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Anomalous dephasing scattering time of Zr80Sn20−xFex alloys at low temperature
Physica E 77 (2016) 7–12
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Physica E journal homepage: www.elsevier.com/locate/physe
Anomalous dephasing scattering time of Zr80Sn20 xFex alloys at low temperature R.N. Jana, S. Sinha, A.K. Meikap n Department of Physics, National Institute of Technology Durgapur, Mahatma Gandhi Avenue, Durgapur 713209, West Bengal, India
H I G H L I G H T S
G R A P H I C A L
A minima has been observed on the resistivity versus temperatures curve. The low temperature resistivity in absence of magnetic field obeys the ρo5/2T1/2 law. The spin–orbit interaction is strong and independent of temperature. The zero temperature scattering rate increases with increase of le . The inelastic scattering rate obeys an anomalous behavior τe−−1ph ∝ T 2le .
The electron–phonon scattering rate shows anomalous behavior and obeys the relation τe−−1ph ∝ T2le , where le is the electron elastic mean free path.
art ic l e i nf o
a b s t r a c t
Article history: Received 20 July 2015 Accepted 19 October 2015 Available online 28 October 2015
We report the results of a comprehensive study of weak electron localization (WEL) and electron– electron interaction (EEI) effects in disordered Zr80Sn20 xFex alloys. The resistivity in absence of magnetic field shows a minimum at temperature T ¼Tm and follows ρo5/2T1/2 law within the temperature range 5 K rT r Tm, which suggests predominant EEI effect. Magnetoresistivity is positive due to strong spin– orbit interaction. The dephasing scattering time is dominated by the electron–phonon scattering. The electron–phonon scattering rate shows anomalous behavior and obeys the relation τe−−1ph ∝ T2le , where le is the electron elastic mean free path. The zero temperature scattering time (τ0 ) strongly depends on the disorder and its magnitude decreases with increasing disorder resistivity ρ0 . Such anomalous behavior cannot be explained in terms of existing theories. & 2015 Elsevier B.V. All rights reserved.
Keywords: A. Zirconium alloy D. Low temperature D. Magnetoresistivity D. Electron phonon scattering
A B S T R A C T
1. Introduction Anomalous temperature dependence of normal state resistivity in disordered solids has become an interesting topic in solid state physics during the last few decades [1–6]. In such disordered solid a critical resistivity (ρc) has been defined by Mooji [7], where temperature coefficient of resistivity (TCR) is zero. He also suggested that the TCR is negative for disordered materials having resistivity ρ 4150 μΩ cm. However, Tsuei [8] reported that the n
Corresponding author. E-mail address: [email protected] (A.K. Meikap).
http://dx.doi.org/10.1016/j.physe.2015.10.020 1386-9477/& 2015 Elsevier B.V. All rights reserved.
weak electron localization (WEL) and the electron–electron interaction (EEI) play a significant role on TCR of disordered alloys. At low temperatures both WEL and EEI effects introduce corrections to temperature dependent resistivity. It is established that in absence of magnetic field conductivity correction is dominated by EEI, but low magnetic field dependent conductivity is dominated by WEL in the case of three-dimensional disordered metals. By fitting the low field magnetoresistivity with the theory of WEL, the different scattering time like temperature dependent inelastic scattering time (τi ) and temperature independent spin–orbit scattering time (τSO ) and zero temperature scattering time (τ0 ) has been calculated. Generally, the two mechanisms like electron–
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R.N. Jana et al. / Physica E 77 (2016) 7–12
electron and electron–phonon scattering have major contributions to the temperature dependence of the inelastic scattering. It is well established that in case of three dimensional systems the electron–phonon scattering is the dominating inelastic scattering process. The temperature dependence of the electron–phonon scattering for clean systems has been well established both theoretically and experimentally [9,10], but in disordered metals it is still in controversy. Theoretically the electron–phonon interaction in disordered metals has explained by many authors [11–17] but they concluded in different predictions. Some experimental works [18–21] already have been done on various disordered conductors on the issue of temperature dependence electron–phonon scattering but they were not able to establish a complete conclusion. Particularly, the disorder or mean free path (le ) dependent zero temperature scattering rate (τ0−1) and electron–phonon scattering rate (τe−−1ph ) of these alloys has not been established properly for different systems. In this work, we have reported the systematic measurement of τe − ph from WEL studies in a series of disordered Zr80Sn20 xFex and have shown how the electron–phonon scattering rate (τe−−1ph ) and the zero temperature
Table 1 Values of relevant physical parameters for disordered Zr80Sn20 xFex alloys, ρo and ρ (300 K) are the resistivity at 10 K and 300 K respectively, le is the mean free path, D is the electron diffusion coefficient, KF is the Fermi wave vector, Tm is the temperature at which resistivity is minimum, τso 1 is the spin–orbit scattering rate and
τϕ−1 (10 K) is the dephasing scattering rate at 10 K. τ0−1 is the zero temperature dephasing scattering rate, p is the temperature exponent of electron–phonon scattering rate (τe−−1ph ) and Ae − ph is the strength of electron–phonon coupling. Parameters
V82Al18Fe0
V82Al17Fe1
V82Al16Fe2
V82Al15Fe3
ρ0 (mΩ cm) ρ (300 K) (mΩ cm) le (Å) D (cm2/s) KF le Tm (K)
91.9 144.2 4.59 2.07 7.80 11.0 2.70 1013
98.2 165.5 4.29 1.93 7.30 11.5 2.73 1013
105.1 182.1 4.01 1.81 6.82 12.0 2.79 1013
114.5 198.3 3.68 1.66 6.26 13.0 2.85 1013 0.91 1011
−1 (s 1) τSO
τϕ−1 (s 1) τ0−1
(s
1
)
P Ae − ph (s 1 K p)
2.70 1013
1.16 1011
1.01 1011
1.07 1011
9.20 1010
8.55 1010
7.83 1010
2.14 2.43 108
2.09 1.87 108
2.01 1.50 108
2.05 1.16 108
scattering rates (τ0−1) depend on disorder of these alloys.
115.2 115.0 114.8 114.6 114.4 105.6
3. Results and discussions We have measured the electrical resistivity of the disordered Zr80Sn20 xFex alloys in the temperature range 5rTr300 K in absence of magnetic field. The variation of electrical resistivity with temperature is shown in Fig. 1 for different samples. The figure shows anomalous behavior, the resistivity initially decreases with decreasing temperature and then increases with further decreases of temperature i.e. the curve shows a resistivity minimum at temperature T¼ Tm, which is listed in Table 1. On the other hand, no superconducting transition has been observed in the investigated
(T)(
Zr80Sn20 xFex alloys with Iron concentration x¼ 0, 1, 2, 3 were prepared by a standard arc melting method. Appropriate amounts of spec-pure Zr, Sn and Fe were melted several times. The melted ingots were subjected to a homogenizing anneal at 800 °C for 48 h. Nominal Iron concentration was used to produce compositional disorder in the samples. The samples were cut into specimens of rectangular shape, typically ¼0.2 0.2 10 mm3, for resistivity measurement. Copper electrodes were spot-welded on to the samples. Resistance and magnetoresistance were measured by using the standard four probe technique and were performed in a closed cycle Helium Cryostat. The temperature was measured and monitored by a lakeshore 335 temperature controller. Low measurement currents were applied and care was taken to avoid any appreciable Joule heating of the electrons. In the case of magnetoresistance measurements, in order to minimize any appreciable contribution from the many body EEI effects, a small magnetic field range was always used. The values of Fermi wave vector (KF) for different samples were evaluated by using the free electron model result [22], KF ¼3.63/(rs/ao) Å 1 taking the values of rs/ao for individual materials where ao is the Bohr radius and rs is the radius of a sphere whose volume is equal to the volume per conduction electron. The electron elastic mean free path ‘le’ is then obtained through the relation le ¼ 3π2ħ/(kF2e2ρo), where ρo (at 10 K) is the impurity resistivity. The values of the electron diffusion constant D ℏK 1 have been found out from the relations D = 3 vF le and vF = mF , where m is 1.46 times the free electron mass [23]. The estimated values of le, D and KFle are listed in Table 1.
