Journal of ELECTRONIC MATERIALS, Vol. 42, No. 7, 2013

DOI: 10.1007/s11664-012-2279-z  2012 TMS

Lorenz Function of Bi2Te3/Sb2Te3 Superlattices N.F. HINSCHE,1,4 I. MERTIG,1,2 and P. ZAHN3 1.—Institut fu¨r Physik, Martin-Luther-Universita¨t Halle-Wittenberg, 06099 Halle, Germany. 2.—Max-Planck-Institut fu¨r Mikrostrukturphysik, Weinberg 2, 06120 Halle, Germany. 3.—HelmholtzZentrum Dresden-Rossendorf, P.O. Box 51 01 19, 01314 Dresden, Germany. 4.—e-mail: nicki.hinsche@ physik.uni-halle.de

Combining first-principles density functional theory and semiclassical Boltzmann transport, the anisotropic Lorenz function was studied for thermoelectric Bi2Te3/Sb2Te3 superlattices and their bulk constituents. It was found that, already for the bulk materials Bi2Te3 and Sb2Te3, the Lorenz function is not a clear function of charge carrier concentration and temperature. For electrondoped Bi2Te3/Sb2Te3 superlattices, large oscillatory deviations of the Lorenz function from the metallic limit were found even at high charge carrier concentrations. The latter can be referred to quantum well effects, which occur at distinct superlattice periods. Key words: Thermoelectric transport, heterostructures, DFT, electronic structure

INTRODUCTION For many decades, thermoelectric (TE) energy conversion has successfully enabled self-supporting energy devices for outer-space missions or integrated electronics.1,2 However, poor conversion efficiency prohibited the breakthrough use of thermoelectrics as an alternative energy source. The conversion performance of a TE material is quantified by the figure of merit (FOM) ZT ¼

rS2 S2 T¼ jph ; jel þ jph L þ rT

(1)

where r is the electrical conductivity, S is the thermopower, and jel = LrT and jph are the electronic and lattice contributions to the thermal conductivity, respectively. L denotes the Lorenz function, 2 which becomes the Lorenz number BÞ in the highly degenerate, metallic limit. L0 ¼ ðpk 3e2 In recent years, nanostructuring concepts3,4 have enabled higher values for ZT by increasing the numerator, called the power factor (PF = rS2), or decreasing the denominator of Eq. 1. The latter is obtained by phonon blocking at superlattice (SL) interfaces or grain boundaries5–8 and leads to a (Received June 21, 2012; accepted September 20, 2012; published online October 19, 2012)

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reduced lattice thermal conductivity jph. Here, the Lorenz function is particularly important for thermoelectrics, providing a measure to separate the electronic and lattice contribution to the thermal conductivity.9 Deviations L = L0 already occur in the degenerate limit for simple metals, semimetals, and semiconductors.10 Hence, assuming incorrect values for the Lorenz number leads to incorrect values for jel and jph and can even result in nonphysical negative values for jph.11 To the best of our knowledge, investigations on the Lorenz function of thermoelectric SLs at ab initio level are lacking so far. In the present work we analyze the anisotropic Lorenz function for Bi2Te3/Sb2Te3 SLs, as well as for the bulk constituents. The two telluride single crystals and the composed p-type SL show the highest values for bulk and nanostructured TE so far.12 On the basis of ab initio density functional theory (DFT) and semiclassical Boltzmann transport equations (BTE), the Lorenz function is in particular studied for different charge carrier concentrations and SL periods at room temperature. METHODOLOGY For both Bi2Te3 and Sb2Te3, as well as for the composed SLs, we used the experimental lattice parameters and relaxed atomic positions13 as

Lorenz Function of Bi2Te3/Sb2Te3 Superlattices

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provided for the hexagonal Bi2Te3 crystal structure. The 15 atomic layers per unit cell are composed of three quintuples Te1–Bi–Te2–Bi–Te1. The hexagonal lattice parameters are equally chosen to be hex ˚ ˚ ahex BiTe = 4.384 A and cBiTe = 30.487 A for Bi2Te3, Sb2Te3, and the SLs, respectively. Preceding studies revealed that a larger in-plane lattice constant, e.g., hex ahex BiTe > aSbTe, is favorable for enhanced cross-plane TE transport.14,15 To introduce SLs with different SL periods, we subsequently substitute Sb at Bi sites, starting with six Bi sites in hexagonal bulk Bi2Te3. Substituting four atomic layers of Bi with Sb leads to a (Bi2Te3)x/(Sb2Te3)1-x SL with x ¼ 26, that is, one quintuple Bi2Te3 and two quintuple Sb2Te3. The ˚ /20 A ˚ )-(Bi2Te3/ latter case coincides with a (10 A Sb2Te3) SL in the experimental notation of Ref. 12. Semiclassical BTE have been used extensively in the past to calculate TE transport properties16,17 and offer high reliability for narrow-gap semiconductors over broad doping and temperature ranges.14,18–20 Within the relaxation time approximation (RTA), the ð0Þ transport distribution function (TDF) L?;k ðl; 0Þ,21 and with this the generalized conductance moments ðnÞ L?;k ðl; TÞ, are defined as Z  2 s X ðnÞ 3 m d k v L?;k ðl; TÞ ¼ k;ð?;kÞ ð2pÞ3 m   (2) @fðl;TÞ ðEmk  lÞn  : @E E¼Em k

