Indications of effects of phonon-roton states on electronic transport

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Materials Science and Engineering, A181/A182 (1994) 916-920

Indications of effects of phonon-roton states on electronic transport Chr. Lauinger Physics Department, Technical University of Chemnitz-Zwickau, D-09009 Chemnitz (Germany) and Physics Institute, University of Karlsruhe, PO Box 6380, D-76131 Karlsruhe (Germany)

J. Feld and J. Rimmelspacher Physics Institute, University of Karlsruhe, PO Box 6380, D-76131 Karlsruhe (Germany)

P. H~iussler Physics Department, Technical University of Chemnitz-Zwickau, D-09009 Chemnitz (Germany)

Abstract In this paper we report strong indications of the effects of phonon-roton states on electronic transport. Under conditions where the wavevector Qp of these states is close to 2kF, inelastic umklapp-scattering effects of the conduction electrons with phonon-roton states can occur. Below a characteristic temperature T0, under which the electronic energy is too low to excite these states, the inelastic umklapp scattering disappears, leaving the elastic umklapp scattering on the static structure and the inelastic scattering on long-wavelength phonons (Debye phonons). The variation in the scattering behaviour with temperature causes pronounced transport anomalies. The thermopower, for example, exhibits a kink at T0 and large deviations from the free-electron behaviour above T0. This effect is responsible for the different sign of the thermopower to the Hall coefficient at elevated temperatures, a long-standing problem which often exists in liquid and amorphous metals. Whereas indications of the inelastic umklapp scattering are also visible in the resistivity, seen as an inflection point where the resistivity which was increasing starts to decrease, the Hall effect seems to be unaffected.

1. Introduction

The electronic transport of disordered metals is still not well understood. Although Ziman's [1] model and its modifications [2] gave a deep understanding, there are still fundamental aspects which await a thorough understanding. Deviations of the Hall coefficient to the free-electron value [3, 4], for example, are one of the most important features requiring understanding [5]. Another serious difficulty exists if the Hall coefficient R H is compared with the thermopower S( T ). When the comparison is, as normally, done at elevated temperatures, the signs of R n and S(T) may be opposite [6], although RH itself may be close to the free-electron value RH° [3, 6] and both should agree. This contradicts the existing theories and has initiated many discussions [6, 7]. Whereas the Hall coefficient, generally, is linear with T and close to constant, the thermopower often shows non-linearities [8]. Attempts to explain these non-linearities were presented by Kaiser [9] using the assumption of many-body effects. Accordingly, mass enhancement effects due to the electron-phonon interaction, renormalization terms of the velocity and relaxation times, and even higher corrections have been 0921-5093/94/$7.00

SSD10921-5093(93)05730-D

introduced. Good numerical fits to the experimental data have been reported for alloys such as Ca-A1 [8]. For binary noble-metal-polyvalent-element alloys, good fits were also reported, but instead of mass enhancement effects a reduction now needs to be understood [10]. In addition, some fitting parameters are very often found to be completely out of the range which is theoretically allowed. Accordingly, one of the authors (P.H.) proposed a different explanation based on characteristic phonon anomalies of disordered systems, the so-called phonon,roton states [11, 12]. During the last decade, phonon rotons have been a main focus of research of the amorphous [13] and liquid state [14]. Phonon-roton states are energetically low-lying collective-density excitations at wavenumbers Qp where the structure factor shows a pronounced m a x i m u m (Qp ~ Kp). They may be thought as the remnants of periodicity. In many liquid and amorphous alloys it was shown that the maxima in the structure factor are electronically induced and hence are positioned at scattering v e c t o r s Kpe ~ 2 kF, with 2 kF the Fermi sphere diameter [4]. Ideal conditions for elastic and inelastic umklappscattering effects therefore exist. The intensity of the © 1994 - Elsevier Sequoia. All rights reserved

Chr. Lauinger et al.

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Effects of phonon-roton states on electronic transport

structure factor influences the energy of the phonon-roton states and, subsequently, the corresponding characteristic temperature T0. The higher the intensity, the lower T0 should become. Thermopower contains information on electronscattering mechanisms. It is a measure of the electric field set-up when electrons diffuse down a thermal gradient with the magnitude of diffusion ultimately limited by the elastic and inelastic scattering of the electrons. Different mechanisms are additive and umklapp scattering plays an important role [15, 16]. Here we give a comparison of different amorphous alloys. Mg-Zn, a model-like amorphous system shows similar effects to noble-metal-polyvalent-element alloys [ 17].

