Electrical resistivity of the mu-Al4Mn giant-unit-cell complex metallic alloys

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Feuerbacher, Marc Heggen, Stefan Roitsch, Janez Dolinsek

To cite this version:

Magdalena Wencka, Simon Jazbec, Zvonko Jaglicic, Stanislav Vrtnik, M. Feuerbacher, et al.. Electrical resistivity of the mu-Al4Mn giant-unit-cell complex metallic alloys. Philosophical Magazine, Taylor &

Francis, 2010, pp.1. �10.1080/14786435.2010.512578�. �hal-00625571�

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Electrical resistivity of the mu-Al4Mn giant-unit-cell complex metallic alloys

Journal: Philosophical Magazine & Philosophical Magazine Letters Manuscript ID: TPHM-10-May-0192.R1

Journal Selection: Philosophical Magazine Date Submitted by the

Author: 13-Jul-2010

Complete List of Authors: Wencka, Magdalena; J. Stefan Institute Jazbec, Simon; J. Stefan Institute

Jaglicic, Zvonko; Institute of Mathematics, Physics and Mechanics Vrtnik, Stanislav; J. Stefan Institute

Feuerbacher, M.; Forschungszentrum Jülich, Institut für Festkörperforschung

Heggen, Marc; Forschungszentrum Juelich GmbH, Institut fuer Festkoerperforschung

Roitsch, Stefan; Forschungszentrum Juelich GmbH, Institut fuer Festkoerperforschung

Dolinsek, Janez; J. Stefan Institute, F5 Keywords: electrical transport, quasicrystals

Keywords (user supplied): Complex metallic alloys, Electrical resistivity, Al-Mn system

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Electrical resistivity of the µ-Al₄Mn giant-unit-cell complex metallic alloy

M. Wencka,^1§ S. Jazbec,¹ Z. Jagličić,² S. Vrtnik,¹ M. Feuerbacher,³ M. Heggen,³ S.

Roitsch,³ J. Dolinšek ¹

1J. Stefan Institute, University of Ljubljana, Jamova 39, SI-1000 Ljubljana, Slovenia

2Institute of Mathematics, Physics and Mechanics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia

3Institut für Festkörperforschung, Forschungszentrum Jülich, Jülich D-52425, Germany

The µ-Al₄Mn complex intermetallic phase with 563 atoms in the giant unit cell shows a complicated temperature dependence of the electrical resistivity that exhibits a broad maximum at about 175 K and a minimum at 13 K. The temperature dependence of the resistivity was reproduced by employing the theory of quantum transport of slow charge carriers, which predicts a crossover from the metallic (Boltzmann-type) positive- temperature-coefficient electrical resistivity at low temperatures to the insulator-like (non-Boltzmann) negative-temperature-coefficient resistivity at elevated temperatures.

The low-temperature resistivity minimum was reproduced by considering it as a magnetic effect due to increased scattering of the conduction electrons by the Mn spins on approaching the spin glass phase that develops below the spin freezing temperature

T

f = 2.7 K.

Short title: Electrical resistivity of µ-Al₄Mn

Keywords: Complex metallic alloys, Electrical resistivity, Al–Mn system

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1. Introduction

Among a vast number of known intermetallic phases, the term "complex metallic alloys" (CMAs) denotes a class of phases with giant unit cells containing some hundred up to several thousand atoms [1]. Examples of CMAs arethe cubic NaCd₂ with 1152 atoms per unit cell (u.c.) [2,3], the Bergman phase Mg₃₂(Al,Zn)₄₉ (162 atoms/u.c.) [4], the orthorhombic ξ'-Al₇₄Pd₂₂Mn₄ (258 atoms/u.c.) and the related Ψ phase (about 1500 atoms/u.c.) [5-8], the cubic β-Al₃Mg₂ (1168 atoms/u.c.) [9,10], the hexagonal λ-Al₄Mn (586 atoms/u.c.) [11], the Al₃₉Fe₂Pd₂₁ (248 atoms/u.c.) [12] and the heavy-fermion compound YbCu_4.5, comprising as many as 7448 atoms in the unit cell [13]. These giant unit cells contrast with elementary metals and simple intermetallics whose unit cells in general comprise from single up to a few tens atoms only. The giant unit cells with lattice parameters of several nanometers provide translational periodicity of the CMA crystalline lattice on the scale of many interatomic distances, whereas on the atomic scale, the atoms are arranged in clusters with polytetrahedral order, where icosahedrally-coordinated environments play a prominent role. The structures of CMAs thus show duality; on the scale of several nanometers, CMAs are periodic crystals, whereas on the atomic scale, they resemble quasicrystals (QCs) [14]. The high structural complexity of CMAs together with the two competing physical length scales—one defined by the unit-cell parameters and the other by the cluster substructure—may have a significant impact on the physical properties of these materials, such as the electronic structure and lattice dynamics. In this paper, we present a study of the electrical resistivity and the magnetic susceptibility of the hexagonal µ-Al₄Mn phase with 563

