The s-d exchange model as the underlying mechanism of magnetoresistance in ZnO doped with alkali metals

High field magnetoresistance has been studied in epitaxial n-type ZnO:Na and ZnO:Li thin films in a temperature range between 4 K and 150 K. The resulting negative magnetoresistance can be well fitted using a semiempirical model of Khosla and Fischer based on third order contributions to the s-d exchange Hamiltonian. The parameters obtained from this model were carefully analyzed. One of these parameters is related to a ratio between electron mobilities at zero field (a non-exchange scattering mobility and an exchange or spin dependent one ). From Hall effect measurements was obtained, displaying a weak temperature dependence in accordance with highly n-doped ZnO while the extracted exhibits an anomalous T-dependence. On the other hand, our magnetoresistance data cannot be properly fitted using Kawabata’s expression based on a weak-localization model.


Introduction
When a few atomic percent of Manganese is replaced by Galium in GaAs, this semiconductor becomes ferromagn etic below a Curie temperature T C ∼ 110 K [1]. This was a remarkable discovery that allowed the development of spin tronic in semiconductors in the following years [2]. Despite there is still a debate about the origin of ferromagnetism in this III-V semiconductor, there is one model that stands among the rest: the p d or sd exchange model. In this picture, the itinerant carriers (which are presumably holes donated by Mn in the case of Ga 1−x Mn x As) experiment an exchange interaction with the localized magnetic moments (the Mn ones in Ga 1−x Mn x As) giving rise to a magnetic order. The effect observed in Ga 1−x Mn x As triggered the research in others potential dilute magnetic semiconductors (DMS) like ZnO or GaN where room temperature ferromagnetism was predicted when they are p doped [3][4][5]. This latter issue becomes a materials science challenge in the II-VI semiconducting ZnO since it naturally becomes ntype and the stability of acceptors in this material is rather low [6,7]. Moreover, the doping with magnetic elements like Mn should be carefully performed [8,9] since impurity clusters, spurious magnetic phases or an inhomogeneous distribution of magnetic impurities in the sample can generate magnetic signals which are the respon sible of the wide variety of T C values found in ZnO ranging from 0 to 1000 K [10,11].
Alternatively, it has been predicted that acceptor doping with nonmagnetic elements like Li or Na stabilizes the for mation of zinc vacancies V Zn [6,[12][13][14]. Moreover, recent works have proposed and also shown evidence of V Zn as local ized magnetic defects in ZnO [15,16]. Doping of ZnO with Na or Li has been previously performed in a wide variety of micro and nanostructures in search of the development of optical and optoelectronic devices [6,[12][13][14][17][18][19][20][21][22]. It has been found that the optimal Li or Na doping that generates substitutional defects like Zn Li or Zn Na (which are the ones that stabilize the V Zn ) without producing a significant amount of interstitial ones (which are donors) is around 5 at.% [13,14]. Above this concentration, interstitial doping starts to increase compensating the acceptor ones.
One of the key effects when itinerant electrons interact with localized magnetic moments is the presence of negative magnetoresistance (MR) [23][24][25], a relevant property for spin tronic applications [2] and a more subtle technique to analyze the DMS. In the case of ZnO, this effect has been observed in ndoped samples [26][27][28][29] with MR values between 3% and 7% at T ∼ 5-10 K. For electron carrier concentrations lower than n ∼ 3 × 10 18 cm −3 , the negative magnetoresistance is over passed by an enhanced positive magnetoresistance [26,28] in accordance to a twoband model (an impurity plus a con duction band) [30,31]. The negative MR in ZnO can be well described using a phenomenological model originally proposed by Khosla and Fischer to account for their experi ments in ntype CdS [31]. In their model, Khosla and Fischer adapt expressions for negative MR previously developed by Appelbaum based on third order terms of the sd Hamiltonian [32] in the context of a tunneling problem between two metals with magnetic impurities at the interface. This adaptation requires an appropriate transformation of tunneling rates to scattering rates [31] in order to provide an adequate inter pretation of the parameters involved in the model. In this work, we provide a detailed study of the physics behind these parameters in MR measurements on ntype ZnO films doped with nonmagnetic elements (Na and Li).

