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Enhanced spin–orbit coupling and charge carrier density suppression in LaAl1−x Crx O3/SrTiO3 hetero-interfaces
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2015 J. Phys.: Condens. Matter 27 125007 (http://iopscience.iop.org/0953-8984/27/12/125007) View the table of contents for this issue, or go to the journal homepage for more
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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 27 (2015) 125007 (9pp)
doi:10.1088/0953-8984/27/12/125007
Enhanced spin–orbit coupling and charge carrier density suppression in LaAl1−xCrxO3/SrTiO3 hetero-interfaces Pramod Kumar, Anjana Dogra, P P S Bhadauria, Anurag Gupta, K K Maurya and R C Budhani CSIR-National Physical Laboratory, Dr. K.S. Krishnan Marg, New Delhi-110012, India E-mail: [email protected] Received 13 December 2014, revised 22 January 2015 Accepted for publication 29 January 2015 Published 6 March 2015 Abstract
We report a gradual suppression of the two-dimensional electron gas (2DEG) at the LaAlO3 /SrTiO3 interface on substitution of chromium at the Al sites. The sheet carrier density at the interface (n ) drops monotonically from ∼2.2 × 1014 cm−2 to ∼2.5 × 1013 cm−2 on replacing ≈60% of the Al sites by Cr and the sheet resistance (R ) exceeds the quantum limit for localization (h/2e2 ) in the concentrating range 40–60% of Cr. The samples with Cr 40% show a distinct minimum (Tm ) in metallic R (T ) whose position shifts to higher temperatures on increasing the substitution. Distinct signatures of Rashba spin–orbit interaction (SOI) induced magnetoresistance (MR) are seen in R measured in out of plane field (H⊥ ) geometry at T 8 K. Analysis of these data in the framework of Maekawa–Fukuyama theory allows extraction of the SOI critical field (HSO ) and time scale (τSO ) whose evolution with Cr concentration is similar as with the increasing negative gate voltage in LAO/STO interface. The MR in the temperature range 8 K T Tm is quadratic in the field with a +ve sign for H⊥ and −ve sign for H . The behaviour of H MR is consistent with Kondo theory which in the present case is renormalized by the strong Rashba SOI at T < 8 K. Keywords: oxide hetero-structures, 2DEG, magnetoresistance (Some figures may appear in colour only in the online journal)
the 3d transition metal series, the LaVO3 /STO interface also turns out to be metallic [14]. But this behaviour ceases to exist in LaCrO3 /STO and LaMnO3 /STO interfaces. However, while LaCrO3 is an antiferromagnetic (TN = 290 K) band insulator, the manganite is a Mott insulator due to the eg1 electron. In both the cases, the interface turns out to be insulating with some magnetic character [15–17]. This dramatic change in the properties of the interface on going from LaTiO3 to LaMnO3 is yet to be understood, although one can attribute it to the increasing magnetic and mixed valent character of the higher 3d transition elements. In the context of doping, it is important to mention that the doping of the interface (δ-doping) of LaAlO3 /SrTiO3 with few monolayers of SrTi1−x Mx O3 (x = 0.02, and M a 3d transition metal) has been investigated and a quenching of the 2DEG behaviour is observed, presumably due to carrier localization at the 3d ion sites [18, 19]. Recently Gray
1. Introduction
Transition metal oxide heterostructures provide new opportunities to understand the effects of broken inversion symmetry on the interaction between spin, charge and orbital degrees of freedom in highly correlated electron systems [1]. One remarkable feature of epitaxial oxide heterostructures is the observation of a two-dimensional electron gas (2DEG) at the interface of LaAlO3 (LAO) and LaTiO3 (LTO) epitaxial films grown on the TiO2 terminated surface of (0 0 1) SrTiO3 [2–5]. While LaAlO3 is a band insulator of 5.2 eV energy gap, the properties of LTO are controlled by Mott physics due to the 3d1 configuration of Ti3+ ions. However, in spite of these fundamental differences, the 2DEG at the LTO/STO and LAO/STO has broadly similar superconducting, magnetic and photoconducting properties [4–13]. As we move from Ti to V in 0953-8984/15/125007+09$33.00
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© 2015 IOP Publishing Ltd Printed in the UK
J. Phys.: Condens. Matter 27 (2015) 125007
P Kumar et al
substrates by ablating these targets with a KrF excimer laser (λ = 248 nm). The STO (0 0 1) substrates were pre-treated with the standard buffered hydrofluoric (NH4 F–HF) solution for 30 s in order to achieve a clean TiO2 terminated surface [34]. Prior to the deposition, the chemically treated STO (0 0 1) were annealed for about an hour at 830 ◦ C in the oxygen pressure of 7.4×10−2 mbar. Thin films of all the compositions were deposited at 800 ◦ C with oxygen partial pressure of 1×10−4 mbar. Deposition was performed by firing the laser at 1 Hz with an areal energy density of 0.56–0.65 J cm−2 on the targets. The film growth mode and thickness were monitored using RHEED (STAIB, 35 keV) during deposition. Once the deposition was completed, the samples were cooled in the same O2 partial pressure at the rate of 10 ◦ C min−1 to ambient temperature. For structural characterization, we have used x-ray diffraction (XRD) and reflectivity measurements carried out in a PANalytical X’Pert PRO MRD x-ray diffractometer equipped with a CuK α1 source. Magneto-transport measurements were performed in the Van der Pauw geometry using the resistivity attachment of a Quantum design magnetic property measurements system. Stable, low resistivity contacts were made with the 2DEG by wire bonding with thin Al wires.