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atoms in the unit cell [15]. Magnetic properties of the µ-Al₄Mn were already studied before [16].

2. Structural considerations and sample description

While the QC phase in the Al–Mn system forms by rapid cooling at a wide composition range, single-phase icosahedral material is produced only at compositions close to Al₄Mn [17].At higher Mn concentrations and lower cooling rates, a decagonal phase is formed. There exist two crystalline phases at the approximate composition Al₄Mn,

µ

and

λ

, both with giant hexagonal unit cells [18]. The µ-Al₄Mn structure belongs to

the P6₃/mmc space group with a = 1.998(1) nm, c = 2.4673(4) nm and 563 atoms in the unit cell (453 Al and 110 Mn, leading to the formula Al_4.12Mn) [15]. The unit cell contains 10 Mn and 32 Al crystallographic sites. Two of the Al sites are partially occupied. The structure consists of interpenetrated icosahedral chains aligned along the [100] crystal direction with pentagonal bi-prisms composed of pentagonal antiprisms (layer thickness 0.235 nm) and pentagonal prisms (layer thickness 0.290 nm) [19].

Along the [021] direction, a pseudo-5-fold symmetry is present. Along the [001]

hexagonal direction, the µ-Al₄Mn structure can be viewed as a layered compound [18].

The layer at z = c/4 (Fig. 1a) is located in a mirror plane and therefore planar; the one centered at z = c/2 (Fig. 1d) is almost planar, although not required to be so by symmetry. The layers centered at z = 0.325c (Fig. 1b) and z = 0.425c (Fig. 1c) are considerably puckered. The layers at z = 0.575c, 0.675c and 3c/4 (not shown) are generated from the layers centered at z = 0.425c, 0.325c and c/4, respectively, by the operation of the twofold axis at z = c/2 in the [110] direction. Since the layer at z =

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0.425c has a pseudo-twofold axis in that direction, the layer at c/2 is a pseudo-mirror with respect to its adjacent layers. A triangular distribution of atoms is evident on the c/4 and c/2 layers. In the µ-Al₄Mn structure, fragments of the Mackay icosahedral cluster occur. The structure is well ordered, showing a very low amount of disorder.

Our µ-Al₄Mn sample with the composition Al_80.7Mn_19.3 was single-phase material grown from an Al-rich melt by the Czochralski technique. A rectangular bar of dimensions 2

×

2

×

8 mm³ with its long edge along the c hexagonal axis was cut from a large, single-grain crystal. Recently, an X-ray photoelectron diffraction study on the 6- fold surface of the same µ-Al₄Mn material was reported [20], where further characterization of the material can be found.

3. Magnetic susceptibility

Magnetic measurements were conducted by a Quantum Design MPMS XL-5 SQUID magnetometer equipped with a 50 kOe magnet. In the measurements, the magnetic field was applied parallel to the hexagonal c axis of the sample. The susceptibility data presented in the units emu/mol are calculated per mol of Al_0.807Mn_0.193 "molecules".

