RESISTIVITY MINIMA IN BULK DISORDERED ALLOYS A. K. Majumdar

1. RESISTIVITY MINIMA IN BULK DISORDERED ALLOYS A. K. Majumdar S. N. Bose National Centre for Basic Sciences, Kolkata & Physics Dept., I. I. T. Kanpur, India Collaborators:S. Chakraborty, T. K. Nath & A. Das (IIT Kanpur) Investigated: A. Amorphous Co-rich magnetic alloys. B. Amorphous Ni/Cu-Zr-Al non-magnetic alloys. C. Crystalline Cu100-xMnx, (36<x<83) cluster-glass/antiferromagnetic alloys. D. CrystallineNi – rich Ni-Fe-Cr ferromagnetic permalloys. High-resolution resistivity data down to 1.2 K, Hall effect and magnetoresistance till 1.4 K and up to 7.5 tesla.

2. Disordered metallic alloys Dilute limit: kF l >> 1, Boltzmann transport holds A. For amorphous alloys: (T) = (T/D)2 I(x), I(x) = Debye integral, x = D/T = 0 + T2, for T << D = 0 + T. for T > D mag(T) ~ T3/2 (higher order T2). B. For crystalline alloys: mag ~ T2 (e-m), ~ T3/2 (SG/CG). In both the above cases  increases with T. INTRODUCTION Eq. (1a) Eq. (1b)

3. Weak-disorder limit, kF l > 1 (Ioffe-Regel criterion) Boltzmann transport breaks down Beyond the Boltzmann Picture Short  => (a) Weak localization Anderson (1958); Abrahams et al. (1979); Altshuler and Aronov (1985) (b) Interaction effect or Coulomb anomaly Reviews: a) Lee and Ramakrishnan, Rev. Mod. Phys. 57, 287 (1985). b) Dugdale, Contemp. Phys. 28, 547 (1987). INTRODUCTION (a) Localization:  = e2 D N(EF) W = | i i |2= | 1 |2 + | 2 |2 + 1* 2 + 1 2* Dephasing: (i) Inelastic scattering at T > 0 (ii) Magnetic Field P(r,t) dr dt = 1/(4Dt)3/2 exp(-r2/4Dt) dr dt

4. 0 = elastic mean free time, i = inelastic mean free time  ~ [(0)-1/2 - (i)-1/2] 0 = 10-15 - 10-16 s For amorphous materials at T << D Electron – phonon scattering => 1/i ~ T2 =>  ~ T. Eq. (2) i = (1010 T2)-1 => i 10-12 s at 10 K Expts. by Howson and Greig: Non-magnetic CuTi/Hf/Zr glasses for 20 < T < 100 K. For T > D , e-ph. scattering => 1/i ~ T =>  ~ T1/2. Eq. (3) Howson and Greig: CuTi, etc., T > 100 K. B = magnetic field induced dephasing time = ћ/4eBD ( 10-12 – 10-13 s for B = 1T) Note: Phase change = e/ћ,  ~ [(0)-1/2 - (B)-1/2] =>  ~ + B1/2 => negative magnetoresistance. INTRODUCTION

5. (b) e-e interaction effect: Energy difference is a source of dephasing Thermal coherence time T ~ h/kBT  10-13 s at 10 K  ~ [(0)-1/2 – (T)-1/2] =>  ~ T1/2. Eq. (4) Expts. by Howson and Greig : CuTi & CuHf , for T < 20 K. Bergmann (1984) gave extensive experimental evidence of both localization & e-e interaction effects in thin films (2-D). INTRODUCTION

6. INTRODUCTION Weak scattering limit for 3d (bulk) materials: (T) = 0 + mT, where D = diffusion constant, F = screening factor. Strong scattering case: McMillan’s scaling theory of metal-insulator transition: (T) = 0[1 + (T/)1/2], Eq. (6)  = correlation gap. Eqs. (5) & (6) => same form (~ T ) as in Eq. (4) Observed only at the lowest temperatures where no other faster dephasing process is present. Eq. (5) MR is + ive due to e-e interaction ~ B2 for low fields & ~ B1/2 for high fields.