et al [20] have studied LaAlx R1−x O3 /STO system where the rare earth element R = Tm and Lu are only 2% per formula unit. At such low doping level, the 2DEG remains more or less unaffected. The other issue of importance in the LAO/STO problem is the co-existence of superconductivity and magnetic ordering. While the former has been established rigorously, the existence of magnetic order is still a matter of debate due to its feeble nature [21–25]. The magnetic order manifests itself in subtle ways through the hysteresis in magnetoresistance (MR) and a characteristic minimum in resistivity. There is also a considerable debate on the source of carriers and magnetic moments whose condensation and ordering respectively lead to these two macroscopic phenomena. Generally, magnetism is attributed to Ti3+ ions which are produced by oxygen vacancies. However, oxygen vacancies also contributes to carriers and it has been noticed that signature of magnetism are absent in high carrier density samples [26]. Furthermore, the carrier concentration in such a system is highly sensitive to growth conditions [27–29]. It is also established that the tetragonal distortion at the interface and confinement of electron gas in 2D lift the degeneracy of Ti (dxy , dyz and dxz ) bands which also show anomalous dispersion, and possibly a Lifshitz transition depending on band filling [30–33]. Since the distortion is intimately linked with the lattice constant of LaAlO3 , which is increased by Cr substitution, the later is likely to affect the dispersion and occupancy of the t2g subbands. To address the critical role of these transition elements in the electronic properties of the interface, we have introduced a controlled site substitution of Al by Cr in LaAlO3 and have studied the electronic transport in LaAlx Cr 1−x O3 /SrTiO3 interfaces. Here, on the one end of the composition phase space we have LAO/STO that shows the confinement of 2DEG at the interface, while for the other end member LaCrO3 /SrTiO3 , no such confinement appears at the interface [15]. With the Cr substitution in LaAlx Cr 1−x O3 /SrTiO3 , one issue of interest is to look for the critical value of Cr at which the interface becomes insulating. But more importantly, we would like to address the evolution of magnetotransport and collective phenomena such as magnetism and superconductivity and their bearing on the nature of the electron donating layer. We note that the Cr substitution reduces the free carrier density at the interface in a rather gradual manner as samples with Cr concentration as high as 40% remain metallic. The measurements of in-plane and out-of-plane field MR in these systems show enhanced Rashba spin–orbit interaction (SOI) with the Cr addition. A comparison of our data with the results of electrostatic gating of LAO/STO interface suggests that the role of Cr addition is similar to that of the negative gate field.
3. Results and discussion
A set of typical RHEED specular beam intensity oscillations observed as a function of growth time during the deposition of x = 0 and x = 0.6 films are shown in figures 1(a) and (b). Such intensity oscillations were used to control the thickness of the deposited films with a precision of one unit cell for all values of x. The observation of sharp oscillations strongly indicates a Frank-van der Merve type [35] layer-by-layer growth of the LaAl1−x Cr x O3 on STO. The RHEED patterns along the (0 0 1) crystallographic direction after film deposition are shown in insets of figures 1(a) and (b). Clear streaks, which are the characteristics of a 2D flat and crystallographically ordered surface, were preserved prior to during and after the growth. The XRD patterns of (x = 0.0, 0.2, 0.4 and 0.6) samples taken in θ –2θ geometry are shown in figure 2(a). A distinct hump in diffracted intensity on the higher 2θ side of the (0 0 2) reflection of the substrate marks the (0 0 2) Bragg line for the film. We have calculated the out-of-plane lattice parameter of the LaAl1−x Cr x O3 films from the position of this hump and the similar ones observed next to the (0 0 1) and (0 0 3) reflections of the substrate. Variation of the out-of-plane ‘c’ lattice parameter of the layers as a function of Cr concentration is shown in the inset of figure 2(a). The increase in the ‘c’ lattice parameter observed on Cr substitution in the film can be attributed to the larger ionic radius of Cr3+ compared to that of Al3+ . The shift in the XRD peak of the film towards the reflection from the substrate with increasing Cr suggests that the top TiO2 plane of STO is under less compressive strain because of a better lattice match between SrTiO3 and LaCrO3 . The Phi scan around the (0 2 2) Miller plane exhibits four-fold symmetry with overlapping of the XRD peaks for the substrate and the film that indicates the cube-on-cube symmetry along (0 1 1) plane. Kiessig fringes were observed
2. Experimental
Dense and compact laser ablation targets of LaAll−x Cr x O3 (x = 0.0, 0.2, 0.4, 0.6, 0.8, 1.0), 25 mm in diameter and 6 mm thick were prepared by solid state reaction method. Films of thickness 20 unit cells (≈8 nm) of all the compositions were deposited on the TiO2 terminated SrTiO3 (0 0 1) single crystal 2
J. Phys.: Condens. Matter 27 (2015) 125007
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Figure 1. Typical RHEED intensity oscillations and RHEED images recorded along the (0 0 1) crystallographic direction during the fabrication of (a) x = 0 and (b) x = 0.6. The clean intensity oscillations reveal a layer-by-layer growth of the LaAl1−x Cr x O3 film on (0 0 1) SrTiO3 .
Figure 2. (a) HRXRD around (0 0 2) peak of single crystal SrTiO3 shows the film peak marked as ‘F’ adjacent to substrate peak, revealing epitaxial growth along the (0 0 1) crystallographic direction of the substrate. Inset of figure 2(a) shows the variation of out-of-plane lattice constant of the thin film as a function of Cr concentration. Figure 2(b) shows Phi scans around the (0 2 2) plane of the substrate and thin film. The four peaks separated at 90◦ show a four-fold symmetry, revealing epitaxial growth along (0 0 1) crystallographic direction of the substrate.
in resistance on cooling below ≈50 K and finally a saturation at the lowest temperature. This behaviour is emphasized in figure 3(b) by plotting the data of metallic samples (x = 0, 0.2 and 0.4) on an expanded scale and in figure 3(c) by normalizing R (T ) with respect to its value at 2 K. We note a shift of the resistivity minimum to higher temperature with the increasing Cr concentration while the magnitude of minimum is nonmonotonic in these samples. A qualitative understanding of the R (T ) can be made by considering various classical and quantum contributions to charge carrier scattering in a 2D metal. However, before we dwell upon detailed interpretation of the R (T ) data, it is worthwhile to show how the n and R at 300 K vary as a function of Cr concentration. These data are shown in figure 4(b). We notice that for the Cr free sample the sheet carrier density n (≈2.2 × 1014 cm−2 ) is close to the number predicted from the polarization catastrophe argument (≈3.2 × 1014 cm−2 ) for the formation of 2DEG in LAO/STO. But it drops monotonically with a slope of 3.35×1013 cm−2 per atomic percentage of the Cr substitution. The Hall resistance
in x-ray reflectivity measurements (not shown here) that also suggest a smooth surface and interface of the grown films. Thus, the two complimentary measurement modes i.e. RHEED patterns along the (0 0 1) crystallographic direction and phi scan near (0 2 2) plane establish excellent crystalline quality of our films. The temperature dependence of the sheet resistance R (T ) of the LaAl1−x Cr x O3 /SrTiO3 structure reveals a clear transition from a delocalized to localized electronic transport with the increasing x as shown in figure 3. While the R (T ) of the samples with x = 0.0, 0.2, 0.4 and 0.6 decreases on lowering the temperature from 300 to ≈50 K, the absolute value of R (T ) at 300 K exceeds the quantum resistance for localization (h/2e2 ) for the samples with x > 0.4. In R (T ) of all the samples, a step like behaviour is noticed at ≈83 and ≈183 K (see figure 3(a)) which also has been reported earlier in the heating cycle of resistance measurements [36, 37]. The metallic behaviour seen in the samples with x 0.6 gives way to a resistivity minimum followed by an increase 3
J. Phys.: Condens. Matter 27 (2015) 125007
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Figure 3. (a) Temperature dependence of the sheet resistance of LaAl1−x Cr x O3 /SrTiO3 heterostructures for x = 0.0, 0.2, 0.4 and 0.6 in the temperature range of 2–300 K. Figure also shows a clear insulating behaviour for sample with x > 0.6. (b) Shows the R (T ) data for x = 0.0, 0.2 and 0.4 samples from 2 to 100 K. The solid lines are fit to equation (2) in the temperature range of 10–100 K. (c) Shows R (T ) of the three samples normalized with respect to its value at 2 K to emphasize the minimum in R (T ). (d) Shows fitting of sheet resistance of x = 0.0, 0.20 and 0.40, to an equation of the type R = R0 + R1 T 2 + R2 T 5 + RK (T /TK ), where the last term is the Kondo contribution which replaces the localization contribution in equation (2).
