Strong anisotropy of electron-phonon interaction in NbP probed by magnetoacoustic quantum oscillations
Clemens Schindler, Denis Gorbunov, Sergei Zherlitsyn, Stanislaw Galeski, Marcus Schmidt, Jochen Wosnitza, Johannes Gooth
SStrong anisotropy of electron-phonon interaction in NbP probed by magnetoacousticquantum oscillations
Clemens Schindler,
1, 2, ∗ Denis Gorbunov, Sergei Zherlitsyn, StanislawGaleski, Marcus Schmidt, Jochen Wosnitza,
2, 3 and Johannes Gooth
1, 2, † Max Planck Institute for Chemical Physics of Solids, 01187 Dresden, Germany Institut f¨ur Festk¨orper- und Materialphysik, Technische Universit¨at Dresden, 01062 Dresden, Germany Hochfeld-Magnetlabor Dresden (HLD-EMFL) and W¨urzburg-Dresden Cluster of Excellence ct.qmat,Helmholtz-Zentrum Dresden-Rossendorf, 01328 Dresden, Germany (Dated: October 15, 2020)In this study, we report on the observation of de Haas-van Alphen-type quantum oscillations(QO) in the ultrasound velocity of NbP as well as ‘giant QO’ in the ultrasound attenuation inpulsed magnetic fields. The difference of the QO amplitude for different acoustic modes reveals astrong anisotropy of the effective deformation potential, which we estimate to be as high as 9 eVfor certain parts of the Fermi surface. Furthermore, the natural filtering of QO frequencies and thetracing of the individual Landau levels to the quantum limit allows for a more detailed investigationof the Fermi surface of NbP as was previously achieved by means of analyzing QO observed inmagnetization or electrical resistivity.
I. INTRODUCTION
Probing the propagation of ultrasound in the quantumregime of electrons yields detailed information on the na-ture of electron-phonon interactions. The ultrasound ve-locity in such regime exhibits quantum oscillations (QO),which can be understood both from a thermodynamic ar-gument [1, 2] as well as from a self-consistent treatment ofultrasound propagation as a stream of acoustic phononsinteracting with an electron gas that is quantized intoLandau levels (LL) [3–6]. Both approaches yield the sameresult, namely, the amplitude of the QO being dependenton the (effective) deformation potential Ξ ki = dE k /dε i ,which is a measure for the change of energy E k of anelectronic band k at a given strain ε i . The connectionto the microscopic picture can be understood intuitivelyby recalling that the probability for an electron in the k -th band of being scattered by a phonon-mode corre-sponding to ε i is proportional to (Ξ ki ) [3–9]. Employingmeasurements of magnetoacoustic QO, the deformationpotential and its anisotropy have been experimentally de-termined for many metals and semimetals (see for exam-ple Refs. 3, 9–14).Recently, the semimetallic transition-metal monopnic-tide NbP is of great interest, mainly due to its symmetry-protected crossings of conduction and valence bandswhich potentially host Weyl fermions [15–17]. It exhibitsa very small and highly anisotropic Fermi surface, con-sisting of intercalated spin-split pairs of electron and holepockets due to spin-orbit coupling [18]. The small Fermisurface gives rise to pronounced QO of relatively low fre-quencies, which have so far been observed in magnetiza-tion [18–20], electrical resistivity [21–24], Hall resistivity[21, 23], thermal conductivity [19], thermopower [19], and ∗ [email protected] † [email protected] heat capacity [19]. The superposition of QO originatingfrom different extremal Fermi-surface orbits yield a richFourier spectrum, especially when H is aligned along the c axis of the tetragonal lattice and the extremal orbits arethe smallest. The peaks in the Fourier spectra could beassigned to orbits via comparison of experimental datato ab initio density functional theory (DFT) calculations[18], however, ambiguities due to the limited resolutionand the broadness of the Fourier peaks remained. In arecent study by some of the authors [23], the evolutionof the Fermi surface upon direct application of uniax-ial stress along the a axis has been probed by meansof Shubnikov-de Haas (SdH) oscillations in the electricalresistivity. These experiments revealed a strong straindependence of the SdH oscillations, which, besides theadditional