RESISTIVITY MINIMA IN BULK DISORDERED ALLOYS

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, (36ltxlt83) cluster-glass/antiferromagne
tic alloys. D. Crystalline Ni 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
INTRODUCTION

• Disordered metallic alloys
• Dilute limit kF l gtgt 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 ltlt ?D
• ?0 ?T. for T gt ?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.

Eq. (1a)
Eq. (1b)
3
INTRODUCTION

• Weak-disorder limit, kF l gt 1 (Ioffe-Regel
  criterion)
• Boltzmann transport breaks down
• Beyond the Boltzmann Picture
• Short ? gt (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).

(a) Localization
? e2 D N(EF) W ?i ?i 2 ?1 2 ?2
2 ?1 ?2 ?1 ?2
Dephasing (i) Inelastic
scattering at T gt 0 (ii) Magnetic
Field P(r,t) dr dt 1/(4?Dt)3/2
exp(-r2/4Dt) dr dt
4
INTRODUCTION

• ?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 ltlt ?D
• Electron phonon scattering gt 1/?i
  T2
• gt ?? T. Eq. (2)
• ?i (1010 T2)-1 gt ?i ? 10-12 s at 10 K
• Expts. by Howson and Greig
• Non-magnetic CuTi/Hf/Zr glasses for 20 lt T lt
  100 K.
• For T gt ?D ,
• e-ph. scattering gt 1/?i T
• gt ?? T1/2. Eq. (3)
• Howson and Greig CuTi, etc., T gt 100 K.
• ?B magnetic field induced dephasing time
• ? /4eBD (? 10-12 10-13 s for B 1T) Note
  Phase change e?/?,
• ? (?0)-1/2 - (?B)-1/2
• gt ?? B1/2 gt negative magnetoresistance.

5
INTRODUCTION

• (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
• gt ?? T1/2. Eq. (4)
• Expts. by Howson and Greig CuTi CuHf , for T
  lt 20 K.
• Bergmann (1984) gave extensive experimental
  evidence of both localization e-e interaction
  effects in thin films (2-D).

6
INTRODUCTION
Weak scattering limit for 3d (bulk)
materials ?(T) ?0 m??T, where D
diffusion constant, F? screening factor. Strong
scattering case McMillans scaling theory of
metal-insulator transition ?(T) ?01
(T/?)1/2, Eq. (6) ? correlation
gap. Eqs. (5) (6) gt 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
INTRODUCTION

• So ?(T) varies in a sequence of ?T, T and ?T at
  low, intermediate and high temperatures.
• gt In expts, only two regions (?T T or T ?T)
  observed in the same sample.

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A. Amorphous Co-rich magnetic alloys

• 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 gt 300 K.
• A1-A4(FM), A5(Mixed FM/SG state for T lt 50 K).
• Depth of minima 0. 1 to 4.4 .

9
A. Amorphous Co-rich magnetic alloys
Fig. 2
Fig. 1

• Figs. 1 - 2 r(T) R(T)/R(30O K) vs. T (raw
  data).
• Resolution(??/?) 1 in 105
• Temperature stability 0.1 K.

10
A. Amorphous Co-rich magnetic alloys
Fig. 3
Fig. 4

• 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. Das A. K. Majumdar, PRB
43, 6042 (1991), PRB (1993), JMMM (1993),
JAP(1991).
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B. Amorphous Ni/Cu-Zr-Al non-magnetic alloys

• 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 lt 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 lt 1.7 K, ?D ? 220 K, ? ? f(T) ? 80 X 10-6
  emu/mole.

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B. Amorphous Ni/Cu-Zr-Al non-magnetic alloys
Fig. 5
Fig. 6

• 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) lt 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.

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B. Amorphous Ni/Cu-Zr-Al non-magnetic alloys

• Fig. 7 Ni-Zr-Al and (CuZr)0.8Al0.2.
• e-e interaction effect, TcltTlt15 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.

14
B. Amorphous Ni/Cu-Zr-Al non-magnetic alloys
Fig. 8
Fig. 9

• Figs. 8 9 ?D/10 lt T lt ?D/3.
• ??
  T. Eq. (8)

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B. Amorphous Ni/Cu-Zr-Al non-magnetic alloys
Fig. 11
Fig. 10

• Figs. 10 11 ?D/3 lt T lt ?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
• gt ?i gtgt ?0 justifies localization.

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B. Amorphous Ni/Cu-Zr-Al non-magnetic alloys
Fig. 12
Fig. 13

• 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).

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 lt T lt 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 (lt 0.12 ).
• Due to e e interaction MR is predicted to be
  ive.

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

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B. Amorphous Ni/Cu-Zr-Al non-magnetic alloys

• Fig. 14 MR vs. external fields between 4 and 20
  K.
• Positive, small (lt 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
C. Crystalline Cu100-xMnx (36 lt x lt 83) CG/AF
alloys

• Resistivity
• Earlier work
• B. R. Coles, Physica (1977)
• For x ltlt 1 , Kondo minimum with ?(T) In T.
• For 0.1 lt x lt 15 , SG (canonical), no
  resistivity minimum.
• For x gt 35 , minima around 20 K of depth lt 1 ,
  but no quantitative analysis.

27
C. Crystalline Cu100-xMnx (36 lt x lt 83) CG/AF
alloys
X83
X36
Fig. 19
Fig. 18

• 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 lt 1/3 , Tmin correlate with ?0 and not x.
• T lt 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).

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C. Crystalline Cu100-xMnx (36 lt x lt 83) CG/AF
alloys

• 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 lt T lt 30 K
• ?(T) ?0 - m?T1/2 ?T3/2
  Bloch-Gruneissen Eq. (18)
• ? ? ?
• e-e CG
  lattice
• Excellent fits, ?2 10-10.

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C. Crystalline Cu100-xMnx (36 lt x lt 83) CG/AF
alloys
30
C. Crystalline Cu100-xMnx (36 lt x lt 83) CG/AF
alloys

• 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).

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C. Crystalline Cu100-xMnx (36 lt x lt 83) CG/AF
alloys

• Magnetoresistance
• TMR LMR measured at 4.2, 20.5 and 63 K till 7.5
  tesla.
• Found
• (i) TMR LMR are lt 0.2 except for x 83 and
  are qualitatively the same.
• (ii) Mn-rich alloys (x gt 60) positive MR,
• x 36 positive MR till 3 tesla and negative
  beyond.
• (iii) x gt 60 High-field ??/? A BH1/2
  C1H2 for h gtgt 1, Eq. (19)
• where h g?BH/kBT, ? (?e2F?/4?2h)(kBT/2Dh)1/2
  , A - 1.3 ?,
• B ?(g?B/kBT)1/2 and 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
  ltlt1, Eq. (20)
• where C 0.053 ?(g?B/kBT)2 is due to e-e
  interaction.

32
C. Crystalline Cu100-xMnx (36 lt x lt 83) CG/AF
alloys
Fig. 21
Fig. 22

• 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).

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C. Crystalline Cu100-xMnx (36 lt x lt 83) CG/AF
alloys

• 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).

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 µ? 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 lt T lt 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) ?01 (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
?H 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-Cr75-13-12, Tmin 14 K
Ni-Fe-Cr70-12-18, Tmin 22 K
Fig. 27 ?H vs. 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
CONCLUSION

• 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.

41

• Thanks