f(l,T) is the Fermi–Dirac distribution, and vmk;ðkÞ ; vmk;ð?Þ? denote the group velocities in the directions in the hexagonal basal plane and perpendicular to it, respectively. Herein, the group velocities were obtained as derivatives along the lines of the Blo¨chl mesh in the whole Brillouin zone (BZ).15 The band structure Emk of band m was obtained by accurate firstprinciples density functional theory (DFT) calculations, as implemented in the fully relativistic screened Korringa–Kohn–Rostoker Greens-function method (KKR).22 Within this approach the Dirac equation is solved self-consistently, thereby including spin–orbit coupling (SOC). Exchange and correlation effects were accounted for by the local density approximation (LDA) as parametrized by Vosko et al.23 Detailed studies on the electronic structure, the thermoelectric transport, and challenges in the numerical determination of the group velocities of Bi2Te3, Sb2Te3, and their SLs have been published before.14,15,24,25 For convenience, the relaxation time s was chosen as 10 fs for the considered systems. Straightforwardly, the temperature- and dopingdependent electrical conductivity r and thermopower S in the in- and cross-plane directions are defined as ð1Þ

r?;k ¼ e

2

ð0Þ L?;k ðl; TÞ

S?;k

1 L?;k ðl; TÞ ; ¼ eT Lð0Þ ðl; TÞ ?;k

(3)

and the electronic contribution to the total thermal conductivity amounts to 0  2 1 ð1Þ L ðl; TÞ ?;k 1 B ð2Þ C (4) jel?;k ¼ @L?;k ðl; TÞ  A: ð0Þ T L?;k ðl; TÞ The second term in Eq. 4 introduces corrections due to the Peltier heat flow that can occur when bipolar conduction takes place.26,27 Using Eqs. 3 and 4 and ð2Þ the abbreviation j0 ¼ T1 L?;k ðl; TÞ;21 we find the Lorenz function as L?;k ¼

j0 r?;k T

 S2?;k :

(5)

Equation 5 clearly shows that in the low-temperature regime L consists of a constant term and a negative term of order T2. RESULTS To introduce our discussions, in Fig. 1 the Lorenz function L and the corresponding electronic thermal conductivity jel in dependence on the chemical potential l are shown for a spherical two-band model (SBM). Varying temperatures and mCB = mVB CB (cf. Fig. 1a) and different effective mass ratios m mVB and fixed temperature T = 300 K (cf. Fig. 1b) are assumed. mCB and mVB are the isotropic effective masses of the conduction band (CB) and valence band (VB), respectively. Setting the valence-band maximum to zero and denoting the bandgap size by pffiffiffiffiffiffiffiffiffiffi ð0Þ Eg, the TDF scales as LVB ðl; 0Þ / mVB ðlÞ3=2 and pffiffiffiffiffiffiffiffiffiffi ð0Þ LVB ðl; 0Þ / mCB ðl  Eg Þ3=2 for the VB and CB, respectively. From Eqs. 4 and 5 it is obvious that, within a SBM, deviations for L and jel from the metallic limit will only occur near the bandgap, where the thermopower S changes significantly. Near the band edges, S increases approximately as S / 1 lT . Thus, L, as well as jel, reach a minimum, and that minimum decreases with decreasing temperature, while shifting towards the middle of the gap (cf. Fig. 1a). At T ¼ 100 K LL0  0:8 at the band edges. In the intrinsic regime LL0 and jel increase, as the thermopower and electrical conductivity are reduced due to bipolar contributions. Figuratively speaking, the additional contribution arises from the fact that electrons and holes can move together in the same direction, transporting energy but not carrying any net charge.26 According to Goldsmid28 and Price,29 the deviation of the Lorenz number from the metallic limit in the intrinsic regime holds to some extent  2 Eg L 1 mCB mVB ¼ 1 þ 2 L0 2 ðm þm Þ kB Tþ4 . Therefore, assuming a CB

VB

fixed charge carrier concentration, LL0 achieves very large values at small temperatures and/or large bandgaps. Adopting the above approaches,28,29 together with mCB = mVB and Eg = 0.1 eV, one

