5 Results and Discussion-Conventional Ferromagnet:
5.2 Theoretical fitting in term of Radovic’s model
A microscopic theoretical model for the interpretation of experimental results with F/S trilayers has been proposed by Radović et al. The detail of theoretical model is presented in Chapter 4. The reduced TC with decreasing dS is associated with the pair-breaking effect within the single-mode approximation. In the framework of this model, the TC is given by Eq. (4.8). The effective pair-breaking parameter, ρ* (T) can be calculated in Eq. (4.12) by Usadel’s equation for the pair amplitude FS in the superconductor by making use of the boundary condition from Eqs 4.9 to 4.11. [5]
This condition implies a high-quantum-mechanical transparency of the F/S interface.
The diffusion coefficient DF of Co can be estimated in terms of the low-temperature conductivity σ and the coefficient of the electronic specific heat γ from the Pippard relation [6]
( 2 2) ( )
F B
v l =
π
k eσ γ
. (5.1)For a single-Co film prepared at conditions identical to the Co layer in our layered structures, the low-temperature resistivity was determined to be ρ= ×7 10−8 Ω cm. Using γ = 4.73 10× −3 J K2 mole for Co [7] the diffusion coefficient is derived from DF = 5 cm2/s. From the spin splitting energy 2I0=1.55 eV [8], the penetration depth of the superconducting pairing function in Co is estimated to beξF = 1.3nm.
The experimental results can be fitted well by equation (4.12) in terms of the Radovic’s model shown as the solid curves in Fig. 5.1. We can extract a critical thickness 30dNbcrit = nm by extrapolating the fit to TC = 0. The parameters for the calculation are TC = 9.1 K, ξS = 16nm, and ε =9.2 ( =η 0.01), For comparison, the superconductor coherence length deduced from the temperature-dependent upper-critical field measurement Hc2⊥(T) is ξS = 12nm. [3] This value is smaller than
the fitted result from proximity effect and should be a result of pair breaking effect.
The bulk resistivities at 10 K measured on sputtered single film of Nb0.4Ti0.6 and Nb0.6Ti0.4 were 40 μΩ cm and 80 μΩ cm, respectively, with errors of about 10 %. The residual resistance ratios (RRR) were larger than 2 for Nb, ~1.25 for Nb0.4Ti0.6 and less than 1.06 for Nb0.6Ti0.4 films, indicating the quality of our Nb0.6Ti0.4 films is not as good as the others. The electron mean free paths estimated from these resistivities were 4.7 nm for pure Nb, 0.9 nm for Nb0.4Ti0.6, and 0.5 nm for Nb0.6Ti0.4 with an assumption that the product ρ = l 3.75 10× −6 Ω μ cm2 remained unchanged. [9, 10]
Moreover, the bulk NbxTi1-x have Tc = 8.8 and 7.0 K for x = 0.4 and 0.6, respectively.
In the same strategy, the critical thickness for the case of Co/ Nb40Ti60, and Co/Nb60Ti40 trilayers are deduced from fitted results, as the solid lines in the Fig. 5.1 (b) and (c) show. Here, the data are within the range of dScrit ξS ≥ ; this ensures the usage of the 2 single-mode approximation, since higher-order modes are substantially short-range modes and strongly damped at dS >ξS. [5]
We looked up the literature and found that the large resistivity and low TC are most likely due to the structure variation as explaining following. Although the critical temperature peaks towards the niobium-rich side of the compositions, i.e. in the range 50-70 at.% Nb, The most widely used superconducting materials are based on Nb-Ti alloys with Ti contents ranging from 46-50 weight % Ti. These alloys of Nb and Ti have both high strength and ductility and can be processed to achieve high critical current densities that make them ideal candidates for magnet and applications.
Nb-Ti based superconductors are commercially produced in long uniform lengths and cost significantly less to produce than other superconductors. [11] The main drawbacks of this material are a low critical temperature, typically requiring cooling
by liquid helium, and a low upper critical field which limits the applied field at which they can be used to below 12 T.
