Importance of Electronic Correlation in the Intermetallic Half-Heusler Compounds
IImportance of Electronic Correlation in the Intermetallic Half-Heusler Compounds
Minjie Lu, Hao Chen, and Glenn Agnolet ∗ Physics Department, Texas A&M University. (Dated: December 9, 2020)Low temperature scanning tunneling spectroscopy of HfNiSn shows a V m ( m <
1) zero biasanomaly around the Fermi level. This local density of states with a fractional power law shape iswell known to be a consequence of electronic correlations. For comparison, we have also measured thetunneling conductances of other half-Heusler compounds with 18 valence electrons. ZrNiPb shows ametal-like local density of states, whereas ZrCoSb and NbFeSb show a linear and V anomaly. Oneinterpretation of these anomalies is that a correlation gap is opening in these compounds. By ana-lyzing the magnetoresistance of HfNiSn, we demonstrate that at low temperatures, electron-electronscattering dominates. The T m ( m <
1) temperature dependence of the conductivity confirms thatthe electronic correlations are a bulk rather than a surface property.
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I. INTRODUCTION
Recently, there has been renewed interest in more effi-cient thermoelectric materials because of the worldwidedemand for sustainable energy sources. One such class ofcandidates are the half-Heusler compounds with 18 va-lence electrons. Among them, the MNiSn (M = Ti, Zror Hf) compounds are narrow gap semiconductors with agap size of 100-300 meV [1, 2]. The narrow gap leads to amoderate electrical resistivity and a large Seebeck coef-ficient. This combination leads to a high ZT , the figureof merit used to evaluate thermoelectric efficiency [1–3]. ZT = α T / ( ρκ ), where α is the Seebeck coefficient, ρ isthe electronic resistivity, κ is the thermal conductivityand T is the absolute temperature. Intensive efforts havebeen made to improve the ZT of MNiSn compounds. Be-cause MNiSn compounds are n-type materials, all of thethree sites, M, Ni and Sn, can be doped by donors to re-duce the electronic resistivity [4–6]. Grain size and grainboundaries can also be engineered to achieve a betterSeebeck coefficient [7, 8]. However, these attempts canbe facilitated by a better understanding of the electronicstructure near the Fermi level.Half-Heusler compounds have the MgAgAs-type crys-tal structure which belongs to the F-43m space group.The crystal structure is shown in Fig. 1(a). It is com-prised of four interpenetrating face-centered cubic sub-lattices. Two sublattices are occupied by transition met-als. The third is occupied by a main group element andthe fourth is occupied by vacancies. The vacancy sub-lattice distinguishes half-Heusler compounds from theirfull-Heusler counterparts as it induces a gap in the elec-tronic structure. The intrinsic disorder observed in manyhalf-Heusler compounds [2, 9, 10] is known to be donorsor acceptors that can generate in-gap states [10–12]. An-nealing can reduce the concentration of this disorder andtransform the compounds from a metal to an insula-tor [9]. Close to the metal-to-insulator transition, disor- ∗ [email protected] der breaks down momentum conservation and electronicscreening becomes inefficient. Therefore, a strong elec-tronic correlation is expected.In this paper, we present scanning tunneling spec-troscopy (STS) of four half-Heusler compounds. Exceptfor ZrNiPb which has a metal-like local density of states(LDOS), the other compounds have different power law V α zero bias anomalies (ZBA) near the Fermi level,around which narrow but hard gaps should be expected.These anomalies are characteristic LDOS of a disorderinduced metal-to-insulator transition (MIT) [13, 14]. De-tails are discussed in Section III.A. These giant anomaliesin the one-electron local density of states play an impor-tant role in determining the electronic and thermal prop-erties of these compounds. Ab-initio approaches gener-ally fail to predict their existence [10, 15] and therefore,the subsequent conclusions from these approaches are un-reliable. These anomalies also make it difficult to inter-pret the results of doping experiments because, insteadof only shifting the chemical potential, doping also leadsto the evolution of the anomaly [16, 17]. In Section III.Band III.C, the magnetoresistance and the temperaturedependence of conductivity of HfNiSn are analyzed toprovide more evidence for a strong electronic correlationin the material. II. EXPERIMENTAL DETAILS