(RH ) for all these samples measured at several temperatures over a field range of ±6 T remains linear in the field as shown in the figure 4(a). In the simplest possible scenario, this suggests a single band electronic transport. The temperature dependence of the sheet resistance of a complex system like this one will have contributions from processes over and above those related to classical scattering by frozen-in disorder, phonons and electrons. One important consequence of the diffusive motion of electrons in a 2D metal at low temperatures is the enhanced quantum backscattering (QBS) as the inelastic scattering events diminish due to phonon freeze out. This effect leads to a ln T growth of R of the form, R (T ) = −K (p/2) ln [T /T0 ] ,
In the presence of magnetic impurities, the classical s–d scattering [42] can contribute to resistance in such a manner that it diminishes quadratically in magnetization (M) in the concentrated impurity limit [43, 44]. At very low impurity concentration where the moments are not ordered, a Kondo-type electronic transport may prevail [45]. A careful examination of Hall resistance RH and MR as a function of magnetic field does not reveal any hysteretic effects, suggesting the absence of direct magnetic scattering in the system down to 2 K. However a higher order s–d scattering in the Kondo sense can contribute to low temperature resistance in a characteristic manner if magnetic impurities are present in the dilute limit. The Kondo contribution leads to a resistance RK ∼ BlnT , where B is a function of exchange integral J between electronic and impurity spins and the density of states at the Fermi energy [45]. At very low temperatures, the logarithmic divergence of RK is cut off due to the formation of Kondo singlets. We, therefore, model the R(T ) data between 10 and 100 K on the basis of the following generalized equation [44];
(1)
where K is a constant and the index p is related to the inelastic scattering time and T0 is order of the Kondo temperature [38, 39]. A similar temperature dependence of sheet resistance is also expected when quantum corrections to conductivity due to electron-electron interactions in a disordered 2D conductor are taken into account [40, 41]. While the ln T dependence of R (T ) seen in figure 3(c), over a limited window of temperature is suggestive of these models, the saturation of R (T ) at the lowest temperature is a deviation from equation (1). Also the temperature range over which the ln T dependence is seen is rather high for the e–e interaction to be operational.
R = R0 + R1 T 2 + R2 T 5 + R3 lnT ,
(2)
where R0 is the residual resistance due to frozen-in disorder, the T 2 term comes from e–e scattering and the T 5 contribution has its origin in electron–phonon scattering. The last term 4
J. Phys.: Condens. Matter 27 (2015) 125007
P Kumar et al
small, positive and isotropic [40]. The Kondo contribution to MR is also small and isotropic but negative in sign. There is ample evidence in literature, particular on GaAs-GaAlAs 2DEG systems where the ‘Dyakonov–Perel’ (DP) mechanism of spin relaxation dominates, that τSO can be a strong function of temperature [50, 51]. In the presence of only k-cubic spin splitting as applicable in the case of SrTiO3 , the magneto-conductance correction due to SOI goes as [52]; e2 [σ (B) − σ (0)] = π h 1 H H 1 1 Hϕ 2Hϕ ϕ SO ψ + + + ψ + + 2 B B 2 2 B B 1 1 Hϕ × − ψ . (3) + 2 2 B − ln Hϕ + HSO − 1 ln Hϕ + 2HSO + 1 ln Hϕ B 2 B 2 B Here Hφ = h/8π eDτφ and HSO = h/8π eDeτSO , and D, τφ and τSO are diffusion constant, dephasing time and spin– orbit scattering time respectively. The τSO , in general, derives contribution from the scattering processes by phonons and impurities in the presence of SOI induced by heavy atoms in the lattice. This is called the Elliott-Yafet τSO [53, 54]. However, the breaking of inversion symmetry at the interface of heterostructures gives rise to a stronger spin–orbit relaxation named the DP mechanism [55]. In the present case we will assume that τSO is entirely dictated by the DP process. We have performed two sets of MR measurements on the LaAl1−x Cr x O3 /STO samples whose sheet resistance is below the threshold for localization (≈12.9 k ). One, the transverse MR was measured with the magnetic field applied perpendicular to the plane (H⊥ ) of the sample. Two, longitudinal MR was measured with the field applied parallel (H ) to the plane of the 2DEG. The MR is defined as [(RH − R0 ) /R0 ] × 100%, where RH and R0 are the sheet resistance in field and in zero-field condition respectively. The MR data for H⊥ configuration in the field range of 0 to ±6 T at 2, 4, 6, 8, 10 and 15 K are shown in figures 5(a) and (b). It is worth mentioning here that widely varying magnitude and sign of MR have been reported in LAO/STO interfaces [56–59]. A comparative study of MR in film grown at high (∼10−4 mbar) and low (∼10−6 mbar) O2 partial pressure suggests that high MR is associated with samples where a significant fraction of carriers is derived from O2 vacancies [60]. The very small values of MR and its sign in H⊥ and H geometries are consistent with results on several other reported 2DEG samples [61, 62]. A careful examination of the four panels in figure 5 reveals that there are three distinct field dependence of the MR depending on the temperature of measurement. At the lowest T (=2 K), the MR increases sharply with field, reaches a peak value and then drops on further increase in the field. In the ±H plots of figure 5, the width of the MR cusp and the field at which it reaches a peak value increases with temperature and Cr concentration in the LaAll−x Cr x O3 /STO films. Once the temperature exceeds a critical value T1 , the positive MR becomes nearly linear in the field. This H -linear behaviour gives way to parabolic field dependence once the temperature exceeds a critical value T2 . The values of T1 and T2 for x = 0
Figure 4. (a) Hall voltage divided by the applied current I as a function of magnetic field measured at 10 K. (b) Variation of sheet resistance (R ) at 300 K and carrier concentration (n ) at 300 and 10 K as a function of Cr concentration in LaAl1−x Cr x O3 /SrTiO3 .