The variations of TC, ρn and the upper critical field Hc2 as a function of alloy composition are plotted in Figure 5.2. The data were adapted from [11] and [12]. The critical temperature shows a mild variation between pure Nb (9.23 K) and Nb50Ti50 (8.5 K), with a weak peak at about Nb70Ti30 (9.8 K). Addition of Ti is more potent at reducing TC for alloys with Ti content above 50 mass%. The critical temperature drops continuously over this range with increasing Ti content. The rate of resistivity increase is concave upward, tending towards the Mott localization limit ( > 100 μΩ cm) for more than 70 at%Ti. Thin-films were used to show that these trends continue for higher Ti content [13], where bulk samples are difficult to make, as shown in the Fig 5.3. Resistivity increases with increase in Ti content. The resistivity of the thin film is found to be larger than typical values found in the corresponding bulk alloy as much as 20 μΩ cm. Except near the endpoints (pure Nb or pure Ti) where higher resistivity ratios can be obtained, residual resistivity ratios of these alloys are close to 1. (see Fig. 5.3)
Figure 5.2: The variation in Hc2 at 4.2 K , TC and resistivity are plotted as a function of the mass fraction of Ti across the binary Nb-Ti alloy system. Hc2 is defined as the linear extrapolation of the high field pinning force to zero. [11]
Figure 5.3: (a) Variation in resistivity as a function of Ti content in Nb-Ti films for 300 K and 10 K, respectively. (b) Variation in resistivity ratios with Ti content in Nb-Ti films for 300K/77K and 300K/10K, respectively. (c) Variation in TC with Ti content in Nb-Ti films. [13]
The upper critical field at 4.2 K exhibits a broad dome-like curve in the range of 40 weight % Ti to 60 weight % Ti with a maximum of about 11.6 T at a composition of 44 weight % Ti. The peak results from a balance of the trends for the resistivity and critical temperature, where the zero-temperature value can be predicted by
(a) (c) (b)
3 0Hc2(0) 3.11 10 TC
μ = × ργ . (5.2)
The above equation is an extension of the Ginzburg–Landau theory [14]. The value of Hc is linked to TC via the condensation energy of the superconducting state and the scattering value of ρ. Thus, the reason for this Hc2 behavior is basically from the anomalous increase in the normal-state resistivity. This increase in ρn is more than compensating for the slight decrease in γ, the electronic specific heat coefficient, and TC, resulting in an enhancement of the upper critical fiels Hc2 for Ti-rich alloys.
The atomic volume difference between Ti and Nb is only about 2 % resulting in a isomorphous system where the β phase has a body-centered cubic structure with a lattice parameter of approximately 0.3285 nm. An important property of the Nb-Ti phase diagram, shown in the Figure 5.4, is that the β phase starts to decompose only well below the melting temperature. [11] The β phase is favored at high temperature, and can be retained by quenching to room temperature. Many β alloys are good superconductors [15], as would be expected from the high transition temperatures of V, Nb and Ta. The other stable α phase in this system is the titanium rich phase which has a low solubility and low-temperature hexagonal close-packed (HCP) structure.
The low Nb content of the α-Ti phase suggests that α-Ti precipitates should have a low TC (approaching the 0.39 K TC of pure Ti) and should be non-superconducting under practical operating conditions. The alpha phase is only stable below 882 °C (at atmospheric pressure) and for the alloy composition range of interest Ti is only stable below 570 °C to 600 °C. In Figure 5.4 the widely used high temperature phase boundaries of Hansen et al. [16] are combined with the calculated low temperature boundaries of Kaufman et al. [17] modified by Moffat and Kattner [18] to provide a composite equilibrium phase diagram that generally reflects production experience.
The competition between these phases and the incipient phase transition of a quenched β alloy to α+β is the origin of many observed physical properties of Nb–Ti
alloy. Thus at room temperature and below, the standard alloy consists of metastable β phase and any phase transition is latent.
In the previous report, the resistivity increases with increasing in Ti content. In our system, even though the TC decreases with increase resistivity following the expected tendency, the resistivity of Nb0.6Ti0.4 is larger than Nb0.4Ti0.6. A. Main et al.
reported that the incipient instability of the β-phase (bcc lattice) in the Ti rich composition region resulting in the dynamical fluctuation of the diffuse phase (ω-phase) and leads to the anomalous increase in the resistivity. [13] In our system, this increase in the resistivity may be due to the stress developed in the film between substrate or ferromagnet layer which influenced the instability in the direction of relieving it in favor of a structural transformation. Another possibility is that the increase in resistivity is due to the small grain size of our polycrystalline films.
Figure 5.4: A hybrid equilibrium phase diagram for Nb-Ti combining the experimentally determined high temperature phase boundaries of Hansen et al [16]
with the calculated low temperature phase boundaries of Kaufman and Bernstein [17]
modified by Moffat and Kattner. [18] Also shown is the martensite transformation curve (Ms) of Moffat and Larbalestier. [19]