Low Effective Mass Leading to High Thermoelectric Performance

Transcription

1 Low Effective Mass Leading to High Thermoelectric Performance 3 Yanzhong Pei, Aaron D. LaLonde, Heng Wang and G. Jeffrey Snyder Materials Science, California Institute of Technology, Pasadena, CA 9, USA. High Seeeck coefficient y creating large density of state (DOS) around the Fermi level through either electronic structure modification or manipulating nanostructures, is commonly considered as a route to advanced thermoelectrics. However, large density of state due to flat ands leads to large effective mass, which results in a simultaneous decrease of moility. In fact, the net effect of high effective mass is a lower thermoelectric figure of merit when the carriers are predominantly scattered y acoustic phonons according to the deformation potential theory of Bardeen-Shockley. We demonstrate the eneficial effect of light effective mass leading to high power factor in n-type thermoelectric PTe, where doping and temperature can e used to tune the effective mass. This clear demonstration of the deformation potential theory to thermoelectrics shows that the guiding principle for and structure engineering should e low effective mass along the transport direction. Bardeen-Shockley[8]. This is ecause µ m -3/ - m c (m c, conduction mass of the carriers) [8-] when the carriers are predominantly scattered y acoustic phonons, as has een found in most of known and good 6 thermoelectrics. As a result, the optimal power factor Increasing the thermoelectric figure of merit (zt) is the most challenging task to enale the widespread use of this method to directly convert heat into electricity. The transport properties including resistivity (ρ), Seeeck coefficient (S), electronic (κ E ) and lattice (κ L ) components of thermal conductivity (κ=κ E +κ L ) determine the figure of merit, zt=s T/ρκ, where T is asolute temperature. Creating phonon scattering centers such as nanostructures[-] to lower κ L, has een proven effective for achieving zt > in many instances. However, κ L in such materials already approaches their amorphous limit[3, ], suggesting strategies targeting increases in zt y improvements of the thermoelectric power factor (S /ρ). The decoupling of S, ρ and κ E in an effort to achieve high zt has een a longstanding challenge as they are strongly coupled with each other through the carrier concentration, scattering and and structure[-7]. However, it is well known that the optimal electronic performance of a thermoelectric semiconductor depends primarily on the weighted moility[7-], µm 3/, which includes oth the density-of-states effective mass (m, with a unit of free electron mass m e ) and the nondegenerate moility (µ) of carriers. More generally, each degenerate carrier pocket makes a contriution to m via m =N /3 v m [7-], where N v and m are the numer of degenerate carrier pockets and the average and mass (density-of-states effective mass for each pocket), respectively. Without explicitly reducing µ, converging many valence (or conduction) ands to achieve high N v and therefore a high m has een proposed as an effective approach to high performance in oth ulk[] and low dimensional[3] thermoelectrics. Without modifying N v and in an attempt to increase the power factor, many efforts have een recently devoted to increasing the Seeeck coefficient (i. e. increasing m through high m ) either y designing[, ] the density of states or manipulating nanostructures[6, 7]. This concept has recently een considered as a criteria[9, ] for otaining good thermoelectrics. However, these methods may reduce the moility significantly[]. In fact, an increase of m resulting from increasing m (i. e. y flattening the and), leads to a significant decrease in moility according to the deformation potential theory of S /ρ µm 3/ N v m - c ecomes inversely proportional to m c, the effective mass along the conducting direction[7, 8,, ]. Since cuic thermoelectric materials such as PTe, SiGe and skutterudites have an isotropic m c that increases with increasing m [3], it is then clear that increasing m actually decreases the optimal power factor in spite of the resulting large Seeeck coefficient. In this paper we demonstrate that a lower effective mass (either y doping or y adjusting the temperature) leads to a high power factor and thus excellent thermoelectric performance in n-pte. When compared with La-doped PTe, a ~% lower effective mass in I-doped PTe results in a ~% higher power factor. A single Kane and (SKB) model[,, ] has een developed to quantitatively understand that the lower effective mass is indeed eneficial for enhancing the thermoelectric performance. This work shows a contrasting example to the commonly utilized strategy for large Seeeck coefficient resulting from heavy and mass for high performance thermoelectrics[9,, 6, 7]. La- and I-doped PTe (La x P -x Te and PTe -x I x with <x<.) were synthesized y the same melting, quenching, annealing and hot pressing method. The synthesis procedure and details of the measurement of transport properties can e found elsewhere[6, 7]. In oth La and I doped PTe the donor states are very deep[8] so that each dopant atom produces one electron[7] in conduction and according to the rules of valence[9].the Hall carrier concentration (n H =, e is electron charge) is determined from the measured Hall coefficient (R H ), and the room temperature values of n H were used to lael the samples. All the samples in this study show n-type conduction. La- and I-doped PTe samples with two important Hall carrier concentrations of ~.8 and ~3 9 cm -3, which respectively enale the highest average zt and peak zt in the temperature range of most interest for thermoelectric applications, were chosen for the discussion of temperature dependent transport properties. To avoid the detrimental effects due