The HfNiSn samples are flux-grown single crystals pro-vided by our collaborator Dr. Lucia Steinke [18], whoalso provided the resistivity and magnetoresistance data.Details of sample growth and measurement of propertiesare described in [18]. The ZrNiPb, ZrCoSb and NbFeSbsamples are polycrystals provided by our collaborator Dr.Fei Tian of Dr. Zhifeng Ren’s group [4]. Tunneling con-ductances are measured with a low temperature scanningtunneling microscope (STM) located in a He cryostat.Atomic resolution images of graphite and self-assembleddodecanethiol monolayer on an atomically flat gold sur-face have been obtained with this STM. Mechanically a r X i v : . [ c ond - m a t . d i s - nn ] D ec cut Platinum-Iridium tips are used. In order to get re-producible tunneling conductance and because of a lackof in-situ tip treatment device, tips are annealed in aBunsen flame [19]. Using the same kind of tip, no ZBAis observed in the tunneling conductance of gold downto 77 K. Samples are cleaved right before experimentsto expose fresh surfaces. Differential conductances aremeasured by a standard lock-in technique with a 0.5 mVmodulation. III. RESULTS AND DISCUSSIONA. Local density of states
Figure. 1(b, c) show a comparison of a scanning elec-tron microscope (SEM) and an STM image of a freshlycleaved HfNiSn surface. Although the STM reveals struc-tures beyond the resolution of the SEM, the rough surfacemakes it difficult to achieve atomic resolution on thesematerials. Such a cratered surface is expected because ofthe vacancy lattice and the intrinsic disorder.The tunneling conductances of two HfNiSn samples aremore systematically studied. One shows a V ZBA, com-mon in disordered conductors with electronic correlation.The other shows a V ZBA. The tunneling conductancesof the latter at different temperatures are plotted in Fig.2(a). All the minima occur at − V shape is clear. Although the tunneling conduc-tance does vary slightly across the surface, the depen-dence on voltage remains the same. Figure 2(b) plots( G [ V, T ] − G [0 , T ]) /T versus ( eV /k B T ) , where G [ V, T ]is the tunneling conductance, and V is the bias voltage.The straight dashed line corresponds to a V depen-dence. Curves at different temperatures collapse to auniversal one that is expected from the finite tempera-ture density of states (DOS) calculated by Altshuler andAronov [20].Tunneling conductance of nominal semiconductors canbe sensitive to surface defects. But the ∼ T m conductiv-ity discussed in Section III.C supports that it is the bulkrather than surface electrons that are correlated. Forfilms thinner than tens of nanometers, the conductivityshould be proportional to ln ( T ) [20, 21], whereas the cor-rugation of the rough HfNiSn surface is around 2 nm fromthe STM topographic image. The √ B dependence ofmagnetoresistance at high fields also demonstrates thatthe transport is 3 dimensional [22].We further measured the tunneling conductances ofother half-Heusler compounds, ZrNiPb, ZrCoSb and Zr-CoSb. These samples are polycrystals. ZrNiPb andHfNiSn are n-type semiconductors [23, 24], whereas Zr-CoSb and NbFeSb are p-type [4, 24]. The results areshown in Fig. 3. ZrNiPb shows a metal-like LDOS,whereas ZrCoSb and NbFbSb show a linear and aquadratic ZBA separately. The quadratic ZBA is likely the correlation gap broadened by both thermal smear-ing and interactions [25, 26]. These measurements aretaken at room temperature simply to observe the giantZBA indicating electronic correlation. Low temperaturemeasurements may reveal other interesting features fordetailed discussion. For each compound, we reproducedthese observations on at least two samples indicating thatthese tunneling conductances are the characteristic of thedifferent compounds. Except for the different electronicstructures of the pure compounds, their chemical bondsmay favor different types and concentrations of the in-trinsic disorder. B. Magnetoresistance