cm)
2. Experimental method
Zr80Sn20-xFex
x=3
105.4
x=2
105.2 105.0 104.8 98.40 98.32
x=1
98.24 98.16 92.02 92.00 91.98 91.96 91.94
x=0
4
6
8
10
12
14
16
18
20
22
T(K) Fig. 1. Resistivity as a function of temperature of different Zr80Sn20 xFex alloys.
temperature range 5rTr300 K. Such behavior of the samples provides an advantage for quantitative study of the electron–phonon scattering time τe–ph in the wide range of temperature from 5 K to 20 K. In order to understand the low temperature electrical transport of the investigated samples, we have studied the temperature and disorder dependence of resistivity in absence of magnetic field. At low temperature (ToTm), the different effects like EEI, WEL and Kondo type scattering [24–29] introduce temperature dependent corrections to resistivity and given by the relation
Δρ (T ) = αT 1/2 + βT p /2 + γ ln T ,
α= −
1.3e2ρ02 ⎡ 4 3F ⎤ kB , ⎢ − ⎥ 4π 2= ⎣ 3 2 ⎦ 2ℏD
(1)
(2)
R.N. Jana et al. / Physica E 77 (2016) 7–12
4.0
0.16
cm )
2.0
4
1.0x10
5
5/2 0
1.2x10
(
cm)
5
1.4x10
5
/
8.0x10
5/2
cm) (T)(
-0.1
X=0
-0.2
X=1
1.5
0.5 0.0 0.0
X=2
-0.4
X=3
Zr 80 Sn 20-x Fe x 2.0
2.5
3.0 1/2
0.1
0.2
0.3
0.4
B(T)
-0.5 -0.6
T=5K T=8K T=10K T=15K T=18K T=20K
1.0
0.0
-0.3
0
-1 -1
2.5
(
0.04
3.0
2
1/2
(T)/T (
0.00
Zr80Sn18Fe2
3.5
Zr 80 Sn 20-x Fe x
0.12
-
cmK
-1/2
)
0.20
0.08
9
3.5
1/2
T (K ) Fig. 2. (a) Resistivity increase Δρ ¼ρ(T) ρo as a function of T1/2 for disordered Zr80Sn20 xFex alloys. The straight solid lines are least square fit to Eq. (1). (b) The slope of resistivity increase Δρ(T)/T1/2 as a function of ρo5/2 for disorder Zr80Sn20 xFex alloys. The straight solid line is a guide to the eye.
where first term of Eq. (1) corresponds to EEI, 2nd term corresponds to WEL and third term due to Kondo type scattering, e is the electronic charge, ρ0 is the resistivity at 10 K, ħ is Planck's constant divided by 2π, kB is the Boltzmann constant, F is a screening factor averaged over Fermi surface, β and γ are two different constants, p is an exponent of inelastic scattering time (τi − T p). A straight line plot of Δρ(T) with T1/2 for different samples has been shown in Fig. 2(a) which ruled out the contribution from Kondo effect and suggested that the contribution due to the EEI is dominating over WEL effect. Apart from the T1/2 dependence, the resistivity correction Δρ(T) is also a function of disorder (Eq. (2)) and proportional to ρ02 /D1/2 or ρ05/2, where D − ρ0−1 is the diffusion constant. Fig. 2 (b) shows the variation of resistivity slope Δρ (T ) /T1/2 with ρ05/2 for Zr80Sn20 xFex alloys. The symbols are the experimental data, and the straight solid line guides the eye. In Fig. 2(b) it is seen that the rise in the slope of the resistivity indeed varies linearly with ρ05/2, firmly supporting the EEI predictions. This observation is an excellent agreement between the theory (EEI) and experiment i.e. Δρ (T ) − ρ05/2 T1/2 − ρ02 (T /D)1/2 and suggested that the EEI is completely controlled by the total amount of disorder (ρ0 ) present in the sample. Magnetoresistivity of investigated alloys was measured under transverse low magnetic fields up to 0.4 T to avoid the contribution due to EEI in the temperature range 5–20 K. Measured magnetoresistivity is positive for all samples, which suggests the strong spin–orbit scattering effects in these alloys. Abrikasiv and Gorkov [30] pointed out that the spin–orbit scattering depends on the atomic number (Z) of the metals and the elastic scattering time
Fig. 3. The variation of the magnetoresistivity with magnetic field at different constant temperature of Zr80Sn18Fe2 alloys. The points are the experimental data and the solid lines are the theoretical predictions from Eq. (3).
−1 (τe), and is given by the relation τso = (βZ )4 τe−1 where β is the fine structure constant. The atomic number of Zr (Z¼ 40) and that of Sn (Z¼46) are moderate and Fe (Z¼28) is low. Therefore, the Zr and Sn atoms introduce the strong spin–orbit scattering in the alloys. On the other hand, the resistivity of the alloys is high, which introduce strong electron elastic scattering rate (τe−1). As a result the investigated samples show strong spin–orbit scattering. It is well established that information for the various electron dephasing scattering time in impure metals has been obtained from analysis of measured magnetoresistivity by the theory of WEL effects. As the investigated samples did not show any superconducting transition within the investigated temperature range, we have analyzed the magnetoresistivity data in the light of three-dimensional WEL theoretical predictions without introducing the superconducting fluctuation correction. The variation of the magnetoresistivity with magnetic field for Zr80Sn20 xFe2 has been shown in Fig. 3. Different points in the figure represent experimental data at different temperature and the solid lines are the theoretical best fitted values obtained using Eq. (3) with fitting parameters Bϕ and Bso [31].
e2 Δρ (B, T ) = 2 2π 2= ρo
⎡ ⎛ ⎞⎤ eB ⎢ 1 ⎛ B ⎞ 3 ⎜ B ⎟⎥ ⎜ ⎟ f f − = ⎢ 2 3 ⎝ Bϕ ⎠ 2 3 ⎜⎝ Bϕ + 4 Bso ⎟⎠ ⎥ ⎦ ⎣ 3
(3)
where Bϕ is the electron dephasing scattering field and Bso is the spin–orbit scattering field. The function f3(x) has already been defined in Baxter et al. [31]. It is shown from the figure that the three-dimensional WEL theory is well suited for the analysis of measured magnetoresistivity, and the electron dephasing scattering field Bϕ and spin–orbit scattering field Bso are reliably extracted from these fits. The different scattering time is related with the scattering field by the relation τx ¼ ħ/(4eDBx). For electron dephasing scattering x¼ ϕ and for spin orbit scattering x ¼so. It is observed that the spin–orbit scattering rate is greater than the electron dephasing scattering rate (τso 1 4 4 τϕ 1) and is independent of temperature and the calculated values of τso 1 and τϕ 1(10 K) are given in Table 1. Generally in disordered
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R.N. Jana et al. / Physica E 77 (2016) 7–12
11
conductors the electron dephasing scattering time consists of zero temperature scattering time (τ0 ) and temperature dependent inelastic scattering time (τin ) and can be presented by the relation
1 1 1 = + . τϕ (T ) τ0 τin (T )
ing scattering rate τϕ−1 (T ) is shown in Fig. 4 for Zr80Sn18Fe2 sample. The symbols in the figure are the experimental data, whereas different lines are the theoretical fit. The least-square fit to Eq. (4) with τo, Ae–ph and p as fitting parameters is given by the solid line, where the best fit value of p is 2.01. On the other hand, the short dashed and dotted curves are also least square fits to Eq. (4) with fixed values of p as 3 and 4 respectively, while τo and Ae–ph as fitting parameters. It is observed from Fig. 4 that the experimental data did not match with the theoretical prediction with p ¼ 3 and 4 but well suited with p ¼2.01. Therefore, we may conclude that the temperature dependent part of the dephasing scattering time can be described by quadratic temperature dependence. For all other samples the variation of τϕ−1 (T ) with temperature has been 11
1.6x10
Zr80Sn18Fe2 11
1.4x10
-1
~T
p
11
1.2x10
p=4
-1
-1
(s )
-1
(S )
(5)
−1 We have calculated the values of τee for Zr80Sn20 xFex alloys at temperature T ¼5 K by using Eq. (5) and the values lie in the range 8.01 107–1.11 108 s 1. It is found that the calculated values of −1 are 732–1426 times smaller than the experimentally measured τee inelastic scattering rate for all the investigated samples. This is because the measurement temperature is above 5 K, where the electron–electron scattering contribution to inelastic scattering is minimized. Therefore, this result rules out any contribution from electron–electron scattering to the inelastic scattering rate. On the other hand, for three-dimensional disordered alloys the inelastic scattering at higher temperature TZ 5 K is dominated by electron– phonon scattering [16,34,35] and can be expressed as τin−1 (T ) − τe−−1ph − Ae − ph T p . The temperature dependence of dephas-
p=3 p = 2.01
11
1.0x10
10
4
11
2x10
-1
3/2 6m ⎛ kB T ⎞ ⎟ . ⎜ 2 2ℏ KF ⎝ KF le ⎠
8.0x10
Zr80Sn20Fe0 Zr80Sn19Fe1 Zr80Sn18Fe2 Zr80Sn17Fe3
(4)
The temperature dependent inelastic scattering time (τin ) arises due to electron–electron scattering (τee ) and electron–phonon scattering (τe − ph ). In three dimensional disordered systems, the electron–electron scattering rate can be expressed as [32–33] −1 τee =
3x10
6
8
10 12 14 16 18 20 22
T(K) Fig. 4. Electron dephasing rate τϕ 1(T) as a function of temperature for the Zr80Sn18Fe2 alloy. The solid, short dashed and dotted curves are least square fit to Eq. (4) with the exponent of temperature p ¼2.01, 3 and 4 respectively.
