J. Phys. Chem. 1990,94, 1123-1 126
1123
Pressure and Concentration Dependence of the Quantum Corrections to the Resistivity in CuTi Alloyst W. Dyckhoff, G. Fritsch,*vi and E. Lhscher Physik- Department, Technische Universitat, Miinchen, 0 - 8 0 4 6 Garching, FRG (Received: June 9, 1989; In Final Form: September 12, 1989) We present a full set of pressure- and temperature-dependent resistivity data for the regions 0 < P < 10 GPa, 1.4 < T < 30 K, and 43 < x < 88. Here, x denotes the concentration of the amorphous alloy Cu,Tilwx. These results are analyzed in terms of a temperature-independent Boltzmann resistivity and additive quantum corrections,such as the quantum interference and the Coulomb correction. It is shown how the various parameters involved in these contributions vary with pressure. Certain arguments are given in an effort to understand qualitatively the observed behavior.
Introduction Quantum corrections to the classical resistivity and to the Hall effect of amorphous alloys are now well established in the three-dimensional This statement is based on an analysis of the temperature and magnetic field dependence of those quantities. However, in order to obtain a consistent picture, the volume dependence should also be in accordance with the relevant theory. Therefore, we would like to present the first systematic interpretation of this effect as a function of volume, temperature, and concentration. The quantum corrections introduce specific structures into the resistivity and the Hall effect in a temperature range below 30 K. Whereas in the classical regime the mean free path is much larger than the mean atomic distance, this is no longer true for amorphous alloys. Due to disorder and to the enhanced elastic scattering when d-electrons are present at the Fermi surface (a situation relevant for CuTi alloys), the elastic mean free path is reduced to the order of the mean atomic distance. However, it is important to realize that the coherence length of the electronic wave function is not destroyed by elastic scattering. Hence, it is much larger than the elastic mean free path and finally determined by inelastic, spin-orbit or magnetic scattering. From these facts the picture of a diffusing electron emerges, which can interfere with itself, because there is a finite probability that the diffusion path will cross itself. The resulting changes in resistivity and Hall effect are called quantum corrections. We observe two main contributions, the quantum interference effect, which is governed by the inelastic (ri) as well as the spin-orbit (T,) scattering times and the Coulomb interaction effect which expresses the fact that the electronic screening is reduced due to the diffusive electronic motion. The second part, therefore, contains the screening constant F and the electronic diffusion constant 0.4 Here, we have assumed that no magnetic impurities are present. In this paper we present resistivity data for CuxTilmx alloys in the concentration range 43 < x < 88 as a function of volume and temperature. The experimental ranges are for the pressure 0 < P < 10 GPa and for the_temperature 1.4 < T < 30 K . The relevant parameters ri. r,, F,and D are extracted by fitting the data to the theory and discussed in terms of their volume, temperature, and concentration dependencies. Some Theory
The quantum corrections to the resistivity can be written as follows: (i) Quantum interference effect
'Dedicated to the 70th birthday of Prof. Dr. H. Drickamer. Fak. BauV/Il/Physik, Universitiit der Bundeswehr Miinchen, D-8014 Neubiberg, FRG.
*
0022-3654/90/2094-1 123$02.50/0
where p is the resistivity at T = 0, Lpo = e 2 / ( 2 a z h ) and , e and h have the usual meanings. The fields B, correspond to the scattering times via
with D the electronic diffusion constant. (ii) Coulomb interaction effect A p c / p = -0.65pLo0[4/3 - ( 3 / 2 ) F j [ k ~ T / ( 2 h D ) ] '( /3~) Here E denotes the screening constant. Sometimes other terms are included in the factor [ 4 / 3 - ( 3 / 2 ) n . l However, since these contributions are not well understood, they are omitted in this analysis. Furthermore, we assume that the Boltzmann resistivity is a constant in the temperature range considered and adds simply to the quantum corrections. Since the latter effects are in the range 10-3p, the second assumption produces no problems, whereas the first one may well be responsible for some scatter in the parameters extracted. The two contributions mentioned above (eq 1 and 3 ) produce a very distinct structure at temperatures below 20 K as long as superconductivity can be set aside. Whereas the Coulomb interaction shows a behavior, the quantum interference part changes sign due to the spin-orbit effect. Assuming T~ = rioTpfor the inelastic scattering time with p = 3, appropriate for the electron-phonon interaction: we get a behavior as depicted in Figure 