information regarding the orbit assignments,also render NbP a promising platform for studying mag-netoacoustic QO. Furthermore, the strong anisotropy ofthe Fermi surface is suggestive of a highly anisotropicelectron-phonon interaction as well, which can be mostconveniently investigated via ultrasonic measurements.In this paper, we report on the measurements of QOin the ultrasound velocity and attenuation in a NbP sin-gle crystal in pulsed magnetic fields H (cid:107) c (or [001]).We have investigated the acoustic modes ( u (cid:107) q (cid:107) [100]),( u (cid:107) q (cid:107) [001]), ( u (cid:107) [001] , q (cid:107) [100]), ( u (cid:107) [010] , q (cid:107) [100]),and ( u (cid:107) [1¯10] , q (cid:107) [110]) corresponding to the elasticmoduli C , C , C , C , and ( C − C ) / u is the displacement vector and q is the direction of propagation of the acoustic wave.Significant differences of the individual QO amplitudesbetween the modes were revealed. A large signal-to-noise ratio, the usage of pulsed magnetic fields beyondthe quantum limit, the high quality of our sample result-ing in peak-shaped QO (presence of higher harmonics ofthe Fourier series), and the natural filtering of certainQO frequencies due to the anisotropic electron-phononinteraction allowed for a detailed analysis of the QO fre- a r X i v : . [ c ond - m a t . o t h e r] O c t quencies and amplitude ratios. Thereby, the anisotropyof Ξ ki and partially also the cyclotron masses, cyclotronmobilities, and phase factors for several extremal Fermi-surface orbits were determined. The QO frequency spec-trum could be analyzed via direct assignments of the LLpeaks rather than Fourier analysis as in previous stud-ies, which allowed for the assignment of formerly elusiveorbits. In addition, the extremal nature (maximum orminimum) of the individual orbits could be deduced fromthe asymmetric shape of the LL peaks. II. METHODS
NbP has a tetragonal crystal lattice (space group I md , no. 109) with the lattice parameters a = b = 3 . (cid:6) A and c = 11 . (cid:6) A [25]. A single-crystalline sample of NbP was grown using chemical va-por transport reactions; the sample has also been used inour previous work [23] for the determination of the elas-tic moduli. For acoustic modes propagating along oneof the main axes, the sample was cut accordingly to acuboid-shape of dimension 1 . × . × .
88 mm . For the( C − C ) / ) trans-ducers (Z-cut for longitudinal waves and X-cut for trans-verse waves) were glued to opposite parallel surfaces forexcitation and detection of acoustic waves. The rela-tive ultrasound-velocity changes ∆ v/v and attenuationchanges ∆ α were measured using an ultrasound pulse-echo phase-sensitive detection technique [9, 26] in pulsedmagnetic fields up to 38 T (test pulses up to 56 T) at tem-peratures ranging from 1.35 to 30 K. Excitation frequen-cies were varied from 27 to 100 MHz with pulse durationsranging from 50 −
200 ns. Strain-induction coupling, i.e.,the Alpher-Rubin effect [2], may be safely neglected atthe used frequencies as the large magnetoresistance inNbP even at moderate magnetic fields ( µ H >
III. RESULTS
The change of sound velocity ∆ v/v and the change ofsound attenuation ∆ α vs magnetic field at T = 1 .
35 K areshown for different acoustic modes in Fig. 1. Here, ∆ v/v refers to the change compared to the sound velocity atzero magnetic field v = (cid:112) C eff /ρ , where C eff is the effec-tive elastic constant governing the respective mode [27]and ρ is the mass density ( ρ = 6 .
52 g cm − for NbP [25]).∆ v/v shows pronounced QO with high harmonic content,whereas dominant frequencies and size of the oscillationamplitudes strongly vary between the modes. Strikingly,the QO amplitude in the C mode is smaller by a factorof ≈
20 compared to the other modes, where for the lastfew LL changes in v by more than one part in a thou-sand are observed. ∆ α exhibits QO with a characteristic -D v / v (cid:1) H ( T )( a ) C C C C ( C - C ) / 2N b P H || [ 0 0 1 ] T = 1 . 3 5 K D (cid:1) (cid:1) H ( T )( b )
3 d B / c m C C C C ( C - C ) / 2 FIG. 1. Magnetoacoustic quantum oscillations in NbP forpulsed magnetic fields H (cid:107) c at T = 1 .