11
1x10
4
6
8 10 12 14 16 18 20 22
T(K)
Fig. 5. Electron dephasing rates τϕ 1(T) as a function of temperature for different disordered Zr80Sn20 xFex alloys. The solid curves are least square fit to Eq. (4).
shown in Fig. 5. In the figure the different points are the experimental data and the solid lines are the fitting lines obtained from Eq. (4) with fitting parameters τo, Ae–ph and p. The best fitted values of the parameters for different samples are τo ¼(0.94–1.28) 10 11 s, Ae–ph ¼(1.16–2.43) 108 s 1/Kp and k T p¼ 2.01–2.14. In quasiballistic limit ( qph le > 1, qph = ℏBv is the wave s
number of the thermal phonons, vs is the sound velocity and equal to 3.61 103 m/s [36,37]), Zhong et al. [38] reported the electron– phonon scattering rate by considering the additive contribution from vibrating defects and impurities. Taking the value of vs and le for the investigated samples we have calculated qph le = (0.013 − 0.017) T . Hence the value of qph le does not meet the quasiballistic criteria (qph le > 1) in the investigated temperature range 5r Tr 20 K for different samples. So the observed quadratic temperature dependence of electron–phonon interaction could not be explained by the above theoretical prediction [38]. On the other hand, the disorder or dirty criteria (qph le < 1) has been satisfied from the above calculation. In such dirty system (qph le < 1), theoretical studies of quadratic temperature dependence of electron–phonon interaction was done by different authors [11–17,39]. Considering all earlier theories Sergeev and Mittin [17,40] have formulated the theory of the electron–phonon scattering time in a disordered conductor, where they take into account both the static and vibrating random scattering potentials. They found that if the random scattering potential is completely dragged by phonons, the effective electron–phonon scattering rate would decrease due to disorder and presented by the relation τe−−1ph − T 4le at low temperature, which is not observed in our investigated samples. However, they also pointed out that for such systems, due to presence of imperfections the scattering potential would not be completely dragged by the phonon. In such incomplete drag system the interaction between electrons and transverse phonons would dominate over the interaction between electrons and longitudinal phonons and the electron–phonon scattering rate is given by the relation [40]
τe−−1ph =
π 2KF2 kB2 N ( EF ) k (1 − k ) 6m2ρut3 le
T2,
(6)
R.N. Jana et al. / Physica E 77 (2016) 7–12 10
11
1.1x10
T = 10K
(S )
10
4x10
-1
Theory
10
-1
3x10
10
2x10
0
-1
e-ph
-1
(T)(s )
5x10
Experiment
Zr80Sn20-xFex
10
9.9x10
10
9.0x10
10
10
1x10
11
0.0
0.5
1.0
1.5
2.0
2.5
3.0
8.1x10
3.5
X
0.35
10
0.40
0.45
0.50
le(nm)
T = 10K
10
3x10
Zr80Sn20-xFex
1.1x10
11
9.9x10
10
9.0x10
10
8.1x10
10
~ le
10
-1
1x10
-1
-1 e-ph
(S )
10
2x10
0.36
0.38
0.40
0.42
0.44
0.46
le(nm) Fig. 6. Variation of electron–phonon scattering rate (τe−−1ph (10K)) with (a) Fe content x and (b) electron elastic mean free path le for different disordered Zr80Sn20 xFex alloys.
where ρ is the density and ut is the transverse velocity and k is a constant. Taking the values of ρ ¼6.88 103 kg/m3, ut ¼2.53 103 m/s, KF ¼1.7 1010 m 1, N(EF)¼1.27 1047 states/J/m3 with k¼ 0.5 (when transverse phonons dominate), the calculated value of τe−−1ph at 10 K lies in the range 3.54 1010–4.41 1010 s 1 for different Zr80Sn20 xFex samples. The variation of theoretical and experimental values of τe−−1ph has been shown in Fig. 6(a). According to the figure it is observed that the theoretically predicted values did not match with the experimental value. So it is concluded that the magnitude of the electron–phonon interaction can not be explained by the existing theory of electron–phonon interaction. On the other hand, the disorder present in the materials is an important factor for the study of the electron–phonon scattering rate. Generally disorder present in the materials is measured by the electron elastic mean free path by the relation (le ∝ 1/ρ0 ). In order to study the disorder dependency of τe−−1ph we have prepared several Zr80Sn20 xFex alloys containing different amount of disorder i.e. different values of ρ0 or different le . A straight line variation of τe−−1ph at a temperature T¼10 K with le for several alloys has been shown in Fig. 6(b). This observation suggests that the temperature and disorder dependence electron–phonon scattering rate follows the relation τe−−1ph ∝ T 2le . Similar results have already been reported in few three dimensional disordered systems [23,41,42]. Hence the observed anomalous behavior of electron–phonon scattering τe−−1ph ∝ T 2le cannot be explained by the existing theories [11,15,16,38,40], where the electron–phonon scattering rate was reported by the relation τe−−1ph ∝ T 2/le . This discrepancy between the experiment and theories may be raised due to the fact that the existing theories were formulated on the basis of the spherical Fermi surface, however the present investigated disorder material (Zr80Sn20 xFex) has complex Fermi surface. So the anomalous experimental result demands a modification of the existing Debye type phonon spectrum to formulate the actual electron–phonon scattering rate in strong disorder system. The variation of zero temperature dephasing scattering rate (τ0−1) with iron content (x) and elastic mean free path (le ) has been
0
-1
e-ph
-1
(T)(s )
4x10
Zr80Sn20-XFeX
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
X Fig. 7. Variation of zero temperature dephasing scattering rate (τ0−1) with (a) Fe content x and (b) electron elastic mean free path le for different disordered Zr80Sn20 xFex alloys.