Experimental details
Starting from ZnO powders doped with 5 at.% of Na or Li synthesized as described in [14], corresponding ceramic targets were fabricated. Then, high quality oxygen defi cient ZnO:Na and ZnO:Li thin films were grown on sap phire (0 0 0 1) substrates (dimensions: (5 × 5 × 0.5) mm 3 ) by pulsed laser deposition (PLD) at a temperature of 550 • C and an oxygen pressure between 0.05 and 0.5 mTorr using a Nd:YAG laser operated with a wavelength of 266 nm, a repetition rate of 10 Hz and an energy density of 2.1 J cm −2 [33]. The deposition rate was ∼0.018 nm s −1 and the film thicknesses are of 54 nm. EDX mapping studies show an homogenous incorporation of Na in the ZnO:Na films while in the case of ZnO:Li films the Lithium was not able to be observed probably due to its low atomic weight. Xray diffrac tion measurements were performed in the films using CuKα radiation (λ = 0.154 06 nm), see θ − 2θ scans in figure 1(a). The resulting films have epitaxially grown along the caxis direction with a wurtzite structure and without any spurious phase, see figure 1(a). For both films, the position of the (0 0 2) peak is shifted to lower angles (2θ 34.28 • and 34.31 • for the ZnO:Na and ZnO:Li films respectively) compared to bulk ZnO (2θ 34.42 • ) [34]. From the position of these peaks, the corresponding c lattice parameter can be determined using the  [35]. The resulting grain size values are similar for both films being D Na 12 nm and D Li 17 nm for the ZnO:Na and ZnO:Li respectively. In figure 1(b), a SEM image of the ZnO:Na film surface shows small grains with similar sizes to the ones estimated above.
Magnetoresistance measurements were carried out in a cryostat equipped with a rotating sample holder at temper atures between 4 and 150 K and magnetic fields up to 16 T. We have employed the fourpointprobe method using Indium ohmic contacts and a configuration with the applied magnetic field H parallel to the electrical current density I, I H. High electron carrier densities (n 10 20 cm −3 ) have been obtained in our films by Hall effect measurements using the Van der Pauw method. No evidence of anomalous Hall effect has been observed [36]. Magnetization measurements have been performed using a SQUID at temperatures ranging from 4 to 300 K and applied magnetic fields up to 5 T. Remarkably, our films do not present a magnetic signal within the experimental error (∼10 −6 emu). For ∼10 −6 emu, a simple estimation gives a total amount of ∼5 × 10 13 localized magnetic moments. This value corresponds to a few at.% of these moments in our sample.

Results and discussion
The field dependence of MR (defined as (ρ − ρ(H = 0))/ ρ(H = 0)) at different temperatures is presented in figure 2 for ZnO:Na and ZnO:Li films. As it can be observed, the nega tive MR signal increases as the temperature is decreased. The negative MR value at T = 4 K and H = 16 T reaches ∼3.4% and ∼4.7% for ZnO:Na and ZnO:Li respectively in agreement with the values reported for ZnO doped with non magnetic elements and carrier densities above ∼3 × 10 18 cm −3 [26,28,29].
MR expressions based on second order terms of the sd Hamiltonian [23][24][25] lead to a MR ∝ −H 2 dependence which can not properly fit our MR data. In order to evaluate the negative MR taking into account third order terms of the sd Hamiltonian, Appelbaum [32] developed a model based on the transmittance of electrons across a tunneling barrier with local magnetic impurities at the interface. In the context of this problem, the exchange and nonexchange interaction with these impurities improves the tunneling of electrons between the two metals. The adaptation of this model to bulk semicon ductors where dilute local magnetic moments are present is not straightforward and it requires an adequate transformation of the tunneling probabilities due to an interaction with magn etic (T 2 J ) and nonmagnetic impurities (T 2 0 ) to corresponding crosssections or scattering rates. As it was already noticed, the first adaptation was proposed by Khosla and Fischer [31] to account for their negative MR data on CdS semiconducting single crystals doped with In. After several approximations they found that MR can be expressed as: where the last term corresponds to a well known positive MR contribution deduced from a two band model [30,31]. The parameters c and d are related to the conductivities and mobil ities of these two bands [37]. On the other hand, the param eters a and b that appear in the negative MR contribution are defined as [31]: where g is the Landé factor, µ B is the Bohr magneton, S is the spin of the localized moment, J is the exchange integral resulting from the interaction between the itinerant electron and the localized magnetic moment and ρ F is the density of states at the Fermi energy level. α is a numerical factor close to α 1 introduced by Appelbaum [32,38,39]. As it can be noticed, the term (bH) 2 is proportional to (gµ B H/k B T) 2 , hence the electronic conduction will be improved as the magnetic energy of the local moments grows in relation to the thermal energy, see equation (1). The term a 2 1 can be expressed as a 2 1 = 2Jρ F A 1 , where A 1 is related to a ratio between the cross section due to exchange scattering with magnetic impurities, σ J , and the crosssection due to other nonexchange scat tering mechanisms, σ 0 , (nonmagnetic impurities, phonons). Appelbaum [32] proposed that M 2 is expressed by: where M is the normalized magnetization and the last two terms constitute a magnetization variance. Using the Brillouin function for M and reordering terms through hyperbolic functions identities equation (4) can be expressed in the case S = 1/2 as: Hence the ratio (gµ B H/k B T) intervenes in M 2 as well as in the term (bH) 2 . At first sight, it can be noticed that M 2 has a weak Tdependence in the range of temperatures explored. The resulting fittings using equation (1) are in excellent agree ment with our data, see figures 2(b) and (c). In what follows our analysis will be focused on the ZnO:Na data even though similar results have been obtained for ZnO:Li.
