Z. Phys. B 92, 383-387 (1993) ZEITSCHRIFT FOR PHYSIK B �9 Springer-Verlag 1993

Quantum corrections in AICuFe quasicrystals R. Haberkern 1'*, G. Fritsch 1, j . Schilling2

1 Universit/it der Bundeswehr Mfinchen M, BauV/I 1 Physik, D-85577 Neubiberg, Germany 2 Washington University, Physics Department, St. Louis 63130, USA

Received: 1 April 1993

Abstract. We report measurements on Hall-Effect, mag- netoresistivity and resistivity for the quasicrystal-system A1CuFe. Data for three different compositions exhibiting electrical resistivities of 4200, 8700 and 9600 gf~cm's, are discussed in terms of quantum corrections in the temper- ature range 5 to 50 K. It will be shown that the Coulomb contribution of these corrections reacts sensitively to the composition of the quasicrystal.

PACS: 61.44.+p; 72.15.Gd; 72.15.Eb

I. Introduction

Quantum corrections to the resistivity are well known for amorphous systems in the two-dimensional [1] or three-dimensional cases [-2, 3]. The most important fac- tor being there is a very short elastic scattering time in the range 10 -15 s. This fact generates an extremely short electronic mean free path in the range of only sever- al atomic distances. However, one should keep in mind that the coherence length of the electronic wave function is much larger, since it is determined by inelastic scatter- ing processes. In amorphous systems the big elastic scat- tering is brought about by scattering from the short range order. This fact can be expressed by the condition qv=2kF with kF the Fermi vector and q, the position of the first peak in the static structure function S(q), indicating the short range order.

In the quasicrystalline state it is a priori not clear where a large elastic scattering should come from, as the dominating Bragg reflexions are sharp and the others in between are very small. In addition, since in good quasicrystals the defect concentration is reduced, this ef- fect is also not responsible for the elastic scattering. How- ever, as we will show in this paper, there are clear indica- tions that quantum-corrections are present in these al- loys. To our opinion only a strong interaction between the Fermi surface and the Pseudo-Jones-Zone can be

* Part of the thesis of R. Haberkern

the reason for the large elastic scattering contribution necessary for this effect. This fact supports the point of view that the quasicrystals are a kind of Hume-Rothery- phases, stabilizing the system electronically.

II. Experimental details

The alloys A162.2Cu25Fe12.s , A163.2Cu24.sFe12 and A162Cuzs.5Fe12.5 were prepared by melt spinning. Weight amounts of the constituents (Al: 99.9999, Cu: 99.9999 and Fe: 99.98 purity) were melted under a pro- tecting atmosphere to form pellets of about 1.4 g. These samples were reheated inductively in a quartz ampoule and shot by argon gas onto a spinning copper wheel (v=22ms -1) into a helium atmosphere of 130mbar pressure. The resulting ribbon pieces have about 10 mm length, 1 mm width and about 25 gm thickness. Succes- sively, these samples were annealed for about 3 h at 810 ~ under vacuum of about 6.10-5 mbar. This proce- dure yields darkly shining quasicrystalline polycrystals, with no indication from X-ray scattering for any crystal- line or amorphous matrix.

Contacts for the transport measurements were made by evaporating 0.1 gm Cu onto certain spots and fixing wires onto them with silver paint. In this manner two current contacts, two voltage contacts and two Hall- contacts opposite to one another were applied. The sam- ples were glued to the sample holder and inserted into a He cryostat, containing also the superconducting mag- net. The sample temperature could be controlled from pumped He (1.2 K) to room temperature. The geometry of the samples was derived from direct measurements using a microscope. The currents applied were 0.1 to 1 mA for resistivity and 1 to 10 mA for the Hall-effect. Further details of the data analysis can be found else- where I-4].