shown in Fig. 7(a) and (b) respectively. It is observed from Fig. 7 (a) that the values of τ0−1 decrease with increase in Iron content x. The zero temperature dephasing scattering rate may be attributed by different mechanisms like magnetic scattering and microwave noise etc. In the case of magnetic scattering, the magnitude of τ0−1 increases with the increase of magnetic impurity concentration. But according to Fig. 7(a) the value of τ0−1 decreases with increasing Fe concentration x. This behavior suggests that the magnetic scattering contribution to τ0−1 is not dominating due to presence of small Fe atom in the investigated samples. On the other hand, Altshuler et al. [43] pointed out that in three dimensional system the zero temperature dephasing scattering rate also depends on the microwave noises and expressed by the relation τ0−1 − D1/3 − le1/3. However, Fig. 7(b) shows a straight line variation of τ0−1 with le which rules out the contribution of microwave noise to τ0−1. As the values of the parameter le represent the amount of disorder present in the samples, the above observation suggests that the disorder dependence of τ0−1 is dominating over the contribution due to magnetic scattering. From the analysis of the experimental results it may be concluded that both the zero temperature dephasing scattering rate (τ0−1) and the temperature dependent electron–phonon scattering rate (τe−−1ph ) follow the similar (linearly increasing) behavior on le . This may be due to the fact that the values of le have been calculated from the concept free electron model with spherical Fermi surface and the experimentally measured impurity resistivity (ρ0 ). But in case of disordered systems, the concept of spherical Fermi surface may be changed and this change may modify the value of rs (radius of a sphere whose volume is equal to the volume per conduction electron). As a result the values of KF and le of the investigated disordered sample will change. However, in the literature, such theoretical calculation has
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R.N. Jana et al. / Physica E 77 (2016) 7–12
not been reported earlier. Therefore, in such disordered dirty systems, more experimental and theoretical studies are needed to clear the actual mechanism present. For the analysis of the experimental magnetoresistivity data, we have used the three dimensional WEL theory. So it is important to check whether the data in the present investigation satisfy the three dimensional criterion or not. Considering the measured values of τϕ(T), we have calculated the electron dephasing length Lϕ ¼(Dτϕ)1/2 for Zr80Sn20 xFex alloys systems. The calculated values of Lϕ(T) varies from 285–451 Å for decreasing the temperature from 20 to 5 K. Therefore, every investigated sample obeys the three dimensional criterion and the analysis of experimental magnetoresistivity data using the three-dimensional WEL theory is justified.
4. Conclusion In summary, we have studied the electrical transport of threedimensional disordered Zr80Sn20 xFex alloys at the temperature range 5 rT r300 K. A minima has been observed on the resistivity versus temperatures curve at T¼ Tm for different samples. The low temperature (5 rT r20 K) resistivity in absence of magnetic field can be explained by the three dimensional electron–electron interaction mechanisms and obeys the ρo5/2T1/2 law. The inelastic scattering time, zero temperature scattering time and spin–orbit scattering time have been calculated from the analysis of low magnetic field magnetoresistivity data by using three dimensional WEL theories. The spin–orbit interaction is strong and is independent of temperature. The disorder dependence of zero temperature scattering rate is dominating over magnetic scattering contribution and its magnitude increases with increase of le . The inelastic scattering rate manifests the direct observation of electron–phonon scattering and obeys an anomalous behavior τe−−1ph ∝ T 2le . Although, the quadratic temperature dependence can be explained by incomplete dragging of random scattering potential by phonons in dirty limit, the disorder or mean free path dependence of electron–phonon scattering rate does not explained by the existing theories.
Acknowledgments This work was supported by the Department of Science and Technology, Govt. of India through Grant no. SR/S2/CMP-18/2012.
References [1] E. Abrahams, P.W. Anderson, D.C. Licciardello, T.V. Ramakrishnan, Phys. Rev. Lett. 42 (1979) 673. [2] H. Suhl, M.B. Maple (Eds.), Superconductivity in d- and f-Band Metals, Academic Press, New York, 1980. [3] G. Bergmann, Phys. Rev. Lett. 48 (1982) 1048. [4] W. Buckel, W. Weber (Eds.), Superconductivity in d- and f-Band Metals, Kernforschungszentrum, Karlsruhe GmbH, Karlsruhe, 1982. [5] P.A. Lee, T.V. Ramakrishnan, Rev. Mod. Phys. 57 (1985) 287. [6] B.L. Altshuler, A.G. Arnov, in: A.L. Efros, M. Pollok (Eds.), Electron–Electron Interactions in Disordered Systems, Elsevier, Amsterdam, 1985. [7] J.H. Mooij, Physica Status Solidi A 17 (1973) 521. [8] C.C. Tsuei, Phys. Rev. Lett. 57 (1986) 1943. [9] M.Yu Reizer, Phys. Rev. B 40 (1989) 5411. [10] M.L. Roukes, M.R. Freeman, R.S. Germain, R.C. Richardson, M.B. Ketchen, Phys. Rev. Lett. 55 (1985) 422. [11] (a) M.Yu. Reizer, A.V. Sergeev, Zh. Eksp. Teor. Fiz. 90 (1986) 1056; (b) M.Yu. Reizer, A.V. Sergeev, Sov. Phys. JETP 63 (1986) 616. [12] G. Bergmann, Phys. Rev. B 3 (1971) 3797. [13] A.V. Sergeev, M. Yu Reizer, Int. J. Mod. Phys. B 10 (1996) 635. [14] T. Takayama, Z. Phys. 263 (1973) 329. [15] D. Belitz, Phys. Rev. B 36 (1987) 2513. [16] J. Rammer, A. Schmid, Phys. Rev. B 34 (1986) 1352. [17] A. Sergeev, V. Mitin, Phys. Rev. B 61 (2000) 6041. [18] K.P. Peters, G. Bergmann, J. Phys. Soc. Jpn. 54 (1985) 3478. [19] C.Y. Wu, W.B. Jian, J.J. Lin, Phys. Rev. B 57 (1998) 11232. [20] J.J. Lin, C.Y. Wu, Europhys. Lett. 29 (1995) 141. [21] J.F. DiTusa, K. Lin, M. park, M.S. Isaacson, J.M. Parpia, Phys. Rev. Lett. 68 (1992) 1156. [22] C. Kittel, Introduction to Solid State Physics, Wiley, New York, 1986. [23] A.K. Meikap, Y.Y. Chen, J.J. Lin, Phys. Rev. B 69 (2004) 212202. [24] (a) B.L. Al’tshuler, A.A. Aronov, Sov. Phys. JETP 50 (1979) 968; (b) B.L. Al’tshuler, A.A. Aronov, Zh. Eksp.Teor. Fiz. 77 (1979) 2028. [25] M. Kavch, N.F. Mott, J. Phys. C.: Solid State Phys. 15 (1982) L707. [26] A. Sahnoune, J.O. Strom-Olsen, Phys. Rev. B 29 (1989) 7561. [27] M.A. Howson, D. Greig, J. Phys. F16 (1986) 989. [28] M.A. Howson, J. Phys. F14 (1984) L25. [29] J. Kondo, Prog. Theor. Phys. 32 (1964) 37. [30] (a) A. Abrikosov, L.P. Gorkov, Zh. Eksp. Theor. Fiz. 42 (1962) 1088; (b) A. Abrikosov, L.P. Gorkov, Sov. Phys. JETP 15 (1962) 752. [31] D.V. Baxter, R. Richter, M.L. Trudeau, R.W. Cochrane, J.O. Strom- Olsen, J. Phys. 50 (1989) 1673. [32] A. Schmid, Z. Phys. B 271 (1974) 251. [33] (a) B.L. Al’tshuler, A.G. Aronov, Pis’ma Zh. Eksp. Teor. Fiz. 30 (1979) 1088; (b) B.L. Al’tshuler, A.G. Aronov, JETP Lett. 30 (1979) 482. [34] A.K. Meikap, A.R. Jana, S.K. De, S. Chatterjee, Solid State Commun. 77 (1991) 249. [35] M.E. Gershenson, Ann. Phys. 8 (1999) 559. [36] G.D. Kneip Jr., J.O. Betterton Jr., J.O. Searbrough, Phys. Rev. 131 (1963) 2425. 11 2 E (u + 2ut ) , dm where E¼ 0.898 10 N/m for Zr [36], vs = l 3 = [2 (1−σp )/(1−2σp )]1/2 , σp = 0. 38 for Zr [36]. Y.L. Zhong, A. Sergeev, C.D. Chen, J.J. Lin, Phys. Rev. Lett. 104 (2010) 206803. G. Bergmann, W. Wei, Y. Zou, R.M. Mueller, Phys. Rev. B 41 (1990) 7386. (a) A. Sergeev, V. Mitin, Superlattice Microst. 27 (2000) 499; (b) A. Sergeev, V. Mitin, Euro. Phys. Lett. 51 (2000) 641. Y.L. Zhong, J.J. Lin, Phys. Rev. Lett. 80 (1998) 588. Y.L. Zhong, J.J. Lin, L.Y. Kao, Phys. Rev. B 66 (2002) 132202. B.L. Al’tshuler, A.G. Aronov, D.E. Khmelnitsky, Solid State Commun. 39 (1981) 619.