The temperature dependence obtained for the parameter b is in good agreement with the predicted by equation (3), as it can be observed in figure 3(a). Taking g = 2 and S = 1/2, which are values corresponding to a local moment formed by a zinc vacancy, V Zn , [15] it is possible to extract Jρ F from equation (3) using the fitting values obtained in figure 3(a). It gives a value of Jρ F = 1.9 in accordance with the ones reported in ZnO which range between ∼1.7 and ∼2.1 [26,29]. As it can be observed in figure 3(b), the resulting parameter a 2 (see equation (1)) has a temperature dependence outside the margins of error. This is an unexpected result considering that previous reports on ZnO suggest that a 2 does not vary with temperature [26,29]. Since the term M 2 is Tdependent (see equation (5)) and it takes part in a 2 (see equation (2)) we have explicitly included this temperature dependence of  and their Tdependence are basically the same within the error. It means that the temperature dependence of M 2 is not a significant contribution to a 2 (see equation (2)) and that it mainly comes from the term a 2 1 . Within the frame of the tunneling problem studied by Appelbaum [32] In the case of bulk semiconductors, Khosla and Fischer [31] express a 2 1 in terms of corresponding crosssec tions as a 2 1 2Jρ F N J σ 2 J /σ 2 0 , where N J is the density of local ized magnetic moments. In the following, we are going to show that if MR is simply expressed in terms of corresponding crosssections using the Drude model and the Matthiessen rule, the resulting a 2 1 differs from the one deduced by Khosla and Fischer [31].
Lets assume that σ(H) = σ 0 + σ J (H) with σ(H = 0) = σ 0 + σ J (H = 0) and ρ = (m * v F /ne 2 ) i N i σ i where i = 0 or J, n is the electron carrier density, m * is the effective mass of the elec tron, v F is the Fermi speed and N i is the corresponding density of scattering centers. If we define σ J (H) = σ J (H = 0)F(H) where F(H) takes into account the magnetic field dependence, MR can be expressed as: where µ i = e/(m * v F N i σ i ) are the electron mobilities and it was assumed that σ 0 σ J (H = 0). If a general comparison between the prefactor of equation (6) and the parameter a 2 in the first term of equation (1) is made, it is possible to express a 2 approximately as: Unlike Khosla and Fischer [31], we find a simple linear relation between a 2 1 and the ratio of corresponding crosssec tions, see equation (7). In addition, equation (7) establishes that a 2 1 represents the ratio between the electronic mobility due to nonexchange scattering mechanisms and the one due to exchange scattering mechanisms (including spin flip and non spin flip events). The previous assumption σ 0 σ J (H = 0) implies that µ 0 µ J (H = 0) hence if the Matthiessen rule is valid, the total electron mobility µ T will be dominated by µ 0 which would be the major contributor to the measured elec tron mobility. Consequently, measuring the electron mobility and using the values of a 2 1 extracted from figure 3(b) it is pos sible to obtain µ J (H = 0) via equation (7).
In figures 4(a) and (b) we present the temperature depend ence of both the electron carrier concentration obtained from Hall effect measurements and the electrical resistivity respec tively. Both of them have a weak temperature dependence in the same way as the resulting electron mobility µ T µ 0 , see figure 4(c). This weak Tdependent mobility at high doping levels in ZnO has been already predicted by Alfaramawi [40]. In his model, scattering with ionized impurities and pho nons are included as well as the electron-electron interaction [40][41][42]. Using the parameters adopted by Alfaramawi [40] in ZnO with an ionized impurity concentration of 1 × 10 20 cm −3 , a good agreement between this model and our experimental values is obtained, see figure 4(c).