In this section, we analyze the magnetoresistance ofHfNiSn to provide another evidence of strong electroniccorrelation.The magnetoresistance of HfNiSn at different temper-atures are shown in Fig. 4(a). The B dependence atlow field and the √ B or a weaker dependence at highfields are indicators of weak anti-localization (WAL) [32–34]. WAL is a quantum phenomenon observed in dis-ordered electronic systems caused by the interference ofself-crossing trajectories of backscattered electrons. Witha strong spin-orbit coupling [35] or a π Berry’s phase[36, 37], the interference is destructive and the current isenhanced. Magnetic field adds additional phases to theelectrons, thereby destroying the interference causing anincrease in the resistance.The formula for 3D WAL is provided by Fukuyamaand Hoshino (F-H) [38, 39]. The correction due to elec-tronic correlation consists of two parts. The orbital partis calculated by Altshuler et al. [40] and the spin partis calculated by Lee and Ramakrishnan [41]. Alternativeexpressions for the orbital part are also available [20, 42].These formula are suspected to fail at high fields [43, 44]and many fits to experimental data show clear deviations[35, 45, 46]. Therefore, we generate the dashed fitted linesin Fig. 4(a) merely to show that the magnetoresistanceof HfNiSn is a typical WAL magnetoresistance, but thefitting parameters will not be discussed here.WAL has become an important tool to extract theelectron dephasing scattering times τ φ [35, 44–46]. Thecritic field B φ = B i + 2 B s contained in the F-H formulais inversely proportional to τ φ . B x = (cid:126) eDτ x , where x stands for ( i ) inelastic, ( s ) spin-flipping or ( so ) spin-orbitscattering respectively, and D is the temperature inde-pendent diffusion coefficient related to elastic scattering[38, 44, 47]. Because of the aforementioned reason, in-stead of fitting the magnetoresistance, we adopt anotherapproach to get the temperature dependence of τ φ .At low field and low temperature, B (cid:28) B φ (cid:28) B so .Utilizing the low field analytic expression of the f func-tion provided by Ousset et al. [48], the F-H formulareduces to a very simple form. (a) (b) (c) FIG. 1. (a) Crystal structure of HfNiSn. 4 Wyckoff positions are labelled. The 4d position is occupied by vacancies. (b) SEMimage of a freshly cleaved surface. (c) STM image of the same surface. −60 −40 −20 0 20 40 60V (mV)0.40.60.81.01.2 d I / d V ( a . u . )
13 K30 K50 K80 K160 K −160 0 1600.51.01.5 ←13 K↓
230 K (a) eVk B T ) G [ V , T ] − G [ , T ] T ( a . u . )
13 K30 K50 K80 K160 K (b)
FIG. 2. (a) Normalized tunneling conductances of a singlecrystal HfNiSn from 13 K to 160 K. The different data setswere normalized by overlapping their values at high positivebias. Inset shows tunneling conductances at 13 K and 290 Kover a larger bias range. (b) Plot of [ G ( V, T ) − G (0 , T )] /T versus ( eV /k B T ) . All curves between 13 K and 160 K col-lapse to a universal one. The black dashed line is a linear fitindicating a V power law dependence. ∆ R [ B ] R [0] = c B φ B (1)where ∆ R [ B ] = R [ B ] − R [0], c = ce π ( e (cid:126) ) and c is aconstant related to the geometry of the sample. B φ provides information on different scattering mech-anisms. B φ = B i + 2 B s ∼ τ i + τ s , where τ s is a tempera-ture independent scattering time due to magnetic impu-rities. Usually, τ i = τ ee + τ ep , where τ ee is the electron-electron scattering time and τ ep is the electron-phononscattering time. It has been established that τ ep ∼ T − x ,where x can be 2, 3 or 4 [46, 49]. In disordered systems,at low temperature, τ ee ∼ T − d , where d is the dimension −200 −150 −100 −50 0 50 100 150 200 V(mV) d I / d V ( a . u . ) (a) −200 −100 0 100 200 V(mV) d I / d V ( a . u . ) −500 0 500 V(mV) d I / d V ( a . u . ) ↓↓ (b) −200 −150 −100 −50 0 50 100 150 200 V(mV) d I / d V ( a . u . ) (c) −200 −150 −100 −50 0 50 100 150 200 V(mV) d I / d V ( a . u . ) (d) FIG. 3. Normalized tunneling conductance of (a) ZrNiPb, (b)HfNiSn, (c) ZrCoSb and (d) NbFeSb. The inset of (b) showsthe the tunneling conductance of HfNiSn from −