[37] vs = [38] [39] [40] [41] [42] [43]
ul ut
Contents lists available at ScienceDirect
Physica E journal homepage: www.elsevier.com/locate/physe
Anomalous dephasing scattering time of Zr80Sn20 xFex alloys at low temperature R.N. Jana, S. Sinha, A.K. Meikap n Department of Physics, National Institute of Technology Durgapur, Mahatma Gandhi Avenue, Durgapur 713209, West Bengal, India
H I G H L I G H T S
G R A P H I C A L
A minima has been observed on the resistivity versus temperatures curve. The low temperature resistivity in absence of magnetic field obeys the ρo5/2T1/2 law. The spin–orbit interaction is strong and independent of temperature. The zero temperature scattering rate increases with increase of le . The inelastic scattering rate obeys an anomalous behavior τe−−1ph ∝ T 2le .
The electron–phonon scattering rate shows anomalous behavior and obeys the relation τe−−1ph ∝ T2le , where le is the electron elastic mean free path.
art ic l e i nf o
a b s t r a c t
Article history: Received 20 July 2015 Accepted 19 October 2015 Available online 28 October 2015
We report the results of a comprehensive study of weak electron localization (WEL) and electron– electron interaction (EEI) effects in disordered Zr80Sn20 xFex alloys. The resistivity in absence of magnetic field shows a minimum at temperature T ¼Tm and follows ρo5/2T1/2 law within the temperature range 5 K rT r Tm, which suggests predominant EEI effect. Magnetoresistivity is positive due to strong spin– orbit interaction. The dephasing scattering time is dominated by the electron–phonon scattering. The electron–phonon scattering rate shows anomalous behavior and obeys the relation τe−−1ph ∝ T2le , where le is the electron elastic mean free path. The zero temperature scattering time (τ0 ) strongly depends on the disorder and its magnitude decreases with increasing disorder resistivity ρ0 . Such anomalous behavior cannot be explained in terms of existing theories. & 2015 Elsevier B.V. All rights reserved.
Keywords: A. Zirconium alloy D. Low temperature D. Magnetoresistivity D. Electron phonon scattering
A B S T R A C T
1. Introduction Anomalous temperature dependence of normal state resistivity in disordered solids has become an interesting topic in solid state physics during the last few decades [1–6]. In such disordered solid a critical resistivity (ρc) has been defined by Mooji [7], where temperature coefficient of resistivity (TCR) is zero. He also suggested that the TCR is negative for disordered materials having resistivity ρ 4150 μΩ cm. However, Tsuei [8] reported that the n
Corresponding author. E-mail address: [email protected] (A.K. Meikap).
http://dx.doi.org/10.1016/j.physe.2015.10.020 1386-9477/& 2015 Elsevier B.V. All rights reserved.
weak electron localization (WEL) and the electron–electron interaction (EEI) play a significant role on TCR of disordered alloys. At low temperatures both WEL and EEI effects introduce corrections to temperature dependent resistivity. It is established that in absence of magnetic field conductivity correction is dominated by EEI, but low magnetic field dependent conductivity is dominated by WEL in the case of three-dimensional disordered metals. By fitting the low field magnetoresistivity with the theory of WEL, the different scattering time like temperature dependent inelastic scattering time (τi ) and temperature independent spin–orbit scattering time (τSO ) and zero temperature scattering time (τ0 ) has been calculated. Generally, the two mechanisms like electron–
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R.N. Jana et al. / Physica E 77 (2016) 7–12
electron and electron–phonon scattering have major contributions to the temperature dependence of the inelastic scattering. It is well established that in case of three dimensional systems the electron–phonon scattering is the dominating inelastic scattering process. The temperature dependence of the electron–phonon scattering for clean systems has been well established both theoretically and experimentally [9,10], but in disordered metals it is still in controversy. Theoretically the electron–phonon interaction in disordered metals has explained by many authors [11–17] but they concluded in different predictions. Some experimental works [18–21] already have been done on various disordered conductors on the issue of temperature dependence electron–phonon scattering but they were not able to establish a complete conclusion. Particularly, the disorder or mean free path (le ) dependent zero temperature scattering rate (τ0−1) and electron–phonon scattering rate (τe−−1ph ) of these alloys has not been established properly for different systems. In this work, we have reported the systematic measurement of τe − ph from WEL studies in a series of disordered Zr80Sn20 xFex and have shown how the electron–phonon scattering rate (τe−−1ph ) and the zero temperature
Table 1 Values of relevant physical parameters for disordered Zr80Sn20 xFex alloys, ρo and ρ (300 K) are the resistivity at 10 K and 300 K respectively, le is the mean free path, D is the electron diffusion coefficient, KF is the Fermi wave vector, Tm is the temperature at which resistivity is minimum, τso 1 is the spin–orbit scattering rate and
τϕ−1 (10 K) is the dephasing scattering rate at 10 K. τ0−1 is the zero temperature dephasing scattering rate, p is the temperature exponent of electron–phonon scattering rate (τe−−1ph ) and Ae − ph is the strength of electron–phonon coupling. Parameters
V82Al18Fe0
V82Al17Fe1
V82Al16Fe2
V82Al15Fe3
ρ0 (mΩ cm) ρ (300 K) (mΩ cm) le (Å) D (cm2/s) KF le Tm (K)
91.9 144.2 4.59 2.07 7.80 11.0 2.70 1013
98.2 165.5 4.29 1.93 7.30 11.5 2.73 1013
105.1 182.1 4.01 1.81 6.82 12.0 2.79 1013
114.5 198.3 3.68 1.66 6.26 13.0 2.85 1013 0.91 1011
−1 (s 1) τSO
τϕ−1 (s 1) τ0−1
(s
1
)
P Ae − ph (s 1 K p)
2.70 1013
1.16 1011
1.01 1011
1.07 1011
9.20 1010
8.55 1010
7.83 1010
2.14 2.43 108
2.09 1.87 108
2.01 1.50 108
2.05 1.16 108
scattering rates (τ0−1) depend on disorder of these alloys.
115.2 115.0 114.8 114.6 114.4 105.6
3. Results and discussions We have measured the electrical resistivity of the disordered Zr80Sn20 xFex alloys in the temperature range 5rTr300 K in absence of magnetic field. The variation of electrical resistivity with temperature is shown in Fig. 1 for different samples. The figure shows anomalous behavior, the resistivity initially decreases with decreasing temperature and then increases with further decreases of temperature i.e. the curve shows a resistivity minimum at temperature T¼ Tm, which is listed in Table 1. On the other hand, no superconducting transition has been observed in the investigated
(T)(
Zr80Sn20 xFex alloys with Iron concentration x¼ 0, 1, 2, 3 were prepared by a standard arc melting method. Appropriate amounts of spec-pure Zr, Sn and Fe were melted several times. The melted ingots were subjected to a homogenizing anneal at 800 °C for 48 h. Nominal Iron concentration was used to produce compositional disorder in the samples. The samples were cut into specimens of rectangular shape, typically ¼0.2 0.2 10 mm3, for resistivity measurement. Copper electrodes were spot-welded on to the samples. Resistance and magnetoresistance were measured by using the standard four probe technique and were performed in a closed cycle Helium Cryostat. The temperature was measured and monitored by a lakeshore 335 temperature controller. Low measurement currents were applied and care was taken to avoid any appreciable Joule heating of the electrons. In the case of magnetoresistance measurements, in order to minimize any appreciable contribution from the many body EEI effects, a small magnetic field range was always used. The values of Fermi wave vector (KF) for different samples were evaluated by using the free electron model result [22], KF ¼3.63/(rs/ao) Å 1 taking the values of rs/ao for individual materials where ao is the Bohr radius and rs is the radius of a sphere whose volume is equal to the volume per conduction electron. The electron elastic mean free path ‘le’ is then obtained through the relation le ¼ 3π2ħ/(kF2e2ρo), where ρo (at 10 K) is the impurity resistivity. The values of the electron diffusion constant D ℏK 1 have been found out from the relations D = 3 vF le and vF = mF , where m is 1.46 times the free electron mass [23]. The estimated values of le, D and KFle are listed in Table 1.