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(a)

slight directional anisotropy. For Sb2Te3 (cf. Fig. 2a) the Lorenz function exhibits only minor anisotropies Lk in a wide doping range, while showing L? L?  1:15Lk at increased electron doping. Reduction L

(b)

Fig. 1. Lorenz function L (thick black lines, left scale) and electronic contribution jel to the total thermal conductivity (thin green lines, right scale) in dependence on position of the chemical potential l within a spherical two-band model. Results are shown for (a) fixed effective masses mVB = mCB and varying temperatures and (b) fixed temperature T = 300 K and varying effective masses. The bandgap is fixed at Eg = 0.1 eV (gray shaded areas), and the Lorenz function is related to the metallic limit L0 ¼ 2:44  108 WX=K2 .

achieves LL0  9 at room temperature for l located deep in the gap. If mVB > mCB, as shown in Fig. 1b, the intrinsic regime Nn = Np and with that the maximal values of LL0 and jel at bipolar conduction shift towards the conduction band minimum (CBM). With increasing mVB, and hence enhanced electrical conductivity r in the VB, it is obvious that jel under hole doping will increase, too. Figure 2 presents first-principles calculations for the Lorenz function L and the related electronic part jel of the thermal conductivity. The dependence on the charge carrier concentration for (a) Sb2Te3, (b) a (Bi2Te3)x/(Sb2Te3)1-x SL at x ¼ 26, and (c) Bi2Te3, is shown, respectively. Due to the high conductivr ity anisotropy r?k > 1 for all of the considered sys14,25 , jel,^ is strongly suppressed compared tems, with jel;k , too. Furthermore, it is obvious that the maximal peak of the Lorenz function is shifted towards the CBM, the latter stemming from a larger density of states at the valence band maximum (VBM) and a higher absolute hole electrical conductivity. Maximal numbers for the Lorenz function L L0 in the intrinsic regime were found to be between 6 and 10 for the considered systems, showing only

of L?k due to bipolar diffusion effects is more apparent at hole doping compared with electron doping, here showing in-plane LL0  0:75 and LL0  0:92 at electron and hole doping of N = 3 9 1019 cm3, respectively. For bulk Bi2Te3 the picture is comparable. However, in the thermoelectrically most interesting range, about 200 meV around the band L edges, L?k is always less than unity, comparable to in previous publications.20 Furthermore, L^ is larger than the metallic limit L0. Very often values of L^  0.5 to 0.630,12 are assumed for the experimental determination of jel,^ in Bi2Te3/Sb2Te3 SLs. In turn, this most probably leads to underestimation of the electrical contribution to the total thermal conductivity in the cross-plane direction. For bulk Bi2Te3, experimental values26 for the in-plane part jel;k are presented for reference in Fig. 2c (green, open circles). We find very good accordance to our calculations in the intrinsic range, while our results slightly overestimate jel;k in the extrinsic regime. Strong deviations of the Lorenz function from the bulk limit could be found for an electron-conducting ˚ /20 A ˚ )– (Bi2Te3)x/(Sb2Te3)1-x SL at x ¼ 26, i.e., (10 A (Bi2Te3/Sb2Te3). In another publication,25 we showed that strong quantum well effects (QWE) in the CB of the SLs lead to enhanced electrical conductivity anisotropy, the latter being pronounced for r the SL at x ¼ 26 showing r?k  20 at electron doping of 19 3 N = 3 9 10 cm . Due to the QWE, the crossplane electrical conductivity r^ is drastically suppressed, and hence L^ is remarkably enhanced. The cross-plane Lorenz function achieves rather large values of LL0  1:5  1:8 at extrinsic carrier concentrations of about N = 3 9 1019 cm3 to 30 9 1019 cm3 for hole and electron doping, respectively. Additionally, oscillations of LL0 with varying doping are found, being much more pronounced than in the bulk materials. Both effects have been proposed within a one-dimensional model for thermoelectric SLs before.31 As expected, in the extrinsic region, at increasing charge carrier concentration, L saturates gradually towards the metallic limit L0 and the thermal conductivity increases with the electrical conductivity. To support our findings, in Fig. 3 the cross-plane Lorenz function L^ at different SL periods is shown. The influence of varying (a) electron and (b) hole doping is shown. Under hole doping, due to the vanishing band offset at the VBM,25 the Lorenz function interpolates smoothly between the bulk limits (cf. Fig. 3b). At lower charge carrier concentrations LL?0 is more suppressed due to a stronger impact of bipolar diffusion. Under varying electron doping we find that LL?0 is remarkably enhanced for

Lorenz Function of Bi2Te3/Sb2Te3 Superlattices

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(a)

(a)