7. So (T) varies in a sequence of T, T and T at low, intermediate and high temperatures. => In expts, only two regions (T & T or T & T) observed in the same sample. INTRODUCTION

8. Resistivity Fe5Co50Ni17-xCrx(BSi)28, x = 0, 5, 10, 15, Mn17(A1 - A5). Measured: ac(T), Mdc(H,T), MR = / (H,T) & (T) using closed-cycle helium refrigerator. Found: 300 = (150 - 254) cm, Tc = (395-180) K, Tmin = 20 K to > 300 K. A1-A4(FM), A5(Mixed FM/SG state for T < 50 K). Depth of minima = 0. 1 to 4.4 %. A. Amorphous Co-rich magnetic alloys

9. Figs. 1 - 2 :r(T)= R(T)/R(30O K) vs. T (raw data). Resolution(/): 1 in 105 Temperature stability: 0.1 K. A. Amorphous Co-rich magnetic alloys Fig. 2 Fig. 1

10. Figs. 3 - 4 : In () vs. In T (again raw data). All three regions observed in each sample Till now only a theoretical prediction Fits to (Eq. (5)) give 2 = 10-10, slope m, D, N(EF), etc. Interpretation questionable due to spontaneous moments Look for cleaner non-magnetic systems. A. Amorphous Co-rich magnetic alloys Fig. 3 Fig. 4 A. Das & A. K. Majumdar, PRB 43, 6042 (1991), PRB (1993), JMMM (1993), JAP(1991).

11. Resistivity (Ni0.5Zr0.5)1-x Alx, x = 0, 0.05, 0.1, 0.15 and 0.2 300 K = (190 – 220) cm, Tcrys. 800 K superconducting with Tc< 1.25 K, D 320 K  independent of temperature (Pauli paramagnet)  80 X 10-6 emu/mole. (Cu0.36Zr0.64)1-x Alx,x = 0, 0.1, 0.15 and 0.2 300 K = (169 – 185) cm, Tcrys. 710 K SC, Tc< 1.7 K, D 220 K,  ≠ f(T)  80 X 10-6 emu/mole. B. Amorphous Ni/Cu-Zr-Al non-magnetic alloys

12. Fig. 5:(T) for Cu-Zr-Al till 300 K, (300 K) increases with Al(inset). Fig. 6: (T) for Ni-Zr-Al from 1.2 – 10 K, Tc(onset) < 2.5 K and is suppressed by disorder (Al). Inset of Fig. 6: Excellent fit to EEI theory (Eq.(5)) shown for x = 0.05 but superconducting fluctuations dominate below 4 K. B. Amorphous Ni/Cu-Zr-Al non-magnetic alloys Fig. 5 Fig. 6

13. Fig. 7:Ni-Zr-Al and (CuZr)0.8Al0.2. e-e interaction effect, Tc<T<15 K (T) = (0) + mT Eq. (7) m = (360 - 460) (m)-1K-1/2 500 (m)-1K-1/2 (near universal value) 2 (fit to Eq. (7)  5 X 10-11, stability of T  10 mK, resolution of / 5ppm m ~ 1/D, D = (3.5 –5.5) X 10-5 m2/s, slopes of Fig. 7 indicate D ~ 1/(300 K) N(EF) ~ 1.0 (atom eV)-1. B. Amorphous Ni/Cu-Zr-Al non-magnetic alloys

14. Figs. 8 & 9: D/10 < T < D/3.  ~ T. Eq. (8) B. Amorphous Ni/Cu-Zr-Al non-magnetic alloys Fig. 8 Fig. 9

15. Figs. 10 & 11 : D/3 < T < D. ~ T Eq. (9) 2 10-10, departure above D. Inelastic mean free path li(T) = 5 X 10-4 T-2 m from Eq. (8)& = 2 X 10-6 T-1 m from Eq. (9) and = 2 2((e2/(h)) li(T)-1/2 l0-1/2. Inelastic scattering time i calculated from L = (e2/(h))(Di)-1/2. Eq. (10) i  10-12 – 10-14 s at higher T, 0 10-15 – 10-16 s => i >> 0 justifies localization. B. Amorphous Ni/Cu-Zr-Al non-magnetic alloys Fig. 11 Fig. 10

16. Fig. 12 & 13: ln  vs. ln T. First convincing observation of 3 regions of QIE T. K. Nath and A. K. Majumdar, PRB 55, 5554(1997), IJMPB (1998). B. Amorphous Ni/Cu-Zr-Al non-magnetic alloys Fig. 12 Fig. 13

17. B. Amorphous Ni/Cu-Zr-Al non-magnetic alloys • Magnetoresistance • (Ni0.5Zr0.5)1-x Alx, x = 0, 0.05, 0.1, 0.15 and 0.2 Already seen that for Tc < T < 15 K,  ~ T (e – e interaction). Measured MR from 2 to 20 K up to 7 T using PPMS. Found that MR is positive and very small even at 2 K & 7 T (< 0.12 %). Due to e – e interaction MR is predicted to be +ive.