cm)
2. Experimental method
Zr80Sn20-xFex
x=3
105.4
x=2
105.2 105.0 104.8 98.40 98.32
x=1
98.24 98.16 92.02 92.00 91.98 91.96 91.94
x=0
4
6
8
10
12
14
16
18
20
22
T(K) Fig. 1. Resistivity as a function of temperature of different Zr80Sn20 xFex alloys.
temperature range 5rTr300 K. Such behavior of the samples provides an advantage for quantitative study of the electron–phonon scattering time τe–ph in the wide range of temperature from 5 K to 20 K. In order to understand the low temperature electrical transport of the investigated samples, we have studied the temperature and disorder dependence of resistivity in absence of magnetic field. At low temperature (ToTm), the different effects like EEI, WEL and Kondo type scattering [24–29] introduce temperature dependent corrections to resistivity and given by the relation
Δρ (T ) = αT 1/2 + βT p /2 + γ ln T ,
α= −
1.3e2ρ02 ⎡ 4 3F ⎤ kB , ⎢ − ⎥ 4π 2= ⎣ 3 2 ⎦ 2ℏD
(1)
(2)
R.N. Jana et al. / Physica E 77 (2016) 7–12
4.0
0.16
cm )
2.0
4
1.0x10
5
5/2 0
1.2x10
(
cm)
5
1.4x10
5
/
8.0x10
5/2
cm) (T)(
-0.1
X=0
-0.2
X=1
1.5
0.5 0.0 0.0
X=2
-0.4
X=3
Zr 80 Sn 20-x Fe x 2.0
2.5
3.0 1/2
0.1
0.2
0.3
0.4
B(T)
-0.5 -0.6
T=5K T=8K T=10K T=15K T=18K T=20K
1.0
0.0
-0.3
0
-1 -1
2.5
(
0.04
3.0
2
1/2
(T)/T (
0.00
Zr80Sn18Fe2
3.5
Zr 80 Sn 20-x Fe x
0.12
-
cmK
-1/2
)
0.20
0.08
9
3.5
1/2
T (K ) Fig. 2. (a) Resistivity increase Δρ ¼ρ(T) ρo as a function of T1/2 for disordered Zr80Sn20 xFex alloys. The straight solid lines are least square fit to Eq. (1). (b) The slope of resistivity increase Δρ(T)/T1/2 as a function of ρo5/2 for disorder Zr80Sn20 xFex alloys. The straight solid line is a guide to the eye.
where first term of Eq. (1) corresponds to EEI, 2nd term corresponds to WEL and third term due to Kondo type scattering, e is the electronic charge, ρ0 is the resistivity at 10 K, ħ is Planck's constant divided by 2π, kB is the Boltzmann constant, F is a screening factor averaged over Fermi surface, β and γ are two different constants, p is an exponent of inelastic scattering time (τi − T p). A straight line plot of Δρ(T) with T1/2 for different samples has been shown in Fig. 2(a) which ruled out the contribution from Kondo effect and suggested that the contribution due to the EEI is dominating over WEL effect. Apart from the T1/2 dependence, the resistivity correction Δρ(T) is also a function of disorder (Eq. (2)) and proportional to ρ02 /D1/2 or ρ05/2, where D − ρ0−1 is the diffusion constant. Fig. 2 (b) shows the variation of resistivity slope Δρ (T ) /T1/2 with ρ05/2 for Zr80Sn20 xFex alloys. The symbols are the experimental data, and the straight solid line guides the eye. In Fig. 2(b) it is seen that the rise in the slope of the resistivity indeed varies linearly with ρ05/2, firmly supporting the EEI predictions. This observation is an excellent agreement between the theory (EEI) and experiment i.e. Δρ (T ) − ρ05/2 T1/2 − ρ02 (T /D)1/2 and suggested that the EEI is completely controlled by the total amount of disorder (ρ0 ) present in the sample. Magnetoresistivity of investigated alloys was measured under transverse low magnetic fields up to 0.4 T to avoid the contribution due to EEI in the temperature range 5–20 K. Measured magnetoresistivity is positive for all samples, which suggests the strong spin–orbit scattering effects in these alloys. Abrikasiv and Gorkov [30] pointed out that the spin–orbit scattering depends on the atomic number (Z) of the metals and the elastic scattering time
Fig. 3. The variation of the magnetoresistivity with magnetic field at different constant temperature of Zr80Sn18Fe2 alloys. The points are the experimental data and the solid lines are the theoretical predictions from Eq. (3).
−1 (τe), and is given by the relation τso = (βZ )4 τe−1 where β is the fine structure constant. The atomic number of Zr (Z¼ 40) and that of Sn (Z¼46) are moderate and Fe (Z¼28) is low. Therefore, the Zr and Sn atoms introduce the strong spin–orbit scattering in the alloys. On the other hand, the resistivity of the alloys is high, which introduce strong electron elastic scattering rate (τe−1). As a result the investigated samples show strong spin–orbit scattering. It is well established that information for the various electron dephasing scattering time in impure metals has been obtained from analysis of measured magnetoresistivity by the theory of WEL effects. As the investigated samples did not show any superconducting transition within the investigated temperature range, we have analyzed the magnetoresistivity data in the light of three-dimensional WEL theoretical predictions without introducing the superconducting fluctuation correction. The variation of the magnetoresistivity with magnetic field for Zr80Sn20 xFe2 has been shown in Fig. 3. Different points in the figure represent experimental data at different temperature and the solid lines are the theoretical best fitted values obtained using Eq. (3) with fitting parameters Bϕ and Bso [31].
e2 Δρ (B, T ) = 2 2π 2= ρo
⎡ ⎛ ⎞⎤ eB ⎢ 1 ⎛ B ⎞ 3 ⎜ B ⎟⎥ ⎜ ⎟ f f − = ⎢ 2 3 ⎝ Bϕ ⎠ 2 3 ⎜⎝ Bϕ + 4 Bso ⎟⎠ ⎥ ⎦ ⎣ 3
(3)
where Bϕ is the electron dephasing scattering field and Bso is the spin–orbit scattering field. The function f3(x) has already been defined in Baxter et al. [31]. It is shown from the figure that the three-dimensional WEL theory is well suited for the analysis of measured magnetoresistivity, and the electron dephasing scattering field Bϕ and spin–orbit scattering field Bso are reliably extracted from these fits. The different scattering time is related with the scattering field by the relation τx ¼ ħ/(4eDBx). For electron dephasing scattering x¼ ϕ and for spin orbit scattering x ¼so. It is observed that the spin–orbit scattering rate is greater than the electron dephasing scattering rate (τso 1 4 4 τϕ 1) and is independent of temperature and the calculated values of τso 1 and τϕ 1(10 K) are given in Table 1. Generally in disordered
10
R.N. Jana et al. / Physica E 77 (2016) 7–12
11
conductors the electron dephasing scattering time consists of zero temperature scattering time (τ0 ) and temperature dependent inelastic scattering time (τin ) and can be presented by the relation
1 1 1 = + . τϕ (T ) τ0 τin (T )
ing scattering rate τϕ−1 (T ) is shown in Fig. 4 for Zr80Sn18Fe2 sample. The symbols in the figure are the experimental data, whereas different lines are the theoretical fit. The least-square fit to Eq. (4) with τo, Ae–ph and p as fitting parameters is given by the solid line, where the best fit value of p is 2.01. On the other hand, the short dashed and dotted curves are also least square fits to Eq. (4) with fixed values of p as 3 and 4 respectively, while τo and Ae–ph as fitting parameters. It is observed from Fig. 4 that the experimental data did not match with the theoretical prediction with p ¼ 3 and 4 but well suited with p ¼2.01. Therefore, we may conclude that the temperature dependent part of the dephasing scattering time can be described by quadratic temperature dependence. For all other samples the variation of τϕ−1 (T ) with temperature has been 11
1.6x10
Zr80Sn18Fe2 11
1.4x10
-1
~T
p
11
1.2x10
p=4
-1
-1
(s )
-1
(S )
(5)
−1 We have calculated the values of τee for Zr80Sn20 xFex alloys at temperature T ¼5 K by using Eq. (5) and the values lie in the range 8.01 107–1.11 108 s 1. It is found that the calculated values of −1 are 732–1426 times smaller than the experimentally measured τee inelastic scattering rate for all the investigated samples. This is because the measurement temperature is above 5 K, where the electron–electron scattering contribution to inelastic scattering is minimized. Therefore, this result rules out any contribution from electron–electron scattering to the inelastic scattering rate. On the other hand, for three-dimensional disordered alloys the inelastic scattering at higher temperature TZ 5 K is dominated by electron– phonon scattering [16,34,35] and can be expressed as τin−1 (T ) − τe−−1ph − Ae − ph T p . The temperature dependence of dephas-
p=3 p = 2.01
11
1.0x10
10
4
11
2x10
-1
3/2 6m ⎛ kB T ⎞ ⎟ . ⎜ 2 2ℏ KF ⎝ KF le ⎠
8.0x10
Zr80Sn20Fe0 Zr80Sn19Fe1 Zr80Sn18Fe2 Zr80Sn17Fe3
(4)
The temperature dependent inelastic scattering time (τin ) arises due to electron–electron scattering (τee ) and electron–phonon scattering (τe − ph ). In three dimensional disordered systems, the electron–electron scattering rate can be expressed as [32–33] −1 τee =
3x10
6
8
10 12 14 16 18 20 22
T(K) Fig. 4. Electron dephasing rate τϕ 1(T) as a function of temperature for the Zr80Sn18Fe2 alloy. The solid, short dashed and dotted curves are least square fit to Eq. (4) with the exponent of temperature p ¼2.01, 3 and 4 respectively.