The zero-field-cooled (zfc) and field-cooled (fc) susceptibilities

χ

=M H were measured in a low magnetic field H = 8 Oe. The low-temperature susceptibilities between 1.9 and 6 K are shown in Fig. 2a. Below about 3 K,

χ

_zfcand

χ

_fc start to distinguish, demonstrating ergodicity breaking on the experimental time scale, typical of a magnetically frustrated spin glass. The spin-freezing temperature

T

f = 2.7 K was determined as the temperature where the zfc–fc splitting occurred (this coincides well with the temperature where

χ

_zfcexhibits a cusp).

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The paramagnetic susceptibility

χ ( )

T up to 300 K in the magnetic field H = 1 kOe is shown in Fig. 2b. The analysis was performed with the Curie-Weiss law,

χ χ θ

+ −

= T

C_CW

0 , (1)

where

χ

₀is the temperature-independent part of the susceptibility,

C

CW the Curie- Weiss constant and

θ

the Curie-Weiss temperature. The constant

C

CW gives information on the magnitude of the Mn moments, whereas the type and strength of the coupling between the spins can be estimated from the magnitude and sign of

θ

. In the inset of Fig. 2b, the

χ ( )

T data are displayed in a

( χ − χ

₀

)

⁻¹versus temperature plot.

The fit of the high-temperature data T > 40 K (solid line) yielded the parameter values

χ

0= 1.06

×

10^–4 emu/mol,

C

CW = 2.95

×

10^–2 emu K/mol and

θ

= – 29 K. The negative Curie-Weiss temperature suggests an antiferromagnetic (AFM)-type antiparallel coupling between the spins. The value of the Curie constant

C

_CW was used to calculate the mean effective magnetic moment

µ

_eff

=

p_eff

µ

_B per Mn ion, where p_eff is the mean effective Bohr magneton number that can be calculated using the formula [21]

CW

eff C

p

=

2.83 (with

C

CW given in units per mol of Mn atoms). We obtained

peff = 1.09 per Mn atom. Comparison of this value to the Bohr magneton number of a bare Mn²⁺ ion (

p

= 5.9) shows that the Mn moments in the µ-Al₄Mn phase are significantly reduced, indicating partial screening by the conduction-electron cloud in a conducting environment. While the reduced mean effective Bohr magneton number cannot distinguish between the situations where a large fraction of Mn atoms carry small magnetic moments or a small fraction of Mn atoms carry full magnetic moments

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(

p

= 5.9), the rest being nonmagnetic, we shall nevertheless estimate the fraction of Mn moments for the second scenario. In that case we obtain the magnetic Mn fraction as

( )

⁼

= p p ²

f _eff 3 %, so that the moments would be quite diluted.

The ac susceptibility was measured in an ac magnetic field of amplitude H₀ = 6.5 Oe at the frequencies

ν

= 1, 10, 100 and 1000 Hz. The real part of the susceptibility

χ

′is displayed in Fig. 3. As expected for a nonergodic spin system, the position of the

cusp in

χ

′is frequency-dependent, shifting to lower temperatures at lower frequencies.

At the lowest frequency of 1 Hz, the cusp in

χ

′occurs at the temperature T = 2.7 K.

Considering the cusp in

χ

′ to occur at the freezing temperature, the T_f

( ) ν

relation was determined from the

χ

′

( ) ν

curves. T_f

( ) ν

normalized to the T_f

(

1Hz

)

= 2.7 K is displayed as an inset in Fig. 3, where a logarithmic dependence (base 10) of T_f on the frequency is evident. The frequency shift of T_f is often quantitatively evaluated by the empirical criterion^∆T_f T_f^∆

(

^log

^ν )

, i.e., by calculating the relative change of T_f per

decade

ν

. For the µ-Al₄Mn, this ratio amounts 0.013, which is in the range found for metallic spin glasses [22] like the AuFe and PdMn. The change of T_f with the frequency is thus small, amounting about 1 % per decade

ν

. The ac susceptibility results also support the spin-glass nature of the µ-Al₄Mn phase.

4. Electrical resistivity

The electrical resistivitywas measured between 300 and 2 K using the standard four- terminal technique. The current was applied along the hexagonal c axis of the sample.

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The

ρ ( )

T data are displayed in Fig. 4. The temperature dependence of the resistivity is quite complex, exhibiting a broad maximum at about 175 K and a minimum at 13 K.