of equation (2) represents quantum correction to resistance, which may possibly come from weak localization, low temperature correction to e–e scattering and the scattering by local magnetic moments in the Kondo sense. It is clear from the fits shown in figure 3(b) that equation (2) expresses the temperature dependence of R well down to 10 K. The fitting parameters are listed in table 1. However, below 10 K, we see the saturation of the resistance. The other phenomenon which can significantly affect the low temperature electron transport in these systems is SOI of the Rashba-type originating from the broken inversion symmetry. The conduction band in doped STO consists of t2g orbital, which under spin–orbit coupling splits into J = 3/2 and J = 1/2 states of energy separation ∼17 meV [46–48]. In the context of the interfacial t2g orbitals in LAO/STO system, the quantum confinement along the z direction pushes the dxy orbitals down in energy compared to the dzx and dyz . Thus dxy can be assigned to J = 1/2 atomic state where as dzx and dyz to J = 3/2. These energy bands are further split by the Rashba SOI, which is expected to be cubic in the wave vector k for J = 3/2 state. It has been shown that the k-linear and k-cubic Rashba splittings leave their distinctive signatures in the MR [49]. The measurements of MR also permit us to differentiate between weak localization, quantum correction to the e–e scattering and the Kondo effect. While WL is an orbital effect and thus a strong negative MR is predicted when the external field is perpendicular to the plane of the 2D gas (H⊥ configuration), the MR due to e–e is 5
J. Phys.: Condens. Matter 27 (2015) 125007
P Kumar et al
Table 1. Obtained fitting parameters on fitting the R versus T data by using equation (2) in the temperature range of 10–100 K.
x (Cr con.)
R0 ( −1 )
R1 ( −1 K−2 )
R2 ( −1 K−5 )
R3 ( −1 )
0 0.20 0.40
1711.72 3620.05 7630.34
0.094 0.179 0.395
−2.14E−8 −2.65E−8 −7.19E−8
−262.56 −445.62 −1265.25
Figure 5. Magnetoresistance at different temperatures in transverse magnetic field (H⊥ ) for LaAl1−x Cr x O3 /SrTiO3 samples (a, a )x = 0.0 and (b, b )x = 0.40. Inset of figure (a , b ) shows Kohler plot at 15 K.
and 0.4 samples are (6 K, 8 K) and (8 K, 10 K) respectively. The transition from linear to parabolic dependence is however gradual. We also note that the peak magnitude of MR at 2 K increases with x. We first discuss the +ve MR at T > T2 which goes as ∼ H α (α > 1). The +ve sign of the MR seen here is inconsistent with the WL and Kondo scattering pictures, and the T2 is too large for quantum contributions of e–e to MR to be effective. This regime of MR can be attributed to the classical elastic and inelastic scattering events as the charge carriers tend to go in a cyclotron orbit in the out-of-plane field. Further the Kohler rule [62, 63] predicts, R (H ) /R0 ∝ a|µH |2 and its applicability however depends on the validity of the classical Boltzmann transport in this 2DEG system. Based on the Hall data and the fact that ωc τ 1, we believe the Kohler rule can be applied to H⊥ geometry MR. In the inset of figures 5(a ) and (b ) we have plotted the average MR (=1/2(MR(+H) + MR(−H)) at 15 K for x = 0 and 0.4 samples as a function of (H /R0 )2 . For H 0.5 T we notice an
approximate quadratic field dependence which is pronounced at the higher fields. From this MR data, the calculated Kohler mobility is ≈0.30 cm2 V−1 s−1 and 0.07 cm2 V−1 s−1 for x = 0.0 and x = 0.4 samples respectively. The data at T 6 K show a clear deviation from the classical orbital effect. We interpreted these results in the framework of the k-cubic Rashba SOI effect which leads to a conductivity correction of the type expressed in equation (3). Figures 6(a) and (b) shows the fits to equation (3) in the field range of ±6 T. The quality of these fits is excellent. The critical fields, time scales and length scales extracted from such analysis are listed in the table 2. For the intermediate temperature T1 < T < T2 , the data indicate a linear temperature dependence of MR. This is an interesting result in itself as linear MR has been seen in doped and disordered semiconductors and also in some topological insulators [64–68]. A tentative explanation for such dependence is the possibility of spatial fluctuations in carrier concentration and carrier mobility which can distort 6
J. Phys.: Condens. Matter 27 (2015) 125007
P Kumar et al
respectively. The magnetic field dependence of sheet resistance in the Kondo regime can be modelled as [39, 72, 73]. R (H|| ) = R (0) + RK (H|| /HK ).
(5)
To extract HK , a magnetic field scale, related to Kondo temperature and g factor of the impurity spin [72], we apply equation (5) to the H field data shown in figures 7(a) and (b) for 10 K. The best fitting obtained is HK = 80.5 T and 85.5 T for x = 0.20 and 0.40 respectively. A generalized expression of the type equation (5) can also be written for the Kondo contribution to the temperature dependence of zero field resistivity RK (T /TK ). Following Goldhaber Gorden [72] we have; s
T TK2 RK = RK (T = 0) , (6) TK T 2 + TK2 where
TK =
(21/s
TK − 1)1/2
assuming only the Kondo contribution to resistance equation (6) dominated, a fit to the data shown in figure 3(d) to a equation of the type R = R0 + R1 T 2 + R2 T 5 + RK (T /TK ) .
Over a temperature range of 2–100 K yield a TK of ≈47.6 K and ≈58.5 K for x = 0 and x = 0.40 respectively. The TK and HK thus derived are in order of magnitude agreement to equation (4). The fitting parameters are listed in table 3. At this stage, we revisit the data given in table 1 and compare the critical fields and length scales for different samples deduced in the frame work of Rashba SO contribution to MR. Caviglia et al [74] has studied the behaviour of Hϕ , HSO , Lϕ , τϕ and τSO as a function of gate voltage in LAO/STO interface. The ratio of HSO /Hϕ in their case is ≈50 at 1.5 K and zero bias, and increases with the increase in carrier concentration. For our undoped samples this ratio is ≈4 at 2 K, and decreases further with the Cr doping. This behaviour is consistent with the results of Caviglia et al as they also see a drop with the decreasing carrier concentration (increase negative gate voltage). The trend of τϕ and τSO with increasing Cr concentration is also consistent with the effect of gating, although absolute numbers differ. It is worthwhile to compare our results with the data on 2D electron gas produced at the top gated (0 0 1) SrTiO3 surface [49]. The absolute values of critical fields in this case are lower by an order of magnitude but their dependence on carrier concentration (gate field) is similar. It needs to be pointed out here that the carrier density in gated (0 0 1) STO is lower by a factor of ≈15 as compared to the sheet carrier concentration at the LAO/STO interface. Much lower critical fields have been reported in high mobility 2D gas in Gax In1−x As quantum well structures [75].