2 3 to minority carriers (Fig. ) and to validate the use of the single and conduction model as discussed elow, we focus on the transport properties from 3 to 6 K for this study. For temperatures or carrier concentration where the scattering is not dominated y acoustic (nonpolar) phonons or the transport properties are not sufficiently descried y a single and model the following conclusions may differ. The measured Seeeck coefficient, resistivity, thermal conductivity and zt are shown in Fig. as a function of temperature. The monotonically increasing Seeeck coefficient and resistivity, as well as the slightly (< %) increased Hall coefficient (which can e expected from a slight loss of degeneracy, not shown), with increasing temperature allows the assumption of single and conduction ehavior at T<~6 K to e made in this study. This assumption is consistent with and structure studies of PTe[]. as ( µv/k) - - c κ,( W/m-K) PTe:I,.9e PTe:I,.9e9 3 6 ρ ( mωcm) d zt.... PTe:I,.9e PTe:I,.9e Fig.. Temperature dependent Seeeck coefficient (a), 7 resisistivity (), thermal conductivity (c) and thermoelectric figure of merit (d) for two groups of La- and I-doped PTe having room temperature Hall carrier concentration of ~.8 and ~3 9 cm -3, respectively. The curves represent the predicted results from the single Kane and model with an effective mass of. m e for I-doping and.3 m e for La-doping, respectively. 7 Comparing with La-doped PTe that has nearly the same carrier concentration, a % lower effective mass in I-doped leads to ~% higher figure of merit over the whole temperature considered. It has een well known that the ands located at the L point (N v =) of the Brillouin zone for PTe are nonparaolic[,, 3, 3] and can e well descried y a Kane and model. Furthermore, the scattering of charge carriers in PTe is known to e dominated y acoustic phonons[, ] in the temperature and carrier concentration range having high thermoelectric performance, as is the case for most good thermoelectric material. This is demonstrated in Fig a 8 showing the hall moility decreases sharply with temperature (µ~t p where p < -.[]). Other scattering mechanisms such as y grain oundaries, polar-optical 6 phonons, ionized impurities, predict p -½ implying that these mechanisms do not dominate the transport properties. In fact, the Hall moility predicted y single Kane and model (SKB), with acoustic scattering theory[8, 9] and temperature dependent m (Fig ), agrees well with the experimental data (Fig. a), for oth La- and I-doped PTe. Such a Kane and model provides the expressions for the transport coefficients[, ] as follows: Hall carrier density 3 / N v (mk BT) 3 / nh = = A F 3 erh 3π (); Hall factor / 3 / 3K( K + ) F F A = (K + ) ( F ) (); Hall moility π ecl 3 F µ H = A 3/ 3/ mc (mk BT ) Edef F (3); Seeeck coefficient kb F S = [ ξ] e F (); and Lorenz numer kb F F L = ( ) [ ( ) ] e F F (); where n F m k has a similar form as the Fermi integral n m f n m k / Fk = ( ) ε ( ε + αε ) [( + αε ) + ] dε (6). ε In the aove equations, k B is the Boltzmann constant, ħ the Boltzmann constant, C l the comined elastic moduli[9], E def the deformation potential coefficient[9] characterizing the strength of carriers scattered y acoustic phonons, ξ the reduced Fermi level, ε the reduced energy of the electron state, α (=k B T/E g ) the reciprocal reduced and separation (E g, at L point of the Brillouin zone in this study) and f the Fermi distriution. This model also considers an ellipsoidal Fermi surface y m taking the ratio of the longitudinal ( ) to transverse ( m ) effective mass components into account via the term K = m / m!. a ( cm /V-s) 3 PTe:I,.9e9 3 6 m( m e ) ln(.3)+.ln( T/3) m=e PTe:I,.9e9 ln(.)+.ln( T/3) m=e 3 6 Fig.. Temperature dependent Hall moility (a) and effective mass () for La- and I-doped PTe. The experimental Hall moility (symols) can e well predicted (curves) y an acoustic scattering mechanism. La-doping leads to a ~% higher effective mass over the entire temperature range. The increase in effective mass with