1 . At very low T the Coulomb interaction dominates, then the spin-orbit part (weak antilocalization) shows up with a maximum -B,I12, and finally the decay of the weak localization due to ri dominates. The maximum ( T m x )and zero crossing (To)temperatures are related by T,,,,,/To = ( 9 / 2 ) - l / p = 0.61 for p = 3 (4) The electronic diffusion constant D can be derived from (5) pDe2N(EF)= 1 where N(EF)describes the density of states of the Fermi surface. N ( E F )may be calculated, for instance, from specific heat data with the help of a correction due to electron-phonon enhancement. The spin-orbit scattering time T~~ depends on the density of states of d-electrons at the Fermi surface N d ( E F )and on the spin-orbit coupling strength. Since in amorphous alloys d- and s-electrons are hybridized strongly just at the Fermi surface,' this quantity is not easily attainable. However, T~~ is independent of temperature. ~~~~~~~~~~~
~~~
(1) Hickey, B. J.; Greig, D.; Howson, M. A. J . Phys. 1986, F26, L13. (2) Schulte, A.; Fritsch, G. J . Phys. 1986, F26, L55. (3) Fritsch, G.; Schulte, A.; Liischer, E. In Amorphous and Liquid Materials, Lhscher, E., Fritsch, G., Jacucci, G., Eds.; Nijhoff Publishers: Dordrecht, 1987; p 368. (4) Lee, P. A.; Ramakrishnan, T. V. Phys. Reu. 1982, B26, 4009; Reu. Mod. Phys. 1985, 57, 287. ( 5 ) Fritsch, G.; Lobl, P.; Liischer, E. In Rapidly Quenched Metals; Steeb, S.,Warlimont, H., Eds.; Elsevier: Amsterdam, 1985; p 1027. ( 6 ) Bergmann, G. Phys. Rev. 1984, 107, 1 . (7) Morgan, G. J.; Weir, G. F. Philos. Mag. 1983, 847, 167.
0 1990 American Chemical Society
1124 The Journal of Physical Chemistry, Vol. 94, No. 3, 1990
Dyckhoff et al.
Q I -JB,d
0 o
A
uLOT 6 0 0 I MPa L 5 GPa 9.8 GPa
-1
'\
Figure 1. Quantum corrections to the resistivity Ap/p: I , quantum
interference effect; T,,,,,, maximum temperature; To,zero crossing temperature; C,Coulomb interaction effect; and A p / p , total quantum correction. TABLE I: Characterization of the Cu,TiIMz Samples Examined'
nominal
measd
concn concn xlatom % x/atom % productn 40 45 50 60 65 70 80
43 48 53 62 64
70 88
ms ms ms ms ms ms sp
pb//.ln
cm 192 200 189 186 183
170 115
Ct e
x
x
K T ~
IO3fGPa-I 1O3/GPa-I -4.0 -2.2 -2.0 -0.1
+1.1 +2.5 +2.2
8.7 8.6 8.5 8.4 8.3 8.2 7.9
ms, melt-spun; sp, sputtered. The experimental concentrations are precise to 2 atom%, the resistivity as well as ap to about 10%. The compressibility values K~ are calculated from averaged elemental values. This procedure has been shown to give results accurate to about Resistivity. 'Pressure coefficient at 100 K. dCompressibility.
I
I
10
20
Temperature T / K Figure 2. Pressure variation of the relative resistivity Ap/p below 30 K for CueTim The data are normalized in the range 20-30 K. Only part of the experimental data is shown for clarity. A 0
Cu65T i35 010 7 G P a A 9.4 G P a 0 6 2 G P a
n
The pressure dependence of these quantities will be analyzed in the discussion.
Experiment The apparatus used and the experimental procedures applied have been described in detail elsewhere.* The pressure setup is of Bridgman type. It consists of two opposed anvils made of sintered alumina. The pressure cell is sealed by stainless steel and pyrophilite gaskets. The pressure is determined in situ, by measuring T, of Pb, to a relative accuracy of 5%. The temperature T can be varied by inserting the device into a He4 cryostat applying different exchange gas pressures. The accuracy in the T determination was 0.1 K, applying calibrated carbon-glass resistors. The samples were produced by melt spinning and by sputtering. Their amorphous nature was checked by X-ray scattering and for concentration by plasma ion analysis. Current and potential leads were Cu wires pressed against the samples. The width of the ribbons amounted to about 0.5 mm and the thickness to about 20-40 gm in case of the melt spun samples. The sputtered ones showed a thickness of 10 pm. Zero-pressure measurements were performed (not in all cases, see Figures 2-4) with larger specimens (area 20 X 3 mmz) in a different sample holder. The current applied was limited to 2 mA. The data obtained in the full T range up to 300 K have been published el~ewhere.~Some aspects of the low T work was discussed previously.I0 Table I shows a summary of the samples examined, together with some characteristic data. Results and Discussion The data obtained should in principle be corrected with a pressure- and temperature-dependent geometrical factor in order At low temperatures, Le., T C 30 K , to deduce the resi~tivity.~ (8) Willer, J.; Moser, J. J . Phys. E . Sc. Instrum. 1979, 12, 886. (9) Dyckhoff, W.; Fritsch, G.; Liischer, E. In Materials under Extreme Conditions: Ahlborn, A., Frederiksson, H., Liischer, E., Eds.; Les Editions de Physique: Les Uli, France, 1985; p 139. (10) Dyckhoff, W.; Fritsch, G.: Ltischer, E. 2. Phys. Chem. 1988, 157, 723.