Figure 6. Variation of magneto-conductance [σ (B) =
σ (B) − σ (0)], σ being the sheet conductance, measured in a magnetic field applied perpendicular to the LaAl1−x Cr x O3 /SrTiO3 interface at different temperatures (a) x = 0.0, (b) x = 0.40. Black line is the fit to SOI theory equation (3).
the carrier path as seen earlier in epitaxial layers of InAs and InAsN [69]. Such inhomogeneities can arise from the two different sources of carriers in LAO/STO and LTO/STO interface [70], in addition, since the three temperature regimes are closely spaced, one has to consider the effect of competition between the contributions of SOI and the classical orbital scattering. A rigorous analysis of the data in this temperature window should allow separation of these two contributions. We see a striking difference in the field dependence of MR when the magnetic field is aligned parallel to the plane of the film. The results of these measurements for x = 0.2 and 0.4 samples performed at 2 K, 6 K, 10 K and 20 K are shown in figures 7(a) and (b) respectively. While the MR curve at 2 K for H has the same shape as in the case of H⊥ , the overall value of MR is smaller by a factor of two. There is a remarkable change in the sign and shape of the MR curve for T > 8 K. Now it appears quadratic in field, but with a negative sign which is not consistent with the Kohler type MR emanating from the classical orbital effect. We attributed it to the Kondo scattering where the magnetic field acts in the same manner as the temperature, the identity between the Kondo temperature (TK ) and Kondo field (HK ) goes as [71]. SgµB HK = kB TK ,
(7)
4. Conclusions
(4)
where S, g, µB and kB are impurity spin quantum number, Lande ‘g’ factor, Bohr magnetron and Boltzmann constant
In summary, we have successfully grown epitaxial LaAl1−x Cr x O3 films on TiO2 terminated SrTiO3 . The sheet 7
J. Phys.: Condens. Matter 27 (2015) 125007
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Table 2. Parameters extracted from fitting the magnetoconductance data of figure 6 to equation (3). Hφ and HSO are the characteristic magnetic field, τφ and τSO the relevant relaxation times and D the diffusion coefficient. The carrier effective mass used in these calculation is 1.5me , me is the mass of electron.
Sample Hϕ LaAl1−x Cr x O3 /STO Temperature (T) x = 0.0
2K 4K 6K 2K 4K 6K
x = 0.40
0.17 0.30 0.43 0.41 0.50 0.51
Lϕ (nm) HSO Phase Spin–orbit field coherence D = vF2 τ/2 (T) length (m2 s−1 ) × 10−4
τϕ (s) × 10−12 De-phasing time
τSO (s) × 10−13 Spin–orbit scattering time
0.72 1.79 2.08 1.06 1.34 1.88
1.18 0.66 0.44 2.06 1.66 1.58
2.79 1.10 0.92 8.00 6.19 4.28
31.25 23.33 19.60 20.15 18.10 17.91
8.22 8.38 8.64 1.95 1.99 2.05
Figure 7. Magnetoresistance at different temperatures in longitudinal field (H ) from 6 to −6 T field for (a) x = 0.20 and (b) x = 0.40. Inset of figure 7(a) shows MR for LaAlO3 /SrTiO3 in longitudinal mode. Table 3. Obtained fitting parameters on fitting the R versus T data by using equation (7) in the temperature range of 2–100 K.
x (Cr con.)
R0 ( −1 )
R1 ( −1 K−2 )
R2 ( −1 K−5 )
RK ( −1 )
Kondo temperature (K)
0 0.20 0.40
186.80 846.01 45.00
0.088 0.184 0.371
−2.33E−8 −3.40E−8 −6.23E−8
1049.3 2041.4 5175.3
47.6 49.6 58.5
resistance in the temperature range 2–300 K increased significantly with substitution of Cr at the Al sites. For x 0.6, the sheet resistance goes beyond the 2D quantum limit (R ≈ h/2e2 ). The sheet carrier density is also suppressed monotonically with the increasing Cr substitution. The magnetoresistance data at T 20 K has been analyzed in the detail to extract various classical and quantum contributions to electronic transport. T 10 K out-of-plane field MR suggests strong Rashba spin–orbit coupling dominated transport. The in plane field MR and characteristic minimum in R (T ) suggest Kondo-like scattering of charge carrier by localized spins associated with Ti3+ and Cr3+ ions. The effect of Cr substitution on magneto-transport characteristics mimics the behaviour of LAO/STO interface on negative gate bias, which reduces the carrier concentration at the interface.