500 meV to500 meV. Arrows point to kinks which possibly indicate bandedges. The inset of (d) shows the corresponding current-voltage curve of NbFeSb. of the electronic transport in the material [20, 46, 50].At low fields, utilizing the analytic expression of the Φ function provided by Ousset et al. [48], the orbital partof the correction due to electronic correlation becomes −8 −6 −4 −2 0 2 4 6 8 B (T) Δ R [ B ] / R [ ] ( M Ω − ) −0.2 0.0 0.2 B(T) Δ R [ B ] / R [ ] (a) B (T ) Δ R [ B ] / R [ ] ( M Ω − ) (b) √ B ( √ T ) Δ R [ B ] / R [ ] ( M Ω − ) (c) FIG. 4. (a) Magnetoresistance ∆ R [ B ] /R [0] = ( R [ B ] − R [ B =0]) /R [ B = 0] of a single crystal HfNiSn at different tempera-tures. The black dashed lines are the best fits generated usinga sum of the F-H formula and the corrections due to electroniccorrelation. (b) Magnetoresistance versus B . Dashed linesare linear fits. (c) Magnetoresistance versus √ B . Data areprovided by our colleague. [18] ∆ R [ B ] R [0] = c ( c − lnT ) T B (2)where c = 0 . × ce π ( De πk B (cid:126) ) , c = λ + ln γT F π , λ isthe temperature independent electron-phonon couplingconstant [51, 52], γ = 0 .
577 is Euler’s constant and T F isa cut off temperature.Utilizing the analytic expression of the g function pro-vided by Ousset et al. [48], the spin part becomes∆ R [ B ] R [0] = c T B (3)Where c = 0 . × c ( egµ B ) F π √ D ( (cid:126) k B ) , g = 2 is the elec-tron spin g-factor, µ B is the Bohr magneton, F (0 < F < T (K) −6 −5 −4 k [ T ]( Ω − T − ) k[T] ∼ τ ϕ ∼ T − FIG. 5. Log-log plot of k [ T ] versus temperature.
1) is the screening parameter for the Coulomb interac-tion.The data clearly follows a B dependence at low fields.Taken together, Eqs. 1, 2 and 3 indicate that ∆ R [ B ] R [0] = k [ T ] B where the slopes of the dashed lines in Fig. 4(b)indicates the values of k [ T ]. The determined values of k [ T ] are very robust. For example, in the analysis of the4 K magnetoresistance, regressions are performed usingincreasing numbers of data points from B < . to B < . . The calculated k [ T ] slowly decreases from1 . × − to 1 . × − which will not qualitativelychange the conclusion. Figure 5 is the log-log plot of k [ T ]versus T . The almost linear relation represents either the T − term in Eq. 2 and 3 or one of the scattering mecha-nisms in B φ dominates. By minimizing the mean squarederror of log ( k [ T ]), the green dashed line in Fig. 5 is gen-erated using only Eq. 1 and supposing that B φ ∼ T ,when only the 3D electron-electron scattering is consid-ered. One could also attempt to fit to an expression thatincludes Eq. 1, 2, 3 and all possible T x terms in B φ .However, when the T − lnT term in Eq. 2 is excluded,no better fitting can be generated than simply includingthe effect of electron-electron scatterings. Therefore, themagnetoresistance supports a strong electronic correla-tion and a 3D electronic transport in HfNiSn. C. Conductivity
According to McMillan [55], Altshuler and Aronov [20],electronic correlation leads to a quantum correction tothe the Boltzmann conductivity in weakly disordered sys-tems, σ I = σ + AT . This relation correctly describesthe typical low temperature conductivity of HfNiSn from1 K to around 140 K.High temperature conductivity of HfNiSn implies agap of 100 meV to 300 meV [9, 53, 54]. Our flux-grownHfNiSn single crystals also show an activated conduc-tivity above 200 K. But the calculated gap is around490 meV. In the tunneling conductance, there is a smallupturn at | V | = 250 mV shown in the inset of Fig. 3(b).If the upturn indicates the band edge, the tunneling con-ductance also indicates a comparable gap of 500 meV andthe ZBA probably lies in an impurity band. IV. CONCLUSION
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