18. B. Amorphous Ni/Cu-Zr-Al non-magnetic alloys Eq. (11) Eq. (12)

19. (12) (11) (14) (13) B. Amorphous Ni/Cu-Zr-Al non-magnetic alloys Eq. (13) Eq. (13) Eq. (14) Eq. (14)

20. B. Amorphous Ni/Cu-Zr-Al non-magnetic alloys Eq. (15) Eq. (16)

21. B. Amorphous Ni/Cu-Zr-Al non-magnetic alloys • Thus we find that • (S - S) & orbital contributions are indistinguishable in the low-field limit. • But in the high-field limit the (S - S) is strongly temperature dependent while the orbital is weakly temperature dependent. • Also MR due to weak localization is – ive for small S – O interaction and + ive for large S – O interaction but with a weak temperature dependence.

22. B. Amorphous Ni/Cu-Zr-Al non-magnetic alloys • Fig. 14 : MR vs. external fields between 4 and 20 K. • Positive, small (< 0.12 %), & strongly temperature dependent.

23. B. Amorphous Ni/Cu-Zr-Al non-magnetic alloys • Fig. 15: Longitudinal and transverse MR vs. external fields at 5K. • Isotropic, consistent with QIE.

24. B. Amorphous Ni/Cu-Zr-Al non-magnetic alloys • Fig. 16:MR vs. external fields data and fit to H (Eq. 15a). • Excellent fits with R2 = 0.998 & 2 consistent with exptal. resolution • Strongly temperature dependent +ive MR implying • dominance of (s – s) contribution to e – e interaction.

25. B. Amorphous Ni/Cu-Zr-Al non-magnetic alloys • Fig. 17: MR vs. T at high fields and fit to - T (Eq. 15a). • Dominance of (s – s) contribution. • A. K. Majumdar, J. Magn. Magn. Mater. 263, 26(2003).

26. Resistivity Earlier work: B. R. Coles, Physica (1977) For x << 1 %, Kondo minimum with (T) ~ In T. For 0.1 < x < 15 %, SG (canonical), no resistivity minimum. For x > 35 %, minima around 20 K of depth < 1 %, but no quantitative analysis. C. Crystalline Cu100-xMnx (36 < x < 83) CG/AF alloys

27. Fig. 18: Magnetic phase diagram of Cu100-xMnx. Fig. 19: (T) data of present work : Resolution in / better than 10 ppm, data every 30 mK. x= 36, 60, 73, 76, and 83; Tmln(K) = 2.5, 16.5, 16.5, 24, 13.5; 0(cm) = 93, 176, 184, 196, 120. Depth < 1/3 %, Tmin correlate with 0and not x. T < Tmin/3:(T) = (0) + A ln T, 2 ~ 10-9 . Eq. (17) = (0) + m T1/2, 2 ~ 10-10 (  Expt. resolution) Eq. (7) Deviations random for (7) & systematic for (17). C. Crystalline Cu100-xMnx (36 < x < 83) CG/AF alloys X=83 X=36 Fig. 19 Fig. 18

28. Support for e-e interaction effect: (a)m = 480, 680, 620 and 560 (m)-1K-1/2. (b) = e2DN(EF), m ~ D-1/2 N(EF) = (1.4 – 2.6) X 1035 erg-1cm-3 Specific heat data :  ~ N(EF) = 2.2 X 1035 erg-1cm-3 Tmin/3 < T < 30 K: (T) = 0 - mT1/2 + T3/2 + Bloch-Gruneissen Eq. (18)    e-e CG lattice Excellent fits, 2 ~ 10-10. C. Crystalline Cu100-xMnx (36 < x < 83) CG/AF alloys

29. 2 K to Tmin/3  = 0 + mT Tmin/3 to 30 K x (Tmin) m 2(10-10) m B(10-3) A 2(10-10) 0(cm) Depth (%) 36 (2.5 K) - - -0.06 5.5 77.2 4.7 93 0.04 60 (16.5 K) - 0.15 0.4 - 0.25 5.2 9.9 0.9 176 0.18 73 (16.5 K) - 0.23 1.7 - 0.40 8.4 16.7 0.4 184 0.26 76 (24 K) - 0.24 2.0 - 0.35 4.2 81 0.3 196 0.33 83 (13.5 K) - 0.08 2.5 - 0.15 4.0 386 3.8 120 0.14 C. Crystalline Cu100-xMnx (36 < x < 83) CG/AF alloys

30. Fig. 20: [(T) - 0] vs. T for each term, their total and the raw data. Note: The change is only 1 in 200  cm for x = 76 with Tm = 24 K. A. Banerjee & A. K. Majumdar, Phys. Rev. B 46, 8958 (1992). S. Chakraborty & A. K. Majumdar, Phys. Rev. B 53, 6235 (1996). C. Crystalline Cu100-xMnx (36 < x < 83) CG/AF alloys