the Indo-French Centre for Promotion Advanced Research. (Project # IFCPAR 4704-I), and PK thanks CSIR for a senior research fellowship. RCB has also been supported by the JC Bose National Fellowship of the department of Science and Technology. References [1] Hwang H Y, Iwasa Y, Kawasaki M, Keimer B, Nagaosa N and Tokura Y 2012 Nat. Mater. 11 103 [2] Ohtomo A and Hwang H Y 2004 Nature 427 423 [3] Ohtomo A and Hwang H Y 2006 Nature 441 120 [4] Ohtomo A, Muller D A, Grazul J L and Hwang H Y 2002 Nature 419 378 [5] Biscaras J, Bergeal N, Kushwaha A, Wolf T, Rastogi A, Budhani R C and Lesueur J 2010 Nat. Commun. 1 89 [6] Reyren N et al 2007 Science 317 1196 [7] Li L, Richter C, Mannhart J and Ashoori R C 2011 Nat. Phys. 7 762 [8] Rastogi A, Kushwaha A K, Shiyani T, Gangawar A and Budhani R C 2010 Adv. Mater. 22 4448 [9] Rastogi A and Budhani R C 2012 Opt. Lett. 37 317
Acknowledgments
We acknowledge CSIR-India for financial support through AQuaRIUS Project. AD and RCB acknowledge funding from 8
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Enhanced spin–orbit coupling and charge carrier density suppression in LaAl1−x Crx O3/SrTiO3 hetero-interfaces
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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 27 (2015) 125007 (9pp)
doi:10.1088/0953-8984/27/12/125007
Enhanced spin–orbit coupling and charge carrier density suppression in LaAl1−xCrxO3/SrTiO3 hetero-interfaces Pramod Kumar, Anjana Dogra, P P S Bhadauria, Anurag Gupta, K K Maurya and R C Budhani CSIR-National Physical Laboratory, Dr. K.S. Krishnan Marg, New Delhi-110012, India E-mail: [email protected] Received 13 December 2014, revised 22 January 2015 Accepted for publication 29 January 2015 Published 6 March 2015 Abstract
We report a gradual suppression of the two-dimensional electron gas (2DEG) at the LaAlO3 /SrTiO3 interface on substitution of chromium at the Al sites. The sheet carrier density at the interface (n ) drops monotonically from ∼2.2 × 1014 cm−2 to ∼2.5 × 1013 cm−2 on replacing ≈60% of the Al sites by Cr and the sheet resistance (R ) exceeds the quantum limit for localization (h/2e2 ) in the concentrating range 40–60% of Cr. The samples with Cr 40% show a distinct minimum (Tm ) in metallic R (T ) whose position shifts to higher temperatures on increasing the substitution. Distinct signatures of Rashba spin–orbit interaction (SOI) induced magnetoresistance (MR) are seen in R measured in out of plane field (H⊥ ) geometry at T 8 K. Analysis of these data in the framework of Maekawa–Fukuyama theory allows extraction of the SOI critical field (HSO ) and time scale (τSO ) whose evolution with Cr concentration is similar as with the increasing negative gate voltage in LAO/STO interface. The MR in the temperature range 8 K T Tm is quadratic in the field with a +ve sign for H⊥ and −ve sign for H . The behaviour of H MR is consistent with Kondo theory which in the present case is renormalized by the strong Rashba SOI at T < 8 K. Keywords: oxide hetero-structures, 2DEG, magnetoresistance (Some figures may appear in colour only in the online journal)
the 3d transition metal series, the LaVO3 /STO interface also turns out to be metallic [14]. But this behaviour ceases to exist in LaCrO3 /STO and LaMnO3 /STO interfaces. However, while LaCrO3 is an antiferromagnetic (TN = 290 K) band insulator, the manganite is a Mott insulator due to the eg1 electron. In both the cases, the interface turns out to be insulating with some magnetic character [15–17]. This dramatic change in the properties of the interface on going from LaTiO3 to LaMnO3 is yet to be understood, although one can attribute it to the increasing magnetic and mixed valent character of the higher 3d transition elements. In the context of doping, it is important to mention that the doping of the interface (δ-doping) of LaAlO3 /SrTiO3 with few monolayers of SrTi1−x Mx O3 (x = 0.02, and M a 3d transition metal) has been investigated and a quenching of the 2DEG behaviour is observed, presumably due to carrier localization at the 3d ion sites [18, 19]. Recently Gray
1. Introduction
Transition metal oxide heterostructures provide new opportunities to understand the effects of broken inversion symmetry on the interaction between spin, charge and orbital degrees of freedom in highly correlated electron systems [1]. One remarkable feature of epitaxial oxide heterostructures is the observation of a two-dimensional electron gas (2DEG) at the interface of LaAlO3 (LAO) and LaTiO3 (LTO) epitaxial films grown on the TiO2 terminated surface of (0 0 1) SrTiO3 [2–5]. While LaAlO3 is a band insulator of 5.2 eV energy gap, the properties of LTO are controlled by Mott physics due to the 3d1 configuration of Ti3+ ions. However, in spite of these fundamental differences, the 2DEG at the LTO/STO and LAO/STO has broadly similar superconducting, magnetic and photoconducting properties [4–13]. As we move from Ti to V in 0953-8984/15/125007+09$33.00
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J. Phys.: Condens. Matter 27 (2015) 125007
P Kumar et al
substrates by ablating these targets with a KrF excimer laser (λ = 248 nm). The STO (0 0 1) substrates were pre-treated with the standard buffered hydrofluoric (NH4 F–HF) solution for 30 s in order to achieve a clean TiO2 terminated surface [34]. Prior to the deposition, the chemically treated STO (0 0 1) were annealed for about an hour at 830 ◦ C in the oxygen pressure of 7.4×10−2 mbar. Thin films of all the compositions were deposited at 800 ◦ C with oxygen partial pressure of 1×10−4 mbar. Deposition was performed by firing the laser at 1 Hz with an areal energy density of 0.56–0.65 J cm−2 on the targets. The film growth mode and thickness were monitored using RHEED (STAIB, 35 keV) during deposition. Once the deposition was completed, the samples were cooled in the same O2 partial pressure at the rate of 10 ◦ C min−1 to ambient temperature. For structural characterization, we have used x-ray diffraction (XRD) and reflectivity measurements carried out in a PANalytical X’Pert PRO MRD x-ray diffractometer equipped with a CuK α1 source. Magneto-transport measurements were performed in the Van der Pauw geometry using the resistivity attachment of a Quantum design magnetic property measurements system. Stable, low resistivity contacts were made with the 2DEG by wire bonding with thin Al wires.
et al [20] have studied LaAlx R1−x O3 /STO system where the rare earth element R = Tm and Lu are only 2% per formula unit. At such low doping level, the 2DEG remains more or less unaffected. The other issue of importance in the LAO/STO problem is the co-existence of superconductivity and magnetic ordering. While the former has been established rigorously, the existence of magnetic order is still a matter of debate due to its feeble nature [21–25]. The magnetic order manifests itself in subtle ways through the hysteresis in magnetoresistance (MR) and a characteristic minimum in resistivity. There is also a considerable debate on the source of carriers and magnetic moments whose condensation and ordering respectively lead to these two macroscopic phenomena. Generally, magnetism is attributed to Ti3+ ions which are produced by oxygen vacancies. However, oxygen vacancies also contributes to carriers and it has been noticed that signature of magnetism are absent in high carrier density samples [26]. Furthermore, the carrier concentration in such a system is highly sensitive to growth conditions [27–29]. It is also established that the tetragonal distortion at the interface and confinement of electron gas in 2D lift the degeneracy of Ti (dxy , dyz and dxz ) bands which also show anomalous dispersion, and possibly a Lifshitz transition depending on band filling [30–33]. Since the distortion is intimately linked with the lattice constant of LaAlO3 , which is increased by Cr substitution, the later is likely to affect the dispersion and occupancy of the t2g subbands. To address the critical role of these transition elements in the electronic properties of the interface, we have introduced a controlled site substitution of Al by Cr in LaAlO3 and have studied the electronic transport in LaAlx Cr 1−x O3 /SrTiO3 interfaces. Here, on the one end of the composition phase space we have LAO/STO that shows the confinement of 2DEG at the interface, while for the other end member LaCrO3 /SrTiO3 , no such confinement appears at the interface [15]. With the Cr substitution in LaAlx Cr 1−x O3 /SrTiO3 , one issue of interest is to look for the critical value of Cr at which the interface becomes insulating. But more importantly, we would like to address the evolution of magnetotransport and collective phenomena such as magnetism and superconductivity and their bearing on the nature of the electron donating layer. We note that the Cr substitution reduces the free carrier density at the interface in a rather gradual manner as samples with Cr concentration as high as 40% remain metallic. The measurements of in-plane and out-of-plane field MR in these systems show enhanced Rashba spin–orbit interaction (SOI) with the Cr addition. A comparison of our data with the results of electrostatic gating of LAO/STO interface suggests that the role of Cr addition is similar to that of the negative gate field.