11
1x10
4
6
8 10 12 14 16 18 20 22
T(K)
Fig. 5. Electron dephasing rates τϕ 1(T) as a function of temperature for different disordered Zr80Sn20 xFex alloys. The solid curves are least square fit to Eq. (4).
shown in Fig. 5. In the figure the different points are the experimental data and the solid lines are the fitting lines obtained from Eq. (4) with fitting parameters τo, Ae–ph and p. The best fitted values of the parameters for different samples are τo ¼(0.94–1.28) 10 11 s, Ae–ph ¼(1.16–2.43) 108 s 1/Kp and k T p¼ 2.01–2.14. In quasiballistic limit ( qph le > 1, qph = ℏBv is the wave s
number of the thermal phonons, vs is the sound velocity and equal to 3.61 103 m/s [36,37]), Zhong et al. [38] reported the electron– phonon scattering rate by considering the additive contribution from vibrating defects and impurities. Taking the value of vs and le for the investigated samples we have calculated qph le = (0.013 − 0.017) T . Hence the value of qph le does not meet the quasiballistic criteria (qph le > 1) in the investigated temperature range 5r Tr 20 K for different samples. So the observed quadratic temperature dependence of electron–phonon interaction could not be explained by the above theoretical prediction [38]. On the other hand, the disorder or dirty criteria (qph le < 1) has been satisfied from the above calculation. In such dirty system (qph le < 1), theoretical studies of quadratic temperature dependence of electron–phonon interaction was done by different authors [11–17,39]. Considering all earlier theories Sergeev and Mittin [17,40] have formulated the theory of the electron–phonon scattering time in a disordered conductor, where they take into account both the static and vibrating random scattering potentials. They found that if the random scattering potential is completely dragged by phonons, the effective electron–phonon scattering rate would decrease due to disorder and presented by the relation τe−−1ph − T 4le at low temperature, which is not observed in our investigated samples. However, they also pointed out that for such systems, due to presence of imperfections the scattering potential would not be completely dragged by the phonon. In such incomplete drag system the interaction between electrons and transverse phonons would dominate over the interaction between electrons and longitudinal phonons and the electron–phonon scattering rate is given by the relation [40]
τe−−1ph =
π 2KF2 kB2 N ( EF ) k (1 − k ) 6m2ρut3 le
T2,
(6)
R.N. Jana et al. / Physica E 77 (2016) 7–12 10
11
1.1x10
T = 10K
(S )
10
4x10
-1
Theory
10
-1
3x10
10
2x10
0
-1
e-ph
-1
(T)(s )
5x10
Experiment
Zr80Sn20-xFex
10
9.9x10
10
9.0x10
10
10
1x10
11
0.0
0.5
1.0
1.5
2.0
2.5
3.0
8.1x10
3.5
X
0.35
10
0.40
0.45
0.50
le(nm)
T = 10K
10
3x10
Zr80Sn20-xFex
1.1x10
11
9.9x10
10
9.0x10
10
8.1x10
10
~ le
10
-1
1x10
-1
-1 e-ph
(S )
10
2x10
0.36
0.38
0.40
0.42
0.44
0.46
le(nm) Fig. 6. Variation of electron–phonon scattering rate (τe−−1ph (10K)) with (a) Fe content x and (b) electron elastic mean free path le for different disordered Zr80Sn20 xFex alloys.
where ρ is the density and ut is the transverse velocity and k is a constant. Taking the values of ρ ¼6.88 103 kg/m3, ut ¼2.53 103 m/s, KF ¼1.7 1010 m 1, N(EF)¼1.27 1047 states/J/m3 with k¼ 0.5 (when transverse phonons dominate), the calculated value of τe−−1ph at 10 K lies in the range 3.54 1010–4.41 1010 s 1 for different Zr80Sn20 xFex samples. The variation of theoretical and experimental values of τe−−1ph has been shown in Fig. 6(a). According to the figure it is observed that the theoretically predicted values did not match with the experimental value. So it is concluded that the magnitude of the electron–phonon interaction can not be explained by the existing theory of electron–phonon interaction. On the other hand, the disorder present in the materials is an important factor for the study of the electron–phonon scattering rate. Generally disorder present in the materials is measured by the electron elastic mean free path by the relation (le ∝ 1/ρ0 ). In order to study the disorder dependency of τe−−1ph we have prepared several Zr80Sn20 xFex alloys containing different amount of disorder i.e. different values of ρ0 or different le . A straight line variation of τe−−1ph at a temperature T¼10 K with le for several alloys has been shown in Fig. 6(b). This observation suggests that the temperature and disorder dependence electron–phonon scattering rate follows the relation τe−−1ph ∝ T 2le . Similar results have already been reported in few three dimensional disordered systems [23,41,42]. Hence the observed anomalous behavior of electron–phonon scattering τe−−1ph ∝ T 2le cannot be explained by the existing theories [11,15,16,38,40], where the electron–phonon scattering rate was reported by the relation τe−−1ph ∝ T 2/le . This discrepancy between the experiment and theories may be raised due to the fact that the existing theories were formulated on the basis of the spherical Fermi surface, however the present investigated disorder material (Zr80Sn20 xFex) has complex Fermi surface. So the anomalous experimental result demands a modification of the existing Debye type phonon spectrum to formulate the actual electron–phonon scattering rate in strong disorder system. The variation of zero temperature dephasing scattering rate (τ0−1) with iron content (x) and elastic mean free path (le ) has been
0
-1
e-ph
-1
(T)(s )
4x10
Zr80Sn20-XFeX
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
X Fig. 7. Variation of zero temperature dephasing scattering rate (τ0−1) with (a) Fe content x and (b) electron elastic mean free path le for different disordered Zr80Sn20 xFex alloys.