2. Experimental details

The samples have been prepared in situ by sequential flash evaporation techniques, with the substrates held at T = 4 K [11]. After deposition they were annealed to a temperature Ta well below the crystallization temperature. Here we discuss those data which were measured below Ta in the well-annealed state. Experimental details of thermopower measurements have been published elsewhere [18]. The Hall coefficient has been measured by a lock-in technique. In order to suppress superconductivity, the resistivities of some samples were measured with very high accuracy within an applied magnetic field.

3. Results and discussion

As representative of many others, Figs. l(a)-l(c) show thermopower and Figs. l(d)-l(f) resistivity data of three different amorphous systems. Of these systems, M g - Z n (and in particular Mgv0Zn30) is one of the best-known amorphous materials. The static and dynamic structural properties [13, 19], the electronic density of states [20] and the electronic transport properties [20, 21] are well known. Noble-metalpolyvalent-element alloys have systematically been studied by one of the authors (P.H.) and are known with similar completeness. Their comprehensive review has been given elsewhere [17]. It is well known that these alloys show concentration regions where electronic transport properties closely approach the free-electron model (FEM) and regions where large deviations exist. The region of agreement with the FEM in the noble-metal-polyvalent-element alloys is in Cu-Sn close to pure Sn (80at.%Sn) [4], in Cu-A1 close to 80 at.% AI [17] and in Mg-Zn close to pure Mg (80at.%Mg) [21]. Approaching Cu or Zn,

917

deviations arise. The d states of Cu have no influence on this behaviour since they are well below EF. It is the height of the structural peak close to 2k F and its relative position to 2 k F which controls deviations from the FEM. Under conditions where a peak in the structure factor is at exactly 2 k v and, in addition, the peak is high, quite a deep pseudogap in the electronic density of states exists at E v. The phonon-roton minimum is at low energies and therefore T0 at low T [17]. Data such as the electronic density of states at E r indicate that Zn-rich Mg-Zn samples are comparable with noblemetal-rich alloys [17]. Zn seems to have the same function as the noble metals. The thermopower signals vs. T are composed of two nearly linear ranges interconnected at T = T0. Therefore we denote the thermopower at low temperatures a S~(T) and the thermopower at high temperatures as Sh( T ). Sl( T ) is well described by Mott's formula n2kB2 s(:r) =

3]elEv

(1)

which is proportional to T. The scattering behaviour of the electrons is described by the thermopower parameter ~ = EF 0{ln[p(E)]}/0 E[EF" In pseudopotential theory, ~ is written as ~=3-2q-1r

(2)

with q a correction term which describes elastic umklapp scattering and r describing effects of the energy dependence of the pseudopotential at 2 kv [15]. For Mg-, Sn- or Al-rich samples, SI(T) is in good agreement with S o, the free-electron value, which is obtained by Mott's formula if ~ is assumed to be 3 [15]. This behaviour is best seen for pure Sn (which is not amorphous but polycrystalline) and for the Sn-rich samples [17]. The thermopower at low T hence supports the Hall effect data which show similar agreement with the corresponding free-electron value in the same range of composition [4, 17]. At low T, obviously, the discrepancy between the sign of the thermopower and the Hall effect disappears. With decreasing content of Mg, Sn or AI, deviations of SI(T) to S O arise (see for example Cu45Sn55). This has been described as an effect of elastic umklapp scattering due to the increasing intensity of the structure factor at 2k F with composition and therefore an enhanced effect of the q factor in eqn. (2). A more detailed description has been published elsewhere [17] and will not be the focus of the present paper. The inelastic scattering of electrons by Debye phonons has no effect on the thermopower [22]. The different effect of phonons of short wavelength (phonon-rotons) and long wavelength (Debye phonons), together with the elastic and inelastic

918

Chr. Lauinger et al.