As it was pointed out, knowing the values of µ 0 and a 2 1 it is possible to extract µ J (H = 0) using equation (7), the results  are showed in figure 5(a). As it can be observed, µ J (H = 0) is weakly Tdependent at T > 70 K which seems to be consistent with the behavior predicted by second order terms of the sd exchange model [23,25]. A departure from this behavior is observed at T < 70 K, that is to say, µ J (H = 0) increases as the temperature is decreased, see figure 5(a). This enhancement of the exchange (or spin) mobility at lowT does not seem to be caused by an increase of the localized magnetic moments order since this has already been taken into account in the term (bH) 2 and M 2 , see equations (1) and (2). Further studies are needed to provide a deeper understanding of this effect. It is worth to compare the MR of our ZnO films doped with alkali metals with the one corresponding to undoped ZnO films which is well established in the literature [26][27][28]43]. In figure 5(b) the MR of our ZnO:Na film and the one for an undoped ZnO film [28] are compared. The resulting fit tings using equation (1) lead to a value of Jρ F 1.9 for the undoped ZnO film which is equal to the value obtained for our ZnO:Na film. This result suggests that the localized magn etic defect that is responsible for the exchange mech anism with the itinerant electrons is the same for both films, pointing to an intrinsic defect (i.e. zinc vacancies). In the case of the undoped ZnO film, the parameter a 2 undoped = 0.007 34 at T 4 K is higher than the one obtained for the ZnO:Na film at the same temperature (a 2 Na = 0.004 43, see figure 3(b)); the ratio between these two values being ∼1.7. Using equa tion (7) for each film and assuming that µ Na 0 µ undoped 0 one obtains that µ Na On the other hand, it is worth to ask if our ZnO films are outside the strong localization regime [44,45]. A simple esti mation using the quantities showed in figure 4 allows us to estimate k F l for our films, where k F is the Fermi wavevector and l is the mean free path. It gives a value of k F l 1.8 > 1, which means that we are in a regime where the weak scat tering theory still works [45] and the Khosla model [31] can be applied. In this regard, Ajimsha et al [46] have considered that their MR data in Gadoped ZnO films can be explained in the context of phase coherent electron transport. They have used a weaklocalization expression [47] which accounts for the negative magnetoresistance plus a positive contribution arising from the electron-electron interaction [46] to fit their data: where: with δ = /(4eH 2 ϕ ). The phasecoherence length is repre sented by ϕ . The function f 3 (δ) can be expressed as [47]: (10) The MR contribution due to electron-electron interaction can be written as [46]: (11) where D = µ 0 k B T/e is the electron diffusion constant, F σ is the Coulomb interaction parameter and h = gµ B H/k B T. The function g 3 (h) is expressed by: where N = 1/(e Ω − 1). We have tried to fit our MR data with equation (8) taking ϕ and F σ as fitting parameters. The best fit to our data was achieved fixing F σ at a value of 0.3 for all T. The electron diffusion constant D was determined at each temper ature using the Hall mobility values from figure 4(c). The resulting fittings are shown in figure 6. As it can be observed, a good agreement is only attained at high temper atures, while at low temperatures it is not possible to obtain a proper fit to our MR data using equation (8), especially at low magnetic fields. This disagreement with the weaklocal ization model at lowT and lowH has already been observed by Andrearczyk et al [43] in ZnO films. (a) Extracted zero field electron mobility due to exchange (or spin) dependent scattering mechanisms for the ZnO:Na film. µ J (H = 0) starts to increase as the temperature is decreased below T 70 K. (b) Field dependent MR at T 4 K for the ZnO:Na film (black circles) and for an undoped ZnO film (blue triangles) adapted from Ghoshal et al [28]. Red lines are the corresponding fittings using equation (1).

Conclusions
In conclusion, we have successfully applied the Khosla model based on the sd exchange Hamiltonian to our negative MR data on oxygen deficient ZnO:Na and ZnO:Li films. The incorporation of these dopants does not seem to affect the MR signal suggesting that intrinsic defects (i.e. zinc vacan cies) play the role of the localized magnetic defects. We have found an anomalous temperature dependence in one of the model's parameter, a 2 . After a careful analysis we have con cluded that this temperature dependence comes from a ratio between a nonexchange and a zero field exchange electron mobility, µ 0 /µ J (H = 0). While the high ndoping level of our films (mainly originated by oxygen vacancies) leads to a weak Tdependent nonexchange mobility, the resulting zero field exchange (or spin) mobility increases when temperature is lowered below T ∼ 70 K.