31. Magnetoresistance TMR & LMR measured at 4.2, 20.5 and 63 K till 7.5 tesla. Found: (i) TMR & LMR are < 0.2 % except for x = 83 and are qualitatively the same. (ii) Mn-rich alloys (x >60): positive MR, x = 36: positive MR till 3 tesla and negative beyond. (iii) x > 60: High-field:/ = A + BH1/2 + C1H2 for h >> 1, Eq. (19) where h = gBH/kBT,  = (e2F/42ħ)(kBT/2Dħ)1/2, A = - 1.3 , B = (gB/kBT)1/2and C1 = (1/2ne0)2 The first two terms are due to e-e interaction and the third term is the normal MR. Also at low fields:/ = CH2 + C1H2 for h <<1, Eq. (20) where C = 0.053 (gB/kBT)2 is due to e-e interaction. C. Crystalline Cu100-xMnx (36 < x < 83) CG/AF alloys

32. Figs. 21 & 22:TMR (/) vs. H (7.5 & 2 tesla) at 4.2 K. Good fits to Eq. (19) and (20). A is negative & B ~ temperature independent (orbital). C. Crystalline Cu100-xMnx (36 < x < 83) CG/AF alloys Fig. 21 Fig. 22

33. Fig. 23: Normal magnetoresistance /0 vs. H/0. It satisfies Kohler's rule till 5 tesla. (iv) x = 36:Additional cluster-glass contribution of - H2 added to Eqs. (19) & (20). It decreases with increase of temperature as in canonical spin glasses. Interpretation of / is consistent with that of (T). S. Chakraborty et al., Int. J. Mod. Phys. B 12, 2263 (1998). C. Crystalline Cu100-xMnx (36 < x < 83) CG/AF alloys

34. D. Ni – rich Ni-Fe-Cr ternary alloys • Fig. 24 : Ternary composition diagram showing: • Ni-rich region (Permalloy) with large µ; s & Rs changing • sign, Constant FAR ridges following Rs 0 line. • Fe – rich (Stainless/heat-resistant alloys) showing exotic magnetic phases. • Alloys in both regions show  minima. Typical  values are  100 µΩ cm.

35. D. Ni – rich Ni-Fe-Cr ternary alloys Resistivity • Fig. 25: (T)/min vs. T for some Ni-rich Ni-Fe-Cr alloys. • S47: 71-8-21, S41: 74-8-18,S34: 73-13-14, S29: 75-13-12 • All show minima at Tmin = 22, 27, 35.5, & 14 K, respectively. • Data every 50 mK below Tmin.

36. D. Ni – rich Ni-Fe-Cr ternary alloys • For 1.2 K < T < Tmin/2 • Very dilute alloys show Kondo minima given by • (T) =0 – m ln T. Eq. (21) • Tm & DOM depend on impurity concentration. • Minima disappear in magnetic fields. • Two level systems also show logarithmic behavior • Spin fluctuations in dilute alloys give • (T) = 0[1 – (T/Tk)2]. Eq. (22) • e - e interaction: (T) = 0 – m’ T. Eq. (23)

37. D. Ni – rich Ni-Fe-Cr ternary alloys • Fig. 26: Deviations from fits to Eqs. (21) & (23). • Random & lower deviations for e – e interaction and systematic & much higher for Kondo-like behaviour. S. Chakraborty & A. K. Majumdar, J. Magn. Magn. Mater. 186, 357 (1998).

38. D. Ni – rich Ni-Fe-Cr ternary alloys Hall resistivity ρH = Ey / jx = R0Bz+ 0RSMS , In ferromagnetic metals and alloys where R0= Ordinary Hall constant, RS= Extra-ordinary or spontaneous Hall constant, B = magnetic induction, and MS = saturation magnetization. Rs~ 2,  = Ohmic resistivity  Minimum in Rs is expected. Ni-Fe-Cr:75-13-12, Tmin = 14 K Ni-Fe-Cr:70-12-18, Tmin = 22 K Fig. 27: Hvs. B for two of them.

39. D. Ni – rich Ni-Fe-Cr ternary alloys Fig. 28 : Rs also show minima since it scales as . Sample 3: 74-8-18, Tmin = 27 K. S. Chakraborty & A. K. Majumdar, Phys. Rev. B 57, 11850 (1998).

40. In 3-d alloys the interpretation of the magnetotransport data at low temperatures in terms of Quantum Interference Effects (QIE) is independent of the nature of disorder, viz, structural or compositional. CONCLUSION

41. Thanks