0
A
v
0"
vaoo
0
t
0
I
I
10
20
30
Temperature T/K Figure 3. The same as Figure 2 for Cu,,Ti35. its temperature dependence is almost zero, because the thermal expansion coefficient5 disappears and the isothermal compressibility K~ is a constant. In addition K~ is taken to be independent of pressure. Such an assumption is a good approximation in the case of alloys containing d-electrons where the compressibility is small. This conclusion is supported by the fact that the pressure dependence of the resistivity turns out to be linear9 at least in the range up to 10 GPa. In order to get the P dependence of the quantum corrections, the resistivities were shifted with respect to the lowest P data set and normalized with respect to an averaged resistivity in the T
Resistivity in CuTi Alloys
The Journal of Physical Chemistry, Vol. 94, No. 3, 1990 1125
TABLE II: Some Parameters Used in the Fits' cu80 Ti20 0 9 3 GPa 0 A
40 45 50 60 65 70 80
3 3 GPa
20
10
N(EF)/
atom % atom
5 8 GPa
~
0
re1 fit
XI
30
Temperature T / K Figure 4. The same as Figure 2 for CuB0Ti2,.
105/m2s-I B,/T 2.4 0.6 2.4 2.7 1.8 3.0 5.7 3.2 6.9 3.7' 5 7.4' 5
eV-'
%-I
Bio X quality 104/T K-' 105a
DX
1.9 1.8 1.7
1.5 1.5 1.5*
0.9*
1.6 1.2 2.1 4.0 4.5 2.9 3.1
3
2.5 2.3 1.6 2 1.2 1.4
'The density of states N(E,) is derived from specific heat data" corrected with the electron-phonon coupling constant.l* (*, extrapolated). D is calculated from the resistivity p and from N(EF). B, indicates the spin-orbit field used for the fits E , # B,(P) and Bio describes the constant used together with p = 3 for the fits Bio # BioV s. (P).According to eq 2, B,s,D = 1.64 X
.-
-----e;::---
-::------a
- 4--------- -&
-----A
I
i o
I
Pressure PIGPa
Pressure P/GPa
Figure 5. Results for the fits B, # B, P);see also Table 11. (a) Pressure dependence of the screening constant (b) Pressure dependence of the inelastic field Bi at 20 K. (0)Cu,Ti,, (V)CuSOTiSO, (A)CuaTi,, (0)
6.
Cu6,Ti3,,and (0)CU,~T~~,,. range 20-30 K, since in this range the quantum contributions are already very small. Therefore, the absolute magnitude of the compressibility and the pressure dependence at high temperatures are eliminated in the A p / p data. The most extreme cases are shown in Figures 2-4. The Pvariation of the Boltzmann resistivity has been analyzed in detail el~ewhere.~ Here, only the overall pressure coefficient of the resistivity ap is included in Table I. An inspection of the data reveals that the position of the maximum in Ap,/p relative to its zero (see Figure 1) approximately remains independent of pressure. Hence, the exponent p characterizing the Tvariation of the inelastic scattering time riremains constant according to eq 4. This fact can be understood easily since p depends only on the dominant scattering mechanism which should no_t change with P in this case. The remaining parameters Bio, B,, F, and D in eq 1 and 3 do not allow a unique fit to the data. For that reason, we took D from the literature by using density of states data. The latter were derived from specific heat measurementsi1and corrected for electron-phonon enhancementsi2 To a first approximation D should remain nearly independent of P. In order to show this, we write for the electronic diffusion constant
0
n wO
I
I
4
8
12
Pressure P/GPa 0
I
-
1
0
\
c U
2 where IeI,,, denotes the elastic mean free path given to a good approximation by the mean atomic distance a and vFis the Fermi velocity. Assuming an extended electronic wave function, i.e., the phase coherence length is much larger than Ielas,-otherwise the quantum corrections will disappear-we have VF
= hkF/m = ( h / m ) ( 3 T 2 n ) ' / 3
(7)
-
with n the electron density, m the electron mass, and kF the V i(V, volume) and a modulus of the Fermi vector. Since n V I 3 ,D should not vary with volume; Le., it should be independent of pressure. Using this result the P variation of the fields
-
(11) Moody,D. E.; Ng, T. K. Physicu B+C 1984, 126, 371. (1 2) Frisch, G.; Dyckhoff, W.; Pollich, W.; Zottmann, W.; Liischer, E. Z. Phys. 1985, 859, 27.