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Fig. 1. (a)-(c) Thermopower and (d)-(f) corresponding resistivity data for amorphous Mg-Zn, Cu-Sn and Cu-A1 alloys with different compositions vs. temperature. The thermopower data are vertically shifted from each other for clarity. The experimental thermopower of polycrystalline pure Sn is included and in some cases the free-electron values. The data on the thermopower S'( T ) at low T as well as on the thermopower sh( T ) at high T are indicated in some of the figures. The parameters denote the Mg, Sn and A1 contents. The resistivity data are normalized at particular temperatures. At very low T the alloys become superconducting (magnetic fields were not applied in these measurements). All measurements were performed in the well-annealed amorphous state.

umklapp scattering of electrons, opens up a new description of S ( T ) over a large range of T. If the scattering mechanism of electrons were independent of T, the t h e r m o p o w e r parameter ~ would be constant and S ( T ) proportional to T (eqn. (1)). This linearity fails because above the characteristic temperature TO a new scattering channel opens owing to the inelastic umklapp scattering of the electrons with p h o n o n - r o t o n states [15]. T h e additional effect of p h o n o n - r o t o n states can be seen nicely if one goes from pure Sn which is polycrystalline with no p h o n o n - r o t o n states to Cu25Sn75 which is amorphous. T h e low temperature region of the amorphous phase stays in good agreement with S O because p h o n o n - r o t o n states cannot b e c o m e excited. This is no longer true for T > T0. P h o n o n - r o t o n states are excited and therefore the kink at TO suddenly occurs, followed by large deviations well above TO[11, 17]. TO itself depends on composition because of the variation in the structural intensity at Kpe = 2 k F. This dependence, obtained from t h e r m o p o w e r data (To sITI) is shown in Fig. 2 for glassy Cu-A1 [23]. Approaching

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40 at.% A1, TOtends to T = 0 K. T h e structure factor at Kpe became maximal for this composition, causing a deep p h o n o n - r o t o n minimum and hence a low T0. T h e question arises: are there p h o n o n - r o t o n effects on other properties as resistivity or the Hall effect?

Chr. Lauinger et al.

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Effects' of phonon-roton states on electronic transport

Figure 1 shows that, indeed, some relations may exist. Those alloys which are close to the free-electron behaviour show positive temperature coefficients at low T which may become negative at elevated temperatures (Mg-Zn and Cu-A1). Those alloys which show large deviations from the free-electron behaviour, i.e. show large positive Sh(T), exhibit negative temperature dependences down to the lowest temperatures. Unfortunately, the transition to the superconducting state and its fluctuations preclude a detailed discussion. Hence resistivity measurements with high accuracy within an applied magnetic field were performed on Cu-A1. In Figs. 3(a), 3(b), 3(c) and 3(d) the normalized resistivity, its second derivative, the thermopower and the Hall coefficient respectively of glassy Cu20Al~0 are shown against T in highly enlarged plots. Only about 0.3% of the total resistivity is drawn, normalized at T= 0 K (Fig. 3(a)). The second derivative of the resistivity is drawn in Fig. 3(b). The inflection point of p( T ) at T0 is obviously closely related to the kink in S(T) (Fig. 3(a)) and therefore to the onset of inelastic umklapp-scattering effects of the electrons with phonon rotons, p ( T ) can be interpreted as follows. Below T0 the main contribution, which is suppressed in Fig. 3(a), is given by elastic umklapp scattering. The increase with decreasing T (p( T ) ocIn T ) is ascribed to electron-electron interaction and is not further discussed here. The increase with rising T is caused by the inelastic scattering of the electrons with Debye phonons. Debye phonons, however, leave the thermopower unaffected [22]. Above T0, finally, the inelastic scattering with phonon-rotons arises. The increasing resistivity starts to decrease, shows a maximum and finally exhibits a negative temperature coefficient. The negative temperature coefficient may basically be seen as resulting from phonon-rotons. The static structure factor decreases owing to phonon-roton excitations and hence the resistivity in Ziman's description. The decrease in the resistivity can be understood if elastic umklapp scattering, which becomes weaker on account of inelastic umklapp scattering, contributes more to the resistivity than the latter. Close to 40 at.% A1 (or 30 at.% Sn or 50-70 at.% Zn), the phonon-roton states become excited at lowest temperatures and, subsequently, the negative temperature coefficient of p( T ) exists down to the lowest temperature. Some characteristic temperatures ToR(T), taken from the inflection point of the resistivity, are included in Fig. 2 and, within experimental resolution, agree well with TosIT). The Hall coefficient (Fig. 3(d)) shows no obvious effects of phonon-roton scattering within experimental resolution, which, on the contrary, is much smaller than for other transport properties.

919

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4. Conclusions

Owing to the reinterpretation of S( T ) and the inclusion of phonon-rotons, the long-standing problem of different signs of R H and S( T ) seems to be solved if R H and S(T) are compared below T0. Phonon-roton states create anomalous features at elevated temperatures.

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Effects of phonon-roton states on electronic transport

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