IIl. Results

In this paper we would like to discuss the data, obtained for resistivity p as a function of temperature Tand mag-

384

. . . . ' . . . . I . . . . ' . . . . I . . . . ' . . . . i . . . . . . . . . I ' ' '

0.99

~ 0"97 f / �9 A163"2Cu2~t'gFe12"~

0.96 [] A16zsCu25"~ I, / + A162.0Cu25.sFe12.5

0.950 :, . ~ . . . . . . . . . , . . . . . . . . . , . . . . . . . . . , . , 5 10 15 20 25 30 35 40 45

T/K

Fig. 1. The normalised electrical resistance as a function of tempera- ture for the three alloys considered. The full lines are fits to the data as described in the text

netic field B and for the Hall-constant RH in the tempera- ture range 1.8 to 35 K. The magnetic field was varied between - 8 and + 8 Tesla. Since the Hall voltage turned out to be linear in the magnetic field, meaningful Hall- constants could be derived. The results for the three dif- ferent alloys examined are summarized in Figs. 1 to 3. Preliminary data have been published elsewhere [4]. The resistivity curves show distinct differences for the various alloys, as can be seen by inspection of Fig. 1. In Fig. 2 the magnetoresistivity is shown for magnetic fields up to 8 T. A quadratic part in the field can be recognized at low fields. At 4.2 K the magnetoresistivity varies from 1.6% up to 5% for the different alloys referring to a field of 8 T. The magnetoresistivity is positive. Finally, in Fig. 3 the Hall-coefficient is reproduced. It shows a turn-up at the lowest temperatures and a structure strongly varying with the alloy type considered in the T-range up to 35 K.

We have also determined the magnetic susceptibility Z of one of our samples (A162.sCu25.oFelz.s). At high temperature the susceptibility is diamagnetic. However, at T< 20 K a paramagnetic contribution is developing, resulting eventually in a paramagnetic state. These data are given in Fig. 4. In addition, in Fig. 5, the dependence of the magnetization on the field is plotted, indicating a very small saturation field. At higher fields diamagnetic behaviour is recovered.

0.04

S 0.03 ,2.,

0.02

0 ~ . . . . . . , , , , . . . . , . . . . , . . . . , . . . . , . . . . , . . . . , .

+ 3.92 K 2 / 0.05 x 4.50 K / * / . , /

o 6.49K A1620Cu255Fetz 5 / . 4 / / . / " �9 12.7K ' ' " , / - ~ / x / o 16.7 K . / ~ ' / ~ �9 244K / / z~ 32.2 K

0.01 .

1 2 3 4 5 6 7 8 a 0q

B/T

0.05

0.04

0.03

0.02

0.01

o 10.0K A1625Cu250Fe125 �9 20.0 K " ' "

1 2 3 4 5 6 7 8

B/T

0.016

0.014 . 4 .0K ' U " j ) '~ o 6.2K A1632Cu248Fe120 / /

0.012 �9 10.5 K . . . . / / o 13.9K / /

0.008

0.006

0.004

0.002

r 0 - 1 2 3 4 5 6 7 8

B/T

Fig. 2. Magnetoresistivities as a function of the magnetic field B with the temperatures as parameter for the three alloys considered. The full lines are fits to data as described in the text

IV. D i s c u s s i o n

We would like to divide the discussion into two parts with reference to the different T-behaviour of the Hall- constant. The first part analyses the rise at the lowest temperatures whereas the other one refers to the behav- iour at higher T's, but still below 35 K.

We attribute the rise of Rn to an anomalous contribu- tion R~, stemming from the additional paramagnetic ef- fect at low temperatures. This R~ should be proportional to the paramagnetic part of the susceptibility [5]. The magnitude of this effect and the very small saturation

magnetization points towards a minor amount of mag- netic moments being present. A rough estimate yields about 10 ppm of them.

We suppose that certain Fe-atoms are responsible for such a behaviour. Whether this is an intrinsic property of the quasicrystalline structure due to different sur- roundings of a Fe-atom or whether this is an indication of the presence of Fe-clusters is unknown. These findings support qualitatively earlier results from Klein et al. [6].

Now let us discuss the region above about 4 K. Here,

the Hall-constants show a clear l /~-behavior . It is sur-

385

1.05 ,+ , , ,

+ c a

+ca � 9

eq 1

0.95

+ +

[3 f~

+ A162.oCu25.sFela.5 o A162.sCu25.oFe12.5 �9 A163.zCu24,sFe12.o

+ +

+ +

t l /2

n ~ [ 3

0"90 ' ' ' �89 ' 3 1 . . . . 4 ;

(T/K) l/z

+

+

[3

; ' +

Fig. 3. Normalised Hall effect data against the square root of the temperature for the three alloys considered. At low temperatures a steep rise is clearly to be seen. The straight line is a fit to the