35 K for differentacoustic modes. (a) Change in the relative ultrasound veloc-ity − ∆ v/v versus magnetic field. (b) Change in ultrasoundattenuation ∆ α versus magnetic field. The curves are shiftedwith respect to each other for better visibility. spike-like shape, also varying in terms of amplitude anddominant frequencies depending on the mode. We re-call that the physical mechanism responsible for the QOin ultrasound attenuation, which are commonly termedas ‘giant QO’ [1, 8], is not related to the Landau tubespassing through the extremal parts of the Fermi surfaceas in the de Haas-van Alphen (dHvA)-type oscillations.Instead, spikes in ∆ α occur when the Landau tubes passthrough the Fermi-surface section, where the componentof the Fermi velocity parallel to q is equal to the phasevelocity of sound [1, 3, 8, 28]. This resonance conditionis the reason for the spike-like shape, as it is only fulfilledfor particular values of the wavevector in contrast to thecontribution of many wavevectors in the dHvA-type oscil-lations. Notably, the resonant Fermi-surface orbits candiffer substantially from the extremal orbits, especiallywhen q ⊥ H . Hence, the position of the observed spikesin ∆ α do not necessarily coincide with the LL peaks in∆ v/v .Above 30 T, all electrons and holes are confined to theirlowest LL; and v ( H ) and α ( H ) exhibit a steady slope inthe investigated field (measured up to 56 T for C ) andtemperature range, showing no signatures for correlation-driven charge instabilities. Such correlation-driven phasetransitions, e.g., a charge density wave, would manifest ina slope change of ∆ v/v and a peak in ∆ α [29], and havebeen predicted to occur in the extreme quantum limit ofWeyl semimetals [30, 31]. Notably, there have been ob-servations of indicative features in the extreme quantumlimit in the electrical resistivity and in the sound velocityand attenuation in the related compound TaAs [32, 33].However, in case of pristine NbP the interaction strengthpresumably is too feeble as to allow for experimental ac-cess to these energy scales within our achievable field andtemperature range. A. Quantum oscillations in the velocity of sound
1. Frequency analysis and orbit assignment
To analyze the QO in the ultrasound velocity, − ∆ v/v is plotted against 1 /H (Fig. 2). The ultrasound velocity,just as any thermodynamic property of a material, ex-hibits singularities upon increasing magnetic field when-ever a cyclotron orbit corresponding to a LL is exactlyequal to an extremal orbit of the Fermi-surface sheet per-pendicular to the applied H . According to the Onsagerrelation [1], these singularities are periodic in 1 /H withthe frequency F = ( (cid:126) / πe ) A ext , where A ext is the areaenclosed by the corresponding extremal orbit, (cid:126) is thereduced Planck constant and e the electron charge. Plot-ting LL number vs 1 /H , F can then be extracted usinga linear fit [see Fig. 2(g)].For a maximum orbit, − ∆ v/v will increase with(1 /H ) − / approaching a LL singularity from a lowerfield and then decrease steeply, once the area of the cor-responding cyclotron orbit exceeds that of the maximumorbit [28]. Accordingly, for a minimum orbit these slopesare reversed and the steep rise appears on the low-fieldside of the LL peak. If smearing due to finite temper-ature and electron scattering is sufficiently suppressed,the QO retain a high harmonic content and approach asawtooth-like shape. The asymmetry of the individualLL peaks then allows for identifying whether the corre-sponding peak is arising from a maximum or minimumorbit of the Fermi surface.Clearly, the dominant frequency of 30 .
89 T in C and C (also very well distinguishable in the ( C − C ) / C mode. As assigned in Ref. 18 based on DFTcalculations and further indicated by comparing experi-mental and calculated strain dependences [23], this fre-quency is likely stemming from the α rather than the γ orbit [hereafter, we use the same labeling for the extremalorbits of NbP as in these Refs., see Fig. 2(a)]. The α oscillation is much less pronounced in C [see Fig. 2(c)],allowing for a clear identification of the 14 .