3 3 increasing temperature is due to the Kane type and structure and is associated with the temperature dependent and gap. Utilizing the aove SKB model, excellent prediction of the Hall carrier concentration dependent Seeeck coefficient and Hall moility can e otained for oth La- and I-doped PTe over a road carrier concentration range as shown in Fig. 3. Literature data from different sources[7, 3-38] show good consistency with the current work. It is seen that the La-doped series shows slightly higher Seeeck coefficient values at oth 3 K (Fig. 3a) and 6 K (Fig. 3), which correspondingly means a higher density-of-states effective mass y % than that in the I-doped samples. Quantitatively, m is found to e.±.3 m e and.3±. m e at 3 K, and.±. m e and.±. m e at 6 K, for I- and La-doped PTe, respectively, where the standard deviations are otained on approximately different samples. Most importantly, only varying m y %, enales an accurate prediction (curves in Fig. 3c and 3d ) of the Hall moility at oth 3 and 6 6 K using the SKB model without any other adjustale parameters. Here, the values K= 3.6[39], C l =7. Pa[], E def = ev[7] and α=k B T/(.8 ev +. ev/k T) [, -] are used for oth I- and La-doped series. The excellent agreement etween the experimental and predicted results confirms the validity of the model itself and additionally indicates that the higher m is indeed responsile for the oserved higher S and lower. a S ( µv/k) c ( cm /V-s) 3 3 K Alekseeva, PTe:La 9 Alekseeva, PTe:La 3 K 8 9 S ( µv/k) d 3 ( cm /V-s) 9 The higher m in La-doped PTe is presumaly due to the conduction and flattening, related to an increase in and 3 6 K 9 6 K Fig. 3. Hall carrier concentration versus Seeeck coefficient (a and ) and Hall moility (c and d) for I- and La-doped PTe at 3 K (a and c) and 6 K ( and d), compared with the predicted results (curves) according to the single Kane and model. Providing a % higher effective mass in La-doped series, oth the increase in Seeeck coefficient and decrease in Hall moility can e well predicted y the SKB model gap[3] according to the Kane dispersion E(k) [, 3]: k E = + (7). E m Eg In a Kane and system, the increase of m with increasing and gap has een theoretically predicted [, ] and experimentally confirmed [3, 39, -7] in lead chalcogenides. Furthermore, the increase of m in PTe and related materials can e induced y either temperature[, 39, ] or chemical sustitution[], making availale an additional tunale parameter for further investigation of m dependent thermoelectric properties. With the knowledge of and separation at L point of the Brillouin zone, one can calculate the reduced Fermi level from the experimental Seeeck coefficient according to Eq.. Consequently, m can e otained from Eqs. -. In this way, we calculate the temperature dependent m (Fig. ) for oth La- and I-doped PTe having room temperature Hall carrier concentrations of ~.8 and ~3 9 cm -3. It is clearly seen that the % higher m in the La-doped series persists throughout the entire temperature range. The m, otained in the manner is related to the density-of-states at the and edge. Largely resulting from the lattice expansion[], the and gap increases with increasing temperature leading to an increase in m as theoretically predicted [, ] and experimentally oserved [3, 39, -7] in Kane and systems (Eq. 7). Therefore, the oserved increase in m with increasing temperature y dlnm/dlnt =. (curves), can e well understood y the SKB model and consistent with literature reports[, 39, ]. PF ( µw/cm-k ) 3 3 K 6 K,,,, 8 9 n H Fig.. Thermoelectric power factor versus Hall carrier concentration for La- and I-doped PTe at the two end temperatures of 3 and 6 K. The % difference in effective mass due to variant dopant leads to the peak power factor differing y ~% at oth temperatures. The ~% increase in effective mass (Fig. ), originating from the temperature increase from 3 to 6 K, results in a ~% decrease in peak power factor in oth La- and I-doped samples. With a comination of the predicted Hall carrier concentration dependent Seeeck and Hall moility, the thermoelectric power factor (PF= S n H e ) is calculated and compared with the experimental data for the La- and

4 3 I-doped series at 3 and 6 K in Fig.. It is now clear that ~% higher m leads to ~% lower maximal PF in La-doped series at oth 3 K (3 vs. 8 µw/cm-k ) and 6 K ( vs. µw/cm-k ). Moreover, the temperature induced ~% increase in m correspondingly results in a ~% decrease in maximal PF for oth La- (8 vs. µw/cm-k ) and I-doped (3 vs. µw/cm-k ) PTe, as temperature rises from 3 to 6 K. An increase of m due to independent mechanisms leads to a reduction of the overall optimal thermoelectric power factor, despite the resulting increase of the Seeeck coefficient. The physics ehind why a higher m without increasing N v has a detrimental effect to thermoelectric performance, is similar for the SKB model as it is for a paraolic and model. Comining Eqs., 3 and, one otains: 7 N k C F PF = v B l ξ F (8). πedef mc F Because the first term only includes fundamental constants or composition independent material parameters in this 8 study, the PF is inversely proportional to m c which is proportional to m in cuic materials. The third term is a function of the reduced Fermi level and reduced and separation. This term has the same maximal value when 8 tuning the doping level at a given reduced and separation (meaning a given temperature in this study). Therefore, a temperature induced increase of m in a Kane and system may lead to a slight deviation from the relationship of PF /m c as predicted in a paraolic and system where the 9 third term is not and separation dependent. With the known temperature dependent m for oth Laand I-doped PTe, use of the SKB model also enales a prediction of the temperature dependent transport properties at any temperature and doping level. Fig. also shows the 9 calculated results (curves) having the same doping level as the actual samples, using the experimentally estimated lattice thermal conductivity. Possessing a comparale thermal conductivity at similar doping levels (Fig. c) the La-doped series shows ~% lower zt due to the ~% higher m over the entire temperature range studied here, even though the Seeeck coefficient is higher, as compared to the I-doped samples. In summary, we show an example of achieving higher thermoelectric performance as a result of lower effective mass, which is contrary to the generally held elief that higher effective mass is eneficial for thermoelectrics ecause of the resulting higher Seeeck coefficient. It is demonstrated that the significant reduction of carrier moility resulting from increased effective mass through and flattening actually reduces the thermoelectric power factor in n-pte and zt. 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