VTbehaviour as suggested by the quantum corrections

20

15

10

5 g

0

qo

-15

i - A16zsCuzs.0Fe12,5 �9 B = 0 . 5 T

%

. . . . ~ . . . . , . . . . t . i l l [ , l l l l l l l l l l l l l [ l l l

0 100 200 300 400

T/K

Fig. 4. Magnetic susceptibility for A162.5Cu25Fe12.5 as a function of temperature. The value of X is calculated from the magnetization at a field of 0.5 T

10

5

-5

-lo -15

-20

= -25

-30

-35

-40

i o o o �9

A162.5Cu25.0Fet 2.5 �9 T = 5 K

1 2 3 4 5

B/T

Fig. 5. Magnetization of the alloys A162.sCu25Fe12.5 as a function of the magnetic field

Table 1. Characteristic values for the three alloys as derived from a fit to the theory

R M S for R(T) p/#Ocm D-105 F~ Z~o.lO ta -104 m2s -1 s

AI63.zCu24.8Fel 2.o 3.45 A162.oCu25.sFe12.5 2.37 A162.sCu2s.oF%2.5 2.06

4200 3.3 0.32 2.1 8700 3.4 0.99 3.4 9600 3.6 0.73 2.6

R M S f o r A . 1 0 lz B,IO lz C-1012 AR(B) . 104 s s K 1"5 s. K 2

A163.2Cu24.sFe12.o 0.44 oo 167 355 A162.oCu2s.sFe12.5 1.93 - 1 5 . 4 48.5 - 9 0 1 A16z.sCu25.oFelz.5 2.43 - 9.7 32.5 - 3 8 6

prising that the pre-factor of this term can either be posi- tive or negative, depending on the alloy considered. Hence, small concentration shifts generate large effect in Rn. For an explanation, let us briefly consider some theoretical aspects of quantum effects in the resistivity.

The quantum corrections contain two contributions, namely the quantum interference and the Coulomb-cor- rection [2, 3]. Whereas both contribute to the resistivity p(B, T), in the Hall-constant only the Coulomb correc- tion occurs. Hence, the relevant parameter F~ can be determined therefrom. This screening constant F~ is the angular average of the electron-ion potential at the Fer- mi-level. It generates a positive ~/T contribution to the Hall-effect and also to resistivity (R (T, B = 0)) when being larger than 8/9. Hence, as can be deduced from Fig. 3, F~ becomes very large for the alloy A162Cuzs . sFe12 .5 . This is an indication of a strong electron-ion potential at the Fermi-surface. The values of F, are given in Ta- ble 1. Using them, a fit to p(B, T) was made applying the usual equations for the resistivity and the magneto- resistivity [7]. Within this procedure the absolute value of the resistivity at zero Kelvin p, the electronic diffusion coefficient D, the spin-orbit scattering time Zso and the inelastic scattering time zi(T) were considered as free parameters. Due to the low T-effect only data above 4 K were taken into account. In order to get a good fit with a mean square deviation between 1 and 4.10 - 4 w e were forced to parametrize zie(T) as:

1 1 T 3/2 T 2

zi~(T) = A + B ---~ C '

where the powers of T are derived from the electron- electron scattering process [8]. Interestingly enough, the parameter A = oo and B as well as C are positive in case of A163.2Cu24.sFe12, the alloy with the lowest resis- tivity of 4200 Ixf~cm. This result shows two physically motivated contributions to zie(T), emphasizing the strong electron-electron interaction in this alloy. The sit- uation is quite different in the other two alloys consid- ered. Here, A and C turn out to be negative. This result is physically not meaningful and can only be interpreted as a limit towards a more complicated temperature de-

3 8 6

',X

' , \ _ _ - + A16z0Cu25.sFelz5

A162.sCu25.0Fe12.5 101 ~ - , , , , . , . , - - � 9 A163.aCu24.8Fe12.0

1 0 o " . . ) -~

. '2 . . - .+

> - .'2"+...

i i i i i i I i i i

1 0 1

T/K

Fig. 6. Ineleastic scattering times zi~(T ) as a function of temperature in a double logarithmic plot for the three alloys considered

the magnetic moment. The resistivity changes dramati- cally between 4000gf~cm and 9600~tf~cm. Hence, con- centration effects influence strongly this parameter. From the discussion above it is, however, clear that this fact may not be blamed on the scattering times or elec- tronic diffusion coefficients. Hence, the only remaining parameter is the concentration of free charge carriers.