Figure 6. Results for the fits Bio # Bio(P);see also Table 11. (a, top) Pressure dependence of the screening constant ?t (b, bottom) Pressure dependence of the spin-orbit field B,. Symbols as Figure 5 and (V)
C ~ , O T ~( 0~ )OC,U ~ S T ~ W B,, mirror inversely that of the corresponding scattering times
7,
(eq 2). Fits of the data to eq 1 and 3 applying those P-independent D values still produce a lot of scatter in the parameters. Therefore, we decided to introduce additional conditions. At first, it was assumed that B, # B,(P), incorporating suitably averaged values for B,. The results obtained are shown in Figure 5; the parameters are summarized in Table 11. This procedure was repeated for Bio # Bio(P)again using some averages. These results are plotted in Figure 6 and listed in Table 11. Comparing both efforts we
The Journal of Physical Chemistry, Vol. 94, No. 3, 1990
Dyckhoff et al.
can draw some conclusions, discussed in the following. The quantity Biodoes not depend on pressure; hence, ri = q0TP, the inelastic scattering time, should only be a function of concentration. With respect to the errors involved in q0(;=20%), we only state an increase of this quantity with rising Ti content (Table 11). This point of view is also supported by the fact that the electron-phonon coupling constant X exhibits only a minor P dependence.I3 Now, let us analyze the screening parameter P. Figures 5a and 6a reveal a decrease with P,independent of the fitting prozedure applied. The relative variation is the smaller the larger F turns out to be. F gets larger with increasing Cu content. In order to get an idea how such a behavior may be understood, the predictions of the NFE model will be examined.I4 Here, we have F = [In (1 x2)]/x2 with x = 2kF l,, where 1, denotes the screening length. Putting I , kF-I/* yields
up also in the variation of the pressure coefficient of the resistivity (Table I). Since one set of fits was made for B, # B,,(P) and the other one for Bio # Bio(P)yielding dB,/dP > 0, we conclude that the real P dependence of B,, should lie in between. Hence, experimentally B, must increase with pressure. Again a discussion of this effect has to consider the P variation of Nd(EF) and of the spin-orbit coupling parameter. The P dependence of Nd(I??F)can be derived from eq 5 by setting D = constant and using the pressure dependence of the Boltzmann resistivity p, given in Table I. This procedure yields
1126
+
-
(i/F) @ l a p
= (K,/3){[E(i + x2)]-I - I )
20 atom %).Is An inspection of Table I exhibits a decreasing resistivity with rising pressure for CumTia (ap< 0). The pressure coefficient apchanges sign at about xcu = 65 atom %. Therefore, we expect the same behavior for 7,;' and for B,, as long as the spin-orbit coupling constant does not vary with pressure. The experiment shows the predicted increase with P and also gives the tendency toward the correct concentration dependence. However, again the magnitude is much too small and the P coefficient of B,, never gets negative. Therefore, we have to conclude that the spin-orbit coupling.strength should also increase with P to some extent.
Conclusion A full set of low-temperature data for the electrical resistivity of amorphous CuTi alloys has been presented and discussed as a function of volume, temperature, and concentration for the first time. The pressure variation of the quantum corrections to the resistivity of amorphous d alloys can be understood qualitatively within the theories given for these effects. The screening parameter F decreases with rising P mainly due to d-electron effects. The inelastic scattering time ri= r i 0 Premains independent of P, since the scattering mechanism (electron-phonon) does not change and the coupling constant X is almost independent of P. Finally, the ~spin-orbit ~ scattering time rW decreases with rising pressure. This behavior may be discussed in terms of the density of states and the spin-orbit coupling constant. The latter quantity should increase with P and decrease with the Ti content of the alloy CuTi. Acknowledgment. We thank Prof. M. Kalvius, TU Munich, for permission to use his pressure equipment.
