnO NWs, for the values of ZnO NWs. ZT of B1, P1, B2 and P2 are plotted in Fig. 4, with κl to be 0.12 W m−1 K−1 for B1 and P1 and 0.30 W m−1 K−1 for B2 and P2. We can find that from B1 to P1 and from B2 to P2, ZT is repressed after surface passivation, mostly resulting from the drop of σ . It is obvious that ZT increases with carrier concentration at the first stage due to increase of σ and begins to drop at a specific carrier concentration due to decrease of S and increase of κe . A closer look will also find that the carrier concentration, where ZT reaches maximum, becomes larger from bare to passivated NWs and when NW size decreases. It has been claimed that quantum confinement lowers
σ , Seebeck coefficient S, thermal conductivity of electron κe and power factor P on free electron
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high ZT against doping difficulty while determining NW diameter. As indicated, κl need be reduced by approximately 2 orders of magnitude comparing to the bulk value in order to compete with conventional TE materials. Although most experimental ZnO NWs are much wider (> 10 nm) than ones investigated here, our calculation supplies a qualitative guide when optimizing the size, carrier doping and surface treatment of ZnO NW TE materials. Acknowledgements
Fig. 4. (Color online.) Dependence of ZT on free electron concentration calculated with κl = 0.12 W m−1 K−1 for B1 and P1, 0.30 W m−1 K−1 for B2 and P2.
valence band and uplifts conduction band, hence increasing the ionization energy and making effective doping more difficult [23]. Khanal [23] also reported that 9 × 1019 cm−3 was the limit of free electron concentration in ZnO NWs with a diameter under 60 nm. That is to say although achievable ZT of smaller NWs (B1 and P1) is higher than that of larger ones (B2 and P2), the higher carrier concentration requirement grows into a new challenge. Especially, ZT of P1 is higher than that of P2 only when carrier concentration is higher than 8.53 × 1019 cm−3 . Thereby, we should balance the dimension of NWs to a satisfactory value from the view of high ZT as well as low carrier concentration requirement. Maximal κl , making highest ZT obtainable larger than 1.0, for B2 and P2 are also calculated (0.316 and 0.157 W m−1 K−1 ), approximately 2 orders of magnitude lower than the bulk value which is 23 W m−1 K−1 [24]. 5. Conclusion In summary, the structural and electronic properties of both bare and passivated ZnO NWs are investigated. Surface Zn atoms of bare NWs relax inward relative to O atoms, as a result elongating the NWs in growth direction, and after surface passivation the relaxation becomes negligible. CBM is a delocalized state in both bare and passivated NWs while VBM of bare NWs is a surface state, successfully removed after pseudo-hydrogen saturation. Then, TE properties are discussed. It is found that σ and κe increase with cross section area but S decreases when the dimension of NWs increases. Furthermore, surface passivation reduces σ and κe while improves S. ZT increases with carrier concentration at the first stage and later begins to drop down. The carrier concentration, in which ZT reaches the peak, is low in large bare NWs. In TE application, bare NW is favored and the highest ZT obtainable is larger in smaller NWs, however requiring heavier n-type doping. In consideration of doping limit in ZnO NWs, we should weigh
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