74 T oscillationas a minimum orbit, assigned to β . After having iden-tified the LL peaks for α and β , the remaining peaksin the high-field range might be assigned to the γ orbitand possibly also the δ orbit [see Fig. 2(e)]. The assign-ment to δ is thereby rather speculative; the second peakat approx. 0 .
06 T − might also stem from the last LL of δ . At low fields, a 0 . C , assigned to β [Fig. 2(d)].Furthermore, by applying a low-pass Fourier filter to C an oscillation of 6 .
81 T is singled out, which was alsoidentified in the Fourier spectra from previous QO stud-ies [18, 22, 23] and assigned to the α orbit [Fig. 2(f)].The extracted frequencies are summarized in Table I. Wenote that we did not observe additional QO patterns pre-dicted to occur in Weyl semimetals when the Fermi levelis near the Weyl points [5].
2. Lifshitz-Kosevich fit
The actual shape of the QO in ∆ v/v can be describedby a Fourier series taking finite-temperature smearingof the Fermi-Dirac distribution and LL broadening dueto electron scattering into account. After Lifshitz andKosevich [1], the oscillatory part of ∆ v/v for a singleQO frequency without spin degeneracy holds˜ v ij v ij = − (cid:18) ∂F∂ε i (cid:19) (cid:18) ∂F∂ε j (cid:19) e Vm c C ij (cid:18) eH (cid:126) π A ext ” (cid:19) × ∞ (cid:88) p =1 p − R T R D cos (cid:20) πp (cid:18) FH − ϕ (cid:19) ± π (cid:21) , (1)where m c denotes the effective cyclotron mass, V the realspace volume, A ext ” is the curvature of the Fermi surfaceat the extremal orbit and ϕ is the phase factor. The ± π/ − ) or minimum (+). Damping of the QO due to thermalsmearing of the Fermi distribution is accounted for by thefactor [1] R T = λ ( T )sinh [ λ ( T )] , with λ ( T ) = p π m c k B Te (cid:126) H . (2)Damping due to electron scattering is taken into accountby the Dingle damping factor [1] R D = exp [ − λ ( T D )] = exp (cid:20) − p πµ c H (cid:21) , (3)where T D is the Dingle temperature and µ c is the mo-bility of an electron exerting a cyclotron motion in anapplied magnetic field (not to be confused with the zero-field transport mobility, which, depending on the currentdirection, can significantly differ from µ c in case of a largeband anisotropy [35]). The β , β , and α oscillationswere clearly distinguishable in C and C , respectively, H || k z D (cid:1) xx ( b )( d ) ( f )( g ) E 1 E 2 H 1 H 2 a b g a b g ( a ) -D v / v (cid:1) H ( T - 1 ) C C a a a a a a a a a a x 0 . 5 ( C - C ) / 2 C C C x 0 . 51 / 3 0 . 8 9 T - 1 -D v / v (cid:1) H ( T - 1 ) ( c ) C ( C - C ) / 2 b b b b b b b b - 1 -D v / v (cid:1) H ( T - 1 ) ( e ) g g g g ( C - C ) / 2( d )( d ) x 2 0 1 / 3 1 . 7 T - 1 - 1 (cid:1) H ( T - 1 ) -D v v b b b d d - 1 -D v v (cid:1) H ( T - 1 ) C a a a a - 1 Landau level (cid:1) H ( T - 1 ) ( d ) g a b a b Landau level (cid:1) H ( T - 1 ) ( d ) g a b a b FIG. 2. Frequency analysis of the quantum oscillations in ultrasound velocity for different modes at T = 1 .