3. Results and discussion
A set of typical RHEED specular beam intensity oscillations observed as a function of growth time during the deposition of x = 0 and x = 0.6 films are shown in figures 1(a) and (b). Such intensity oscillations were used to control the thickness of the deposited films with a precision of one unit cell for all values of x. The observation of sharp oscillations strongly indicates a Frank-van der Merve type [35] layer-by-layer growth of the LaAl1−x Cr x O3 on STO. The RHEED patterns along the (0 0 1) crystallographic direction after film deposition are shown in insets of figures 1(a) and (b). Clear streaks, which are the characteristics of a 2D flat and crystallographically ordered surface, were preserved prior to during and after the growth. The XRD patterns of (x = 0.0, 0.2, 0.4 and 0.6) samples taken in θ –2θ geometry are shown in figure 2(a). A distinct hump in diffracted intensity on the higher 2θ side of the (0 0 2) reflection of the substrate marks the (0 0 2) Bragg line for the film. We have calculated the out-of-plane lattice parameter of the LaAl1−x Cr x O3 films from the position of this hump and the similar ones observed next to the (0 0 1) and (0 0 3) reflections of the substrate. Variation of the out-of-plane ‘c’ lattice parameter of the layers as a function of Cr concentration is shown in the inset of figure 2(a). The increase in the ‘c’ lattice parameter observed on Cr substitution in the film can be attributed to the larger ionic radius of Cr3+ compared to that of Al3+ . The shift in the XRD peak of the film towards the reflection from the substrate with increasing Cr suggests that the top TiO2 plane of STO is under less compressive strain because of a better lattice match between SrTiO3 and LaCrO3 . The Phi scan around the (0 2 2) Miller plane exhibits four-fold symmetry with overlapping of the XRD peaks for the substrate and the film that indicates the cube-on-cube symmetry along (0 1 1) plane. Kiessig fringes were observed
2. Experimental
Dense and compact laser ablation targets of LaAll−x Cr x O3 (x = 0.0, 0.2, 0.4, 0.6, 0.8, 1.0), 25 mm in diameter and 6 mm thick were prepared by solid state reaction method. Films of thickness 20 unit cells (≈8 nm) of all the compositions were deposited on the TiO2 terminated SrTiO3 (0 0 1) single crystal 2
J. Phys.: Condens. Matter 27 (2015) 125007
P Kumar et al
Figure 1. Typical RHEED intensity oscillations and RHEED images recorded along the (0 0 1) crystallographic direction during the fabrication of (a) x = 0 and (b) x = 0.6. The clean intensity oscillations reveal a layer-by-layer growth of the LaAl1−x Cr x O3 film on (0 0 1) SrTiO3 .
Figure 2. (a) HRXRD around (0 0 2) peak of single crystal SrTiO3 shows the film peak marked as ‘F’ adjacent to substrate peak, revealing epitaxial growth along the (0 0 1) crystallographic direction of the substrate. Inset of figure 2(a) shows the variation of out-of-plane lattice constant of the thin film as a function of Cr concentration. Figure 2(b) shows Phi scans around the (0 2 2) plane of the substrate and thin film. The four peaks separated at 90◦ show a four-fold symmetry, revealing epitaxial growth along (0 0 1) crystallographic direction of the substrate.
in resistance on cooling below ≈50 K and finally a saturation at the lowest temperature. This behaviour is emphasized in figure 3(b) by plotting the data of metallic samples (x = 0, 0.2 and 0.4) on an expanded scale and in figure 3(c) by normalizing R (T ) with respect to its value at 2 K. We note a shift of the resistivity minimum to higher temperature with the increasing Cr concentration while the magnitude of minimum is nonmonotonic in these samples. A qualitative understanding of the R (T ) can be made by considering various classical and quantum contributions to charge carrier scattering in a 2D metal. However, before we dwell upon detailed interpretation of the R (T ) data, it is worthwhile to show how the n and R at 300 K vary as a function of Cr concentration. These data are shown in figure 4(b). We notice that for the Cr free sample the sheet carrier density n (≈2.2 × 1014 cm−2 ) is close to the number predicted from the polarization catastrophe argument (≈3.2 × 1014 cm−2 ) for the formation of 2DEG in LAO/STO. But it drops monotonically with a slope of 3.35×1013 cm−2 per atomic percentage of the Cr substitution. The Hall resistance
in x-ray reflectivity measurements (not shown here) that also suggest a smooth surface and interface of the grown films. Thus, the two complimentary measurement modes i.e. RHEED patterns along the (0 0 1) crystallographic direction and phi scan near (0 2 2) plane establish excellent crystalline quality of our films. The temperature dependence of the sheet resistance R (T ) of the LaAl1−x Cr x O3 /SrTiO3 structure reveals a clear transition from a delocalized to localized electronic transport with the increasing x as shown in figure 3. While the R (T ) of the samples with x = 0.0, 0.2, 0.4 and 0.6 decreases on lowering the temperature from 300 to ≈50 K, the absolute value of R (T ) at 300 K exceeds the quantum resistance for localization (h/2e2 ) for the samples with x > 0.4. In R (T ) of all the samples, a step like behaviour is noticed at ≈83 and ≈183 K (see figure 3(a)) which also has been reported earlier in the heating cycle of resistance measurements [36, 37]. The metallic behaviour seen in the samples with x 0.6 gives way to a resistivity minimum followed by an increase 3
J. Phys.: Condens. Matter 27 (2015) 125007
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Figure 3. (a) Temperature dependence of the sheet resistance of LaAl1−x Cr x O3 /SrTiO3 heterostructures for x = 0.0, 0.2, 0.4 and 0.6 in the temperature range of 2–300 K. Figure also shows a clear insulating behaviour for sample with x > 0.6. (b) Shows the R (T ) data for x = 0.0, 0.2 and 0.4 samples from 2 to 100 K. The solid lines are fit to equation (2) in the temperature range of 10–100 K. (c) Shows R (T ) of the three samples normalized with respect to its value at 2 K to emphasize the minimum in R (T ). (d) Shows fitting of sheet resistance of x = 0.0, 0.20 and 0.40, to an equation of the type R = R0 + R1 T 2 + R2 T 5 + RK (T /TK ), where the last term is the Kondo contribution which replaces the localization contribution in equation (2).