The consistency of our interpretation is shown in Fig. 7. Here, the prefactor of the Bg-dependence in the magnetoresistivity, occuring at low fields, is derived from the full fit and compared with measured data. Thus, as can be seen from Fig. 6 deriving Zie(T ) from the B2-behaviour of the magnetoresistivity alone yields no meaningful results. It is also interesting to note that the behaviour of the resistance can be understood by assum- ing an almost constant quantum interference part and a strongly varying Coulomb-contribution.

+ + u _ _ _ + Al6z0Cu/s . sFe l2 . 5

10 -2 o + . . . . . . a AI62.sCu2s.oFe12.5

�9 - - �9 A 1 6 3 . 2 C u 2 4 . s F e 1 2 . 0

1 0" 3 " Q . e ~ ~ - " , , , ~ +

10a ~ - g + 10-5 , , . . . . ~ ,

100 101

T/K

Fig. 7. Prefactor of the B2-behaviour for low fields as derived from the full fit and measured values as a function of temperature

pendence of z~e(T). Inspection of Fig. 6 shows this fact. Whereas for A16a.2Cuz4.sFelz the behaviour of z~e(T) can be roughly described by power law Zie = %eo T - v with p= 1.81, this is no longer true for the other two alloys. In this case a positive curvature is to be seen pointing towards a special behaviour of zie(T). We will comment on this later. The picture emerging from this discussion is, that the quantum interference part is nearly indepen- dent of the concentration whereas the Coulomb part shows a dramatic variation.

The numbers given in Table 1 indicate an electronic diffusion constant which doesn't vary strongly with alloy composition. It is around 3.4.10 - 5 m2/s, which is small as compared to the values obtained from amorphous alloys of roughly 3.10 .4 m2/s. The zle-data feature an inelastic scattering time at 10 K of around 1 ps, also not very much dependent on the alloy composition. The spin-orbit scattering time does vary between 0.21 and 0.34 ps. But, this is not considered to to be significant. Such a short spin-orbit time is characteristic for d-levels being close to Ev. We will argue that these belong to d-band of Fe, being shifted below EF, thus quenching

V. Conclusions

The high value of F, and the strong electron-electron scattering as well as the high resistivities indicate strong electron-electron and electron-ion interactions. If we as- sume a model of Hume-Rothery type, where the Fermi- sphere touches the Pseudo-Jones-Zone, we can under- stand both. Strong eleastic scattering will be present, yielding dramatic electron-ion interactions. In addition, the valence band will be filled up or almost filled up and the conduction band will be empty or almost so. Hence, the number of free carriers will be reduced dra- matically, producing weak electron-electron shielding and therefore, strong electron-electron interactions. Moreover, the resistivity will be high due to the reduced number of mobile charge carriers. In addition, we suggest that the strange curvature in the Z~e(T)-curves observed for the two high resistivity alloys may be caused by the tendency of the electrons to get localized near the gap. The strong dependence on the chemical composition can now also be explained within this model. The delicate balance of almost filled and almost empty band is dis- torted easily by adding atoms with less or more than average number of electrons per atom. This average number should be around 1.8. Also the defect structure should be of great influence. The Hume-Rothery interac- tion generates a pseudo-gap in the density of states at the Fermi-surface. The position and width of the gap depends on the direction in reciprocal space. The gaps should be small, around several tens of a meV [9]. Hence, we expect more the behaviour of a semi-metal than that of a semiconductor. Nevertheless, electronic states in the gaps produced by defects may mask the true behaviour of the quasicrystal. Results from anneal- ing of quasicrystals at temperatures above 810 ~ for prolonged times support this point of view. Hence, we may state, the more perfect the quasicrystal, the higher is the resistivity at low temperatures and the stronger is its temperature dependence.

The arguments derived from the quantum corrections favor the Hume-Rothery model of a perfect quasicrystal. The effect should stabilize this phase due to a gain in

387

electronic energy. The possibil i ty of filling and shifting d -bands close to EF al lows big electronic energy gains and m a y explain why quasicrystals with d-electron com- ponen t s show stable regions in the phase d iagrams.

The authors would like to thank DFG for its support under con- tract number Fr 529/3-2. We also would like to thank P. Lindqvist for valuable discussions.

References

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Plenum Press 1972 6. Klein, T., Berger, C., Mayou, D., Cyrot-Lackmann, F.: Phys.

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