35 K. (a) Projectionsof the electron pockets, E1 and E2, and the hole pockets, H1 and H2, of NbP parallel to the k x - k z plane (or similarly to the k y - k z plane due to the fourfold rotational symmetry). Extremal orbits for H (cid:107) c are shown in red. For an illustration of the fullFermi surface in the first Brillouin zone see for instance Refs. 23 and 34. (b) Top: Shubnikov-de Haas oscillations subtractedfrom the magnetoelectrical resistivity ρ xx at T = 2 K for comparison. Bottom: Landau-level peaks assigned to the maximumorbit α . (c) Landau-level peaks assigned to the minimum orbit β . (d) Low-frequency oscillation visible in the C modeassigned to the minimum orbit β . (e) Assignment of the remaining peaks in the high-field range to the second maximumorbit of E1, γ , and possibly the maximum orbit δ . (f) Oscillation assigned to the maximum orbit α visible in the C mode,emphasized by applying a low-pass Fourier filter. (g) Assigned Landau levels plotted versus inverse magnetic field. Solid linesrepresent linear fits. The inset enlarges the high-field range. and could be approximated using the first 20 harmonicsof Eq. (1). From fits to the QO for different temper-atures (Fig. 3), the damping factors R D and R T couldbe extracted, allowing for the determination of m c , ϕ , µ c , and T D (summarized in Table I). The fitting proce-dure was performed globally for all temperatures with the shared parameters F (fixed), m c , ϕ , and µ c , and anindependent amplitude prefactor. We note that the di-rect fitting of the naturally filtered QO yields a greaterreliability for the m c values compared to the analysis ofFourier spectra, as there the field-dependent amplitudedamping usually leads to a systematic underestimation TABLE I. Experimental results extracted from the analysis of quantum oscillations in the ultrasound velocity for H (cid:107) c . Thecalculated orbits are denoted as in Ref. 18 and experimentally extracted frequencies F are assigned as in Refs. 18 and 23,considering also the asymmetry of the Landau level peaks due to the extremal nature of the orbit (maximum or minimum).The cyclotron mass m c , cyclotron mobility µ c , Dingle temperature T D , phase factor ϕ and the effective deformation potentialΞ i with respect to Ξ are given if possible (absolute values only). Ξ s denotes the deformation potential corresponding to the( C − C ) / F theo (T) a F exp (T) m c ( m ) µ c (10 cm V − s − ) T D (K) ϕ Ξ (eV) b Ξ / Ξ Ξ / Ξ Ξ / Ξ Ξ s / Ξ Electron pocket E1 α Max 32.8 30.89(5) 0.06(1) 25(5) 1.4(6) 0.27(1) 2.1(5) 0.9(1) 0.7(1) 0.24(4) 2.0(2) β Min 11.3 14.74(4) 0.12(2) 9(1) 2.0(5) 0.23(1) 1.4(3) 1.2(1) 6.3(5) 0.8(1) 5.1(4) γ Max 31.1 31.7(5) - - - 0.20(2) - 1.6(1) 3.0(2) 0.6(1) 3.2(3)Electron pocket E2 α Max 7.92 6.81(7) - - - 0.5(1) - - - - - β Min ≈ γ Max 4.7 - - - - - - - - - -Hole pocket H1 δ Max 41.4 42(1) - - - - - 0.8(1) ≈ ≈ ≈ δ Max 22.1 - - - - - - - - - - a The calculated frequencies were obtained from density functional theory in our previous study [23]. b Ξ has been estimated with Eq. (5) using the averaged ∂F/∂ε values from Ref. 23. - D v v (cid:1) H ( T - 1 ) ( a ) - D v / v (cid:1) H ( T - 1 )( b )( d )( c ) L K F i t b L K F i t a ( e ) ( f ) - D v v (cid:1) H ( T - 1 ) - D v / v (cid:1) H ( T - 1 ) - D v v (cid:1) H ( T - 1 ) b - D v / v (cid:1) H ( T - 1 ) FIG. 3. Temperature evolution of the quantum oscillations in the ultrasound velocity for the C [(a) and (c)] and C mode(e) and Lifshitz-Kosevich fit for the frequencies β (b), β (d), and α (f) dominant in C and C , respectively. of m c [23, 36]. Our fits yielded an effective cyclotronmass of 0 . m for α and 0 . m for β , which islarger than the values extracted from Fourier analysis ofdHvA oscillations [18] [0 . m and 0 . m ]. Theextracted m c are also in better agreement with the calcu-lated values from Ref. 18 (0 . m and 0 . m ) comparedto previous methods, albeit this does not necessarily im- ply improved accuracy.