The room-temperature resistivity amounts

ρ

_300K =360 µΩcm, the value in the maximum is

ρ

_175K =365 µΩcm, whereas the minimum value is

ρ

_13K =353 µΩcm.

Apart from the low-temperature resistivity minimum, which will be described in the following as a magnetic effect, we proceed with the resistivity analysis in a semi- quantitative way by employing the theory of quantum transport of slow charge carriers [23], applicable to metallic alloys and compounds in which the electron mean free path

τ

v

l= (where

v

is the electron velocity and

τ

the electron scattering time) is small compared to the extension of the conduction-electron wave packet, in which case the electron propagation is non-Boltzmann. This theory has recently been successfully applied to the Al₄(Cr,Fe) decagonal approximant [24,25], where the temperature- dependent resistivity exhibited a maximum similar to that shown in Fig. 4.

According to the Einstein relation, the conductivity

σ

depends on the electronic density of states (DOS)

g ( ) ε

and the spectral diffusivity D

( ) ε

within the thermal interval of a few k_BTaround the Fermi level

ε

_F. In the case of slowly varying metallic DOS around

ε

_Fit is permissible to replace g

( ) ^ε

by g

( ) ε

_F . For the diffusion constant of slow charge carriers it was shown [23]that it can be written as ^D=v2

^τ

^+L2

( ) ^{τ τ}

^,

where L²

( ) τ

is the non-Boltzmann contribution to the square of spreading of the quantum state at energy

ε

due to diffusion, averaged on a time scale

τ

. L

( ) ^τ

is bounded by the unit cell length and saturates to a constant value already for short averaging time. The dc conductivity of the system can be written as [23]

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( ) ( ) ( )

τ ε τ τ

ε σ

σ

σ

= _B + _NB =e²g _F v² +e²g _F ^L2 , (2)

where

σ

_B is the Boltzmann contribution and

σ

_NBis the non-Boltzmann contribution.

The scattering rate

τ

⁻¹will generally be a sum of a temperature-independent rate

1 0

τ

− due to scattering by quenched defects and a temperature-dependent rate

τ

⁻_p¹due to

scattering by phonons and the magnetic scattering by localized paramagnetic spins. In the simplest case,

τ

_pcan be phenomenologically written as a power-law of

temperature,

τ

_p =

β

T^α , at least within a limited temperature interval. Assuming that

( ) ^τ

L2 can be replaced by its limiting value, a constant L², Eq. (2) yields a minimum in the conductivity

σ

as a function of

τ

or temperature (or equivalently, there is a maximum in the resistivity) at the condition

τ

=L v. At temperatures below the resistivity maximum,

τ

is long enough that the system is in the Boltzmann regime (

σ

≈

σ

_B) with a power-law positive-temperature-coefficient (PTC) metallic resistivity.

At high temperatures above the resistivity maximum,

τ

is short enough that the system is in the non-Boltzmann regime and the resistivity exhibits a negative temperature coefficient (NTC).

Defining A=e²g

( ) ε

_F v²

τ

₀ , B=e²g

( ) ε

_F L²

τ

₀and C =

τ

₀

β

, Eq. (2) can be rewritten as

(

^α

)

σ

α B CT

CT A

SC + +

= + 1

1 ^{, (3)}

where the subscript "_SC" denotes "slow carriers".

σ

_SC contains four fit parameters A,

B

,

C

and

α

(where the last two always appear in a product CT^α).

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The low-temperature resistivity minimum can be attributed to a magnetic effect.

The magnetic measurements presented in Figs. 2 and 3 clearly demonstrate slowing- down of the Mn spin fluctuations upon cooling and their freeze-out on the experimental time scale below the freezing temperature

T

f = 2.7 K. Static spins provide strong magnetic scattering mechanism for the conduction electrons, so that the electrical resistivity increases quite dramatically on approaching the spin glass phase.