shown in Fig. 7(a) and (b) respectively. It is observed from Fig. 7 (a) that the values of τ0−1 decrease with increase in Iron content x. The zero temperature dephasing scattering rate may be attributed by different mechanisms like magnetic scattering and microwave noise etc. In the case of magnetic scattering, the magnitude of τ0−1 increases with the increase of magnetic impurity concentration. But according to Fig. 7(a) the value of τ0−1 decreases with increasing Fe concentration x. This behavior suggests that the magnetic scattering contribution to τ0−1 is not dominating due to presence of small Fe atom in the investigated samples. On the other hand, Altshuler et al. [43] pointed out that in three dimensional system the zero temperature dephasing scattering rate also depends on the microwave noises and expressed by the relation τ0−1 − D1/3 − le1/3. However, Fig. 7(b) shows a straight line variation of τ0−1 with le which rules out the contribution of microwave noise to τ0−1. As the values of the parameter le represent the amount of disorder present in the samples, the above observation suggests that the disorder dependence of τ0−1 is dominating over the contribution due to magnetic scattering. From the analysis of the experimental results it may be concluded that both the zero temperature dephasing scattering rate (τ0−1) and the temperature dependent electron–phonon scattering rate (τe−−1ph ) follow the similar (linearly increasing) behavior on le . This may be due to the fact that the values of le have been calculated from the concept free electron model with spherical Fermi surface and the experimentally measured impurity resistivity (ρ0 ). But in case of disordered systems, the concept of spherical Fermi surface may be changed and this change may modify the value of rs (radius of a sphere whose volume is equal to the volume per conduction electron). As a result the values of KF and le of the investigated disordered sample will change. However, in the literature, such theoretical calculation has
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not been reported earlier. Therefore, in such disordered dirty systems, more experimental and theoretical studies are needed to clear the actual mechanism present. For the analysis of the experimental magnetoresistivity data, we have used the three dimensional WEL theory. So it is important to check whether the data in the present investigation satisfy the three dimensional criterion or not. Considering the measured values of τϕ(T), we have calculated the electron dephasing length Lϕ ¼(Dτϕ)1/2 for Zr80Sn20 xFex alloys systems. The calculated values of Lϕ(T) varies from 285–451 Å for decreasing the temperature from 20 to 5 K. Therefore, every investigated sample obeys the three dimensional criterion and the analysis of experimental magnetoresistivity data using the three-dimensional WEL theory is justified.
4. Conclusion In summary, we have studied the electrical transport of threedimensional disordered Zr80Sn20 xFex alloys at the temperature range 5 rT r300 K. A minima has been observed on the resistivity versus temperatures curve at T¼ Tm for different samples. The low temperature (5 rT r20 K) resistivity in absence of magnetic field can be explained by the three dimensional electron–electron interaction mechanisms and obeys the ρo5/2T1/2 law. The inelastic scattering time, zero temperature scattering time and spin–orbit scattering time have been calculated from the analysis of low magnetic field magnetoresistivity data by using three dimensional WEL theories. The spin–orbit interaction is strong and is independent of temperature. The disorder dependence of zero temperature scattering rate is dominating over magnetic scattering contribution and its magnitude increases with increase of le . The inelastic scattering rate manifests the direct observation of electron–phonon scattering and obeys an anomalous behavior τe−−1ph ∝ T 2le . Although, the quadratic temperature dependence can be explained by incomplete dragging of random scattering potential by phonons in dirty limit, the disorder or mean free path dependence of electron–phonon scattering rate does not explained by the existing theories.
Acknowledgments This work was supported by the Department of Science and Technology, Govt. of India through Grant no. SR/S2/CMP-18/2012.
References [1] E. Abrahams, P.W. Anderson, D.C. Licciardello, T.V. Ramakrishnan, Phys. Rev. Lett. 42 (1979) 673. [2] H. Suhl, M.B. Maple (Eds.), Superconductivity in d- and f-Band Metals, Academic Press, New York, 1980. [3] G. Bergmann, Phys. Rev. Lett. 48 (1982) 1048. [4] W. Buckel, W. Weber (Eds.), Superconductivity in d- and f-Band Metals, Kernforschungszentrum, Karlsruhe GmbH, Karlsruhe, 1982. [5] P.A. Lee, T.V. Ramakrishnan, Rev. Mod. Phys. 57 (1985) 287. [6] B.L. Altshuler, A.G. Arnov, in: A.L. Efros, M. Pollok (Eds.), Electron–Electron Interactions in Disordered Systems, Elsevier, Amsterdam, 1985. [7] J.H. Mooij, Physica Status Solidi A 17 (1973) 521. [8] C.C. Tsuei, Phys. Rev. Lett. 57 (1986) 1943. [9] M.Yu Reizer, Phys. Rev. B 40 (1989) 5411. [10] M.L. Roukes, M.R. Freeman, R.S. Germain, R.C. Richardson, M.B. Ketchen, Phys. Rev. Lett. 55 (1985) 422. [11] (a) M.Yu. Reizer, A.V. Sergeev, Zh. Eksp. Teor. Fiz. 90 (1986) 1056; (b) M.Yu. Reizer, A.V. Sergeev, Sov. Phys. JETP 63 (1986) 616. [12] G. Bergmann, Phys. Rev. B 3 (1971) 3797. [13] A.V. Sergeev, M. Yu Reizer, Int. J. Mod. Phys. B 10 (1996) 635. [14] T. Takayama, Z. Phys. 263 (1973) 329. [15] D. Belitz, Phys. Rev. B 36 (1987) 2513. [16] J. Rammer, A. Schmid, Phys. Rev. B 34 (1986) 1352. [17] A. Sergeev, V. Mitin, Phys. Rev. B 61 (2000) 6041. [18] K.P. Peters, G. Bergmann, J. Phys. Soc. Jpn. 54 (1985) 3478. [19] C.Y. Wu, W.B. Jian, J.J. Lin, Phys. Rev. B 57 (1998) 11232. [20] J.J. Lin, C.Y. Wu, Europhys. Lett. 29 (1995) 141. [21] J.F. DiTusa, K. Lin, M. park, M.S. Isaacson, J.M. Parpia, Phys. Rev. Lett. 68 (1992) 1156. [22] C. Kittel, Introduction to Solid State Physics, Wiley, New York, 1986. [23] A.K. Meikap, Y.Y. Chen, J.J. Lin, Phys. Rev. B 69 (2004) 212202. [24] (a) B.L. Al’tshuler, A.A. Aronov, Sov. Phys. JETP 50 (1979) 968; (b) B.L. Al’tshuler, A.A. Aronov, Zh. Eksp.Teor. Fiz. 77 (1979) 2028. [25] M. Kavch, N.F. Mott, J. Phys. C.: Solid State Phys. 15 (1982) L707. [26] A. Sahnoune, J.O. Strom-Olsen, Phys. Rev. B 29 (1989) 7561. [27] M.A. Howson, D. Greig, J. Phys. F16 (1986) 989. [28] M.A. Howson, J. Phys. F14 (1984) L25. [29] J. Kondo, Prog. Theor. Phys. 32 (1964) 37. [30] (a) A. Abrikosov, L.P. Gorkov, Zh. Eksp. Theor. Fiz. 42 (1962) 1088; (b) A. Abrikosov, L.P. Gorkov, Sov. Phys. JETP 15 (1962) 752. [31] D.V. Baxter, R. Richter, M.L. Trudeau, R.W. Cochrane, J.O. Strom- Olsen, J. Phys. 50 (1989) 1673. [32] A. Schmid, Z. Phys. B 271 (1974) 251. [33] (a) B.L. Al’tshuler, A.G. Aronov, Pis’ma Zh. Eksp. Teor. Fiz. 30 (1979) 1088; (b) B.L. Al’tshuler, A.G. Aronov, JETP Lett. 30 (1979) 482. [34] A.K. Meikap, A.R. Jana, S.K. De, S. Chatterjee, Solid State Commun. 77 (1991) 249. [35] M.E. Gershenson, Ann. Phys. 8 (1999) 559. [36] G.D. Kneip Jr., J.O. Betterton Jr., J.O. Searbrough, Phys. Rev. 131 (1963) 2425. 11 2 E (u + 2ut ) , dm where E¼ 0.898 10 N/m for Zr [36], vs = l 3 = [2 (1−σp )/(1−2σp )]1/2 , σp = 0. 38 for Zr [36]. Y.L. Zhong, A. Sergeev, C.D. Chen, J.J. Lin, Phys. Rev. Lett. 104 (2010) 206803. G. Bergmann, W. Wei, Y. Zou, R.M. Mueller, Phys. Rev. B 41 (1990) 7386. (a) A. Sergeev, V. Mitin, Superlattice Microst. 27 (2000) 499; (b) A. Sergeev, V. Mitin, Euro. Phys. Lett. 51 (2000) 641. Y.L. Zhong, J.J. Lin, Phys. Rev. Lett. 80 (1998) 588. Y.L. Zhong, J.J. Lin, L.Y. Kao, Phys. Rev. B 66 (2002) 132202. B.L. Al’tshuler, A.G. Aronov, D.E. Khmelnitsky, Solid State Commun. 39 (1981) 619.
[37] vs = [38] [39] [40] [41] [42] [43]
ul ut