3. Discussion of the phase factor
The phase factors extracted from fitting Eq. (1) to the∆ v/v data are around 0 . α and β on the electron pocket E2, and vary from 0 .
27 to 0 . α , β and γ on E1. According to re-cent theory works by Alexandradinata et al. [37, 38], thephase factor generally consists of three contributions ϕ = ϕ M − ϕ B − ϕ d , (4)where ϕ M is the Maslov correction ( ϕ M = 1 / ϕ B is the geometric phase, i.e., Berryphase [39], that an electron acquires upon encircling theorbit in reciprocal space, and ϕ d is the dynamic phasefactor which accounts for the generalized Zeeman inter-action of the intrinsic and orbital magnetic moment. Themain interest in analyzing the phase contributions lies inthe extraction of ϕ B , as it potentially allows to identifytopologically non-trivial bands, such as Weyl or Diracbands [39]. Indeed, under certain symmetry constraints(for details, see Refs. 37 and 38) ϕ d vanishes or can onlytake quantized values ± /
2, which then allows to drawconclusions about ϕ B . As all orbits in NbP for H (cid:107) c canbe mapped onto themselves in k space upon applying amirror operation (mirror planes k x = 0 or k y = 0, seeRef. 34), they belong to the classification (II-A, u = 1, s = 0) of Tab. I in Ref. 37, and ϕ d can be either 0 or1 / ϕ from 0or 1 / ϕ B .However, it was shown by Klotz et al. [18] that the Fermi-surface pockets in NbP intersecting with the Weyl bands,E1 and H1, always encompass a pair of Weyl points andshould thus exhibit a trivial phase shift of ϕ B = 1 or 0.Hence, the extracted phase factors of α , β and γ areat odds with the possible values predicted by theory. It israther speculative why this is the case, the reason mightbe slight misalignment of the magnetic field, wrong or-bit assignment or, more generally, inaccuracy of the DFTcalculations, although the latter two are highly improb-able given the otherwise good agreement. The extracted ϕ of E2 are not contradicting theory, but are also notparticularly informative regarding the topological natureof the bands.
4. Extraction of the deformation potentials
Comparing the amplitudes of the same orbit for dif-ferent modes, the ratio of the C − ii ( dF/dε i ) values canbe extracted. With the known elastic constants from ourprevious study [23], the ratio of the effective deformationpotentials can then be calculated via [2]Ξ i = dEdε i = dEdA ext dA ext dε i = (cid:126) em c ∂F∂ε i . (5)The amplitude ratios for the individual orbits have beenextracted by selecting well distinguishable LL peaks(near the quantum limit) and divide their top-to-bottomheights. In case there was no separate LL peak, as for - D v v (cid:1) H ( T - 1 )( c ) ( d ) g - D v v (cid:1) H ( T - 1 )( a ) g ( d ) ( b )( d ) - D v v (cid:1) H ( T - 1 ) b a - D v v (cid:1) H ( T - 1 ) b a FIG. 4. Extraction of the oscillation amplitude for superim-posed peaks in the high-field range by fitting two Lorentzianfunctions (green) at fixed inverse-field values. (a) and (c)Extraction of the height for γ and δ from the ultrasonicquantum oscillations in the C (a) and C mode (c). (b)and (d) Extraction of the height for β for the C (b) and C mode (d). example for the β orbit in C and C , the height wasestimated via fitting of two Lorentzian functions withfixed centers (Fig. 4), whereas the center positions wereextracted from comparison with other modes (see Fig. 2).The resulting deformation potentials w.r.t. Ξ are sum-marized in Table I. They are strongly anisotropic - mea-surable Ξ values vary by up to a factor of ≈ ∂F/∂ε values gathered from Ref. 23, Ξ can be estimated via Eq. (5) to be 2 . . α and 1 . . β taking experimentally (cal-culated) values, respectively. For β , this results in aneffective deformation potential of 9 eV (14 eV) for shearstrain along c . This potential is among the highest re-ported values [10, 41, 42] and illustrates how electrons inthe narrow part of the electron pocket are extremely sus-ceptible to interaction with phonon modes correspondingto such shear strain. We note that upon applying strainalong an axis perpendicular to the c axis, the breakingof the rotational symmetry leads to a degeneracy liftingof the Fermi pockets and ∂F/∂ε actually splits into apositive and a negative branch [23]. As in Eq. (1) thesign of ∂F/∂ε is canceled due to the square, we tookthe average of the absolute values in order to estimateΞ . B. ‘Giant’ quantum oscillations in ultrasoundattenuation D (cid:1) (cid:1) H ( T - 1 )( a ) C C C C ( C - C ) / 2F F F F F F F F F F F F F F Landau level (cid:1) H ( T - 1 )( b ) F F FIG. 5. Frequency analysis of the ‘giant quantum oscillations’in ultrasound attenuation for different modes at T = 1 .