In alloys containing magnetic transition-metal impurities, the

s

-

d

exchange interaction between the

s

moments of the conduction electrons and the

d

moments of the transition-metal impurities contributes a term to the resistivity [26]

[ ( ) ] (

+

)

_ +

( )

_^

= D

T Jg k

S S J g

c _{F F} ^B

spin

ρ

₀

π

²

ε

^{2 2} 1 1 4

ε

log

ρ

. (4)

Here

c

is the magnetic impurity concentration,

S

the impurity spin,

J

the

s

-

d

exchange constant,

ρ

₀ the residual resistivity at T →0 ^and g

( ) ε

_F is the DOS at

ε

_F in a flat band of width

2D

.

The total resistivity can be expressed as a sum of the slow-charge-carrier contribution

ρ

_SC =1

σ

_SC and the magnetic contribution

ρ

_spinand can be conveniently written in the form

T

SC +dlog

=

ρ

ρ

. (5)

For a negative exchange constant,

J

< 0 (and hence

d < 0

), a minimum in the resistivity occurs at low temperatures. Eq. (5) was used to fit the resistivity of the µ-Al₄Mn and the resulting theoretical curve is shown in Fig. 4. Good agreement with the experiment was found in the entire investigated temperature range from 300 to 2 K, reproducing both the maximum and the minimum of the resistivity. The fit parameters are

A

=

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1.93

×

10^–3(µΩcm)^–1,

B

= 8.43

×

10^–4(µΩcm)^–1,

C

= 4.83

×

10^–3,

α

= 0.87 and

d

= – 5.05 µΩcm, where the units of

C

and

d

are chosen such that the temperature in the

expressions CT^α and dlogTappears dimensionless.

We should like to stress that, besides the model of slow-charge-carrier resistivity of Eq. (3), there exist other models that yield a maximum in the electrical resistivity. An example is the Baym-Meisel-Cote theory of amorphous metals [27] that considers the inelastic electron-phonon interaction to be of prime importance for the electron transport at temperatures below the Debye temperature

θ

_D.

5. Conclusions

The µ-Al₄Mn complex intermetallic phase with 563 atoms in the giant unit cell shows complicated temperature dependence of the electrical resistivity that exhibits a broad maximum at about 175 K and a minimum at 13 K. Away from the low-temperature minimum, the temperature dependence of the resistivity was reproduced theoretically by employing the theory of quantum transport of slow charge carriers, which predicts a crossover from the metallic (Boltzmann-type) PTC electrical resistivity at low temperatures to the insulator-like (non-Boltzmann) NTC resistivity at elevated temperatures that yields a resistivity maximum. The low-temperature resistivity minimum was reproduced by considering it to be a magnetic effect due to an increased scattering of the conduction electrons by the Mn spins on approaching the spin glass phase. The spin freeze-out below the spin freezing temperature

T

f = 2.7 K is evident from the zfc–fc magnetic susceptibility splitting below

T

f in low magnetic field and in the frequency-dependent cusp in the ac susceptibility.

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Acknowledgement: This work was done within the 6^th Framework EU Network of Excellence "Complex Metallic Alloys" (Contract No. NMP3-CT-2005-500140). J.D.

acknowledges support from the Centre of Excellence EN→FIST, Dunajska 156, SI- 1000 Ljubljana, Slovenia.

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References:

§On leave from the Institute of Molecular Physics, Polish Academy of Sciences, Smoluchowskiego 17, 60-179 Poznań, Poland.

[1] See, for a review, K. Urban and M. Feuerbacher, J. Non-Cryst. Solids 334&335 143 (2004).

[2] L. Pauling, J. Am. Chem. Soc. 45 2777 (1923).

[3] L. Pauling, Am. Sci. 43 285 (1955).

[4] G. Bergman, J. L. T. Waugh, L. Pauling, Acta Crystallogr. 10 254 (1957).

[5] M. Boudard, H. Klein, M. de Boissieu, et al., Philos. Mag. A 74 939 (1996).

[6] H. Klein, M. Audier, M. Boudard, et al., Philos. Mag. A 73 309 (1996).

[7] J. Dolinšek, P. Jeglič, P. J. McGuiness, et al., Phys. Rev. B 72 064298 (2005).