(b) (b) Fig. 3. Cross-plane component of the Lorenz function L^ for (Bi2Te3)x/(Sb2Te3)1-x superlattices in dependence on the superlattice period. The temperature is fixed to 300 K, and results for three different charge carrier concentrations (in units of cm3) are compared for (a) electron and (b) hole doping. The Lorenz function is related to the metallic limit L0 ¼ 2:44  108 WX=K2 . Lines are provided as a guide to the eye.

r

(c)

and r?k  14 for SL periods of x ¼ 26 and x ¼ 46 are reported, respectively.25 We state that, due to quantum confinement effects in the electronconducting SLs, unexpected deviations of the Lorenz function from L0 can occur also at higher values of doping, which is counterintuitive. The latter could lead to wrong estimations for the electronic part of the thermal conductivity jel and consequently for the lattice thermal conductivity jph. CONCLUSIONS

Fig. 2. Lorenz function L (solid lines, left scale) and electronic contribution jel to the total thermal conductivity (dashed lines, right scale) in dependence on position of the chemical potential l for (a) bulk Sb2Te3, (b) (Bi2Te3)x/(Sb2Te3)1-x SL at x ¼ 26, and (c) bulk Bi2Te3. The in-plane (thick lines) and cross-plane (thin lines) transport directions are compared. The Lorenz function is related to the metallic limit L0 ¼ 2:44  108 WX=K2 . Plotted on the graph of the Lorenz function in the in-plane direction is a color code indicating the charge carrier concentration. The red cross emphasizes the change from n to p doping. The temperature was fixed at 300 K. Thin vertical dash-dotted lines emphasize the position of the chemical potential for charge carrier concentration of N = 3 9 1019 cm3 under p and n doping (red and blue color). The grey shaded areas show the bandgap. Green open circles in (c) show experimental results from Ref. 26 for jel;k for an n-type Bi2Te3 single crystal.

SL periods of x ¼ 26 and x ¼ 46, respectively. For these SL periods, large suppressions of the cross-plane electrical conductivity r^ were found, too. At r N = 3 9 1019 cm3 anisotropies as large as r?k  20

We present first-principles calculations for the Lorenz function of electron- and hole-conducting Bi2Te3/Sb2Te3 superlattices and the related bulk materials at varying charge carrier concentrations. As expected, due to bipolar conduction, the Lorenz function increases to large values within the intrinsic doping regime. More significantly, the Lorenz function L of the superlattices does not change monotonically at extrinsic charge carrier concentrations. While at increased doping asymptotic convergence of L towards the metallic limit L0 is found, a distinct oscillatory behavior of L is also observed. This is most pronounced under electron doping, being caused by quantum well effects in the conduction bands of the superlattices. This counterintuitive effect has consequences for the determination of the thermal conductivity, as L is generally used to separate jel and jph. At thermoelectrically profitable charge carrier concentrations, the application of the metallic value L0 to determine the electronic thermal conductivity could lead to a deviation of a factor of two in either direction in the

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worst case. Consequently, this leads to wrong estimations of the lattice thermal contribution and the figure of merit. A similar behavior was found theoretically for p-type SiGe superlattices,32 and this behavior could be a general effect in thermoelectric superlattices influenced by quantum well effects. ACKNOWLEDGEMENTS This work was supported by the Deutsche Forschungsgemeinschaft, SPP 1386 ‘‘Nanostrukturierte Thermoelektrika: Theorie, Modellsysteme und kontrollierte Synthese.’’ N.F. Hinsche is member of the International Max Planck Research School for Science and Technology of Nanostructures. REFERENCES 1. B. Sales, Science 295, 1248 (2002). 2. T. Tritt and M. Subramanian, MRS Bull. 31, 188 (2006). 3. H. Bo¨ttner, G. Chen, and R. Venkatasubramanian, MRS Bull. 31, 211 (2006). 4. G. Nolas, D. Morelli, and T. Tritt, Ann. Rev. Mater. Sci. 29, 89 (1999). 5. T. Borca-Tasciuc, Superlattices and Microstruc. 28, 199 (2000). 6. S. Lee, D. Cahill, and R. Venkatasubramanian, Appl. Phys. Lett. 70, 2957 (1997). 7. G. Pernot, M. Stoffel, I. Savic, F. Pezzoli, P. Chen, G. Savelli, A. Jacquot, J. Schumann, U. Denker, and I. Mo¨nch, Nat. Mater. 9, 491 (2010). 8. R. Venkatasubramanian, Phys. Rev. B 61, 3091 (2000). 9. C. Uher and H. Goldsmid, Phys. Status Solid B 65, 765 (1974). 10. G.S. Kumar, G. Prasad, R.O. Pohl, J. Mater. Sci. 28, 4261 (1993). 11. J. Sharp, E. Volckmann, and H. Goldsmid, Phys. Status Solid A 185, 257 (2001). 12. R. Venkatasubramanian, E. Siivola, and T. Colpitts, Nature 413, 597 (2001).

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