35 K.(a) Landau level peaks assigned to the resonant orbits F and F . (b) Assigned Landau levels plotted vs inverse magneticfield. Solid lines represent linear fits. The ‘giant QO’ in ∆ α are less straightforward to an-alyze, as the position of the resonant orbits in recipro-cal space is rather complicated to determine for eachcorresponding phonon mode. If plotted against 1 /H [Fig. 5(a)], two periodic series of spikes are very welldistinguishable, labeled as F and F . The Onsager re-lation is valid for the ‘giant QO’ as well; linear fits tothe spike positions vs LL number yield F = 29 . F = 14 . α and β . A puz-zling feature is the observation of the same frequenciesin two modes with perpendicular q , e.g., F in both C and C . This observation might be explained by the pe-culiar shape of the Fermi surface in NbP, where fourfolddegenerate sickle-like pockets are located near the edgesof the first Brillouin zone. In this particular case, theresonant condition might be fulfilled for the same orbitfor elastic waves propagating both along a and c .In contrast to the QO in ∆ v/v , the exact shape of thespikes in ∆ α is rather difficult to fit. Each δ function corresponding to a spike must be convoluted with vari-ous distribution functions accounting for the effects of fi-nite temperature and electron scattering [1]. In our case,this did not seem viable as multiple frequencies superim-pose each other and similar information on the electronicproperties has already been extracted from the QO in∆ v/v , where also the signal-to-noise ratio was more fa-vorable. Nevertheless, the slight asymmetry of the spikescan be attributed to an indirect effect of electron scatter-ing, where the smearing of the LL relaxes the resonancecondition [1]. The spikes of F and F are broader to-wards the low-field side [see Fig. 1(b)], which is indicativeof a convex curvature of the Fermi surface at the resonantorbit ( A ” < IV. SUMMARY
In summary, we studied the QO in ultrasound veloc-ity and attenuation in NbP in pulsed magnetic fields.Thereby, fields with H (cid:107) c beyond the quantum limitwere applied. We compared the QO for several acous-tic modes, revealing significant differences as to whichorbits are dominant. By extracting the amplitudes ofthe QO in the ultrasound velocity, the anisotropy of thedeformation potentials has been determined for severalextremal orbits. Thereby, a large deformation potentialof approximately 9 eV for the minimum orbit β undershear strain along the c axis has been revealed, suggest-ing that electrons in this part of the Fermi surface arevery susceptible to interactions with the phonon modescorresponding to C . Furthermore, the high harmoniccontent of the QO and the large field range allowed fora more reliable determination of the frequencies, effec-tive cyclotron masses, and mobilities as was previouslyachieved by means of Fourier analysis. On a side note, wedid not find any signatures for correlated electron statesin the quantum limit of (pristine) NbP. ACKNOWLEDGMENTS
C. S. would like to thank A. Alexandradinata for en-gaging in helpful discussions. C. S. acknowledges fi-nancial support by the International Max Planck Re-search School for Chemistry and Physics of QuantumMaterials (IMPRS-CPQM). The work was supported byDeutsche Forschungsgemeinschaft (DFG) through SFB1143 and the W¨urzburg-Dresden Cluster of Excellence onComplexity and Topology in Quantum Matter— ct.qmat (EXC 2147, project-id 390858490), and by Hochfeld-Magnetlabor Dresden (HLD) at HZDR, member of theEuropean Magnetic Field Laboratory (EMFL). [1] D. Shoenberg,
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