[8] M. Feuerbacher, C. Thomas, K. Urban, Quasicrystals, Structure and Physical Properties, ed. H.-R. Trebin (Wiley-VCH, Weinheim, 2003), p. 2.

[9] S. Samson, Acta Crystallogr. 19 401 (1965).

[10] S. Samson, Developments in the Structural Chemistry of Alloy Phases, ed. B. C.

Giessen (Plenum, New York, 1969), p. 65.

[11] G. Kreiner, H. F. Franzen, J. Alloys Compd. 261 83 (1997).

[12] F. J. Edler, V. Gramlich, W. Steurer, J. Alloys Compd. 269 7 (1998).

[13] R. Cerny, M. Francois, K. Yvon, et al., J. Phys.: Condens. Matter 8 4485 (1996).

[14] See, e.g., C. Janot, Quasicrystals, 2^nd edition (Clarendon, Oxford, 1994).

[15] C. B. Shoemaker, D. A. Keszler, D. P. Shoemaker, Acta Cryst. B 45 13 (1989).

[16] V. Simonet, F. Hippert, M. Audier, et al., Phys. Rev. B 58 R8865 (1998).

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[17] R. J. Schaefer, L. A. Bendersky, D. Shechtman, et al., Metall. Trans. A 17 2117 (1986).

[18] J. L. Murray, A. J. McAlister, R. J. Schaefer, et al., Metall. Trans. A 18 385 (1987).

[19] C. B. Shoemaker, Phil. Mag. B 67 869 (1993).

[20] R. Widmer, R. Maeder, M. Heggen, et al., Philos. Mag. 88 2095 (2008).

[21] F. E. Mabbs and D. J. Machin, in Magnetism and Transition Metal Complexes (Chapman and Hall, London, 1973), p.7.

[22] J. A. Mydosh, Spin glasses: an experimental introduction (Taylor & Francis, London, 1993), p. 67.

[23] G. Trambly de Laissardière, J.-P. Julien, D. Mayou, Phys. Rev. Lett. 97 026601 (2006).

[24] J. Dolinšek, P. Jeglič, M. Komelj, et al., Phys. Rev. B 76 174207 (2007).

[25] J. Dolinšek, S. Vrtnik, A. Smontara, et al., Phil. Mag. 88 2145 (2008).

[26] U. Mizutani, Electron Theory of Metals (Cambridge University Press, 2001), 416.

[27] U. Mizutani (ref. [26]), p. 479.

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Figure captions:

Fig. 1: (Color online) Layer description of the µ-Al₄Mn crystal structure perpendicular to the z axis according to the structural model by Shoemaker et al. [15]. The layer at z = c/4 (a) is located in a mirror plane and therefore planar; the one centered at z = c/2 (d) is almost planar, although not required to be so by symmetry. The layers centered at z = 0.325c (b) and z = 0.425c (c) are considerably puckered. Other layers are derived by symmetry operations. Triangular distribution of Mn atoms is shown encircled on the c/4 and c/2 layers.

Fig. 2: (Color online) (a) The zfc and fc susceptibilities

χ

=M H of µ-Al₄Mn in the temperature range 1.9 – 6 K in a magnetic field H = 8 Oe, applied along the hexagonal c axis of the monocrystalline sample. (b) Magnetic susceptibility

χ ( )

T between 1.9 and

300 K in H = 1 kOe. The inset shows the

( χ

−

χ

₀

)

⁻¹versus temperature plot. Solid line is the Curie-Weiss fit with Eq.(1) of the high-temperature data T > 40 K.

Fig. 3: (Color online) Real part

χ

′ of the ac susceptibility measured in an ac magnetic field of amplitude H0 = 6.5 Oe at frequencies

ν

= 1, 10, 100 and 1000 Hz. The inset shows T_f

( ) ν

normalized to the T_f

( ν

=1Hz

)

value.

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Fig. 4: (Color online) Temperature-dependent electrical resistivity of µ-Al₄Mn for the current along the hexagonal c axis of the monocrystalline sample. Solid curve is the fit with Eq. (5) and the fit parameters are given in the text.

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Fig. 1

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Fig. 2

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Fig. 3

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

For Peer Review Only

Fig. 4

4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60