(RH ) for all these samples measured at several temperatures over a field range of ±6 T remains linear in the field as shown in the figure 4(a). In the simplest possible scenario, this suggests a single band electronic transport. The temperature dependence of the sheet resistance of a complex system like this one will have contributions from processes over and above those related to classical scattering by frozen-in disorder, phonons and electrons. One important consequence of the diffusive motion of electrons in a 2D metal at low temperatures is the enhanced quantum backscattering (QBS) as the inelastic scattering events diminish due to phonon freeze out. This effect leads to a ln T growth of R of the form, R (T ) = −K (p/2) ln [T /T0 ] ,
In the presence of magnetic impurities, the classical s–d scattering [42] can contribute to resistance in such a manner that it diminishes quadratically in magnetization (M) in the concentrated impurity limit [43, 44]. At very low impurity concentration where the moments are not ordered, a Kondo-type electronic transport may prevail [45]. A careful examination of Hall resistance RH and MR as a function of magnetic field does not reveal any hysteretic effects, suggesting the absence of direct magnetic scattering in the system down to 2 K. However a higher order s–d scattering in the Kondo sense can contribute to low temperature resistance in a characteristic manner if magnetic impurities are present in the dilute limit. The Kondo contribution leads to a resistance RK ∼ BlnT , where B is a function of exchange integral J between electronic and impurity spins and the density of states at the Fermi energy [45]. At very low temperatures, the logarithmic divergence of RK is cut off due to the formation of Kondo singlets. We, therefore, model the R(T ) data between 10 and 100 K on the basis of the following generalized equation [44];
(1)
where K is a constant and the index p is related to the inelastic scattering time and T0 is order of the Kondo temperature [38, 39]. A similar temperature dependence of sheet resistance is also expected when quantum corrections to conductivity due to electron-electron interactions in a disordered 2D conductor are taken into account [40, 41]. While the ln T dependence of R (T ) seen in figure 3(c), over a limited window of temperature is suggestive of these models, the saturation of R (T ) at the lowest temperature is a deviation from equation (1). Also the temperature range over which the ln T dependence is seen is rather high for the e–e interaction to be operational.
R = R0 + R1 T 2 + R2 T 5 + R3 lnT ,
(2)
where R0 is the residual resistance due to frozen-in disorder, the T 2 term comes from e–e scattering and the T 5 contribution has its origin in electron–phonon scattering. The last term 4
J. Phys.: Condens. Matter 27 (2015) 125007
P Kumar et al
small, positive and isotropic [40]. The Kondo contribution to MR is also small and isotropic but negative in sign. There is ample evidence in literature, particular on GaAs-GaAlAs 2DEG systems where the ‘Dyakonov–Perel’ (DP) mechanism of spin relaxation dominates, that τSO can be a strong function of temperature [50, 51]. In the presence of only k-cubic spin splitting as applicable in the case of SrTiO3 , the magneto-conductance correction due to SOI goes as [52]; e2 [σ (B) − σ (0)] = π h 1 H H 1 1 Hϕ 2Hϕ ϕ SO ψ + + + ψ + + 2 B B 2 2 B B 1 1 Hϕ × − ψ . (3) + 2 2 B − ln Hϕ + HSO − 1 ln Hϕ + 2HSO + 1 ln Hϕ B 2 B 2 B Here Hφ = h/8π eDτφ and HSO = h/8π eDeτSO , and D, τφ and τSO are diffusion constant, dephasing time and spin– orbit scattering time respectively. The τSO , in general, derives contribution from the scattering processes by phonons and impurities in the presence of SOI induced by heavy atoms in the lattice. This is called the Elliott-Yafet τSO [53, 54]. However, the breaking of inversion symmetry at the interface of heterostructures gives rise to a stronger spin–orbit relaxation named the DP mechanism [55]. In the present case we will assume that τSO is entirely dictated by the DP process. We have performed two sets of MR measurements on the LaAl1−x Cr x O3 /STO samples whose sheet resistance is below the threshold for localization (≈12.9 k ). One, the transverse MR was measured with the magnetic field applied perpendicular to the plane (H⊥ ) of the sample. Two, longitudinal MR was measured with the field applied parallel (H ) to the plane of the 2DEG. The MR is defined as [(RH − R0 ) /R0 ] × 100%, where RH and R0 are the sheet resistance in field and in zero-field condition respectively. The MR data for H⊥ configuration in the field range of 0 to ±6 T at 2, 4, 6, 8, 10 and 15 K are shown in figures 5(a) and (b). It is worth mentioning here that widely varying magnitude and sign of MR have been reported in LAO/STO interfaces [56–59]. A comparative study of MR in film grown at high (∼10−4 mbar) and low (∼10−6 mbar) O2 partial pressure suggests that high MR is associated with samples where a significant fraction of carriers is derived from O2 vacancies [60]. The very small values of MR and its sign in H⊥ and H geometries are consistent with results on several other reported 2DEG samples [61, 62]. A careful examination of the four panels in figure 5 reveals that there are three distinct field dependence of the MR depending on the temperature of measurement. At the lowest T (=2 K), the MR increases sharply with field, reaches a peak value and then drops on further increase in the field. In the ±H plots of figure 5, the width of the MR cusp and the field at which it reaches a peak value increases with temperature and Cr concentration in the LaAll−x Cr x O3 /STO films. Once the temperature exceeds a critical value T1 , the positive MR becomes nearly linear in the field. This H -linear behaviour gives way to parabolic field dependence once the temperature exceeds a critical value T2 . The values of T1 and T2 for x = 0
Figure 4. (a) Hall voltage divided by the applied current I as a function of magnetic field measured at 10 K. (b) Variation of sheet resistance (R ) at 300 K and carrier concentration (n ) at 300 and 10 K as a function of Cr concentration in LaAl1−x Cr x O3 /SrTiO3 .
of equation (2) represents quantum correction to resistance, which may possibly come from weak localization, low temperature correction to e–e scattering and the scattering by local magnetic moments in the Kondo sense. It is clear from the fits shown in figure 3(b) that equation (2) expresses the temperature dependence of R well down to 10 K. The fitting parameters are listed in table 1. However, below 10 K, we see the saturation of the resistance. The other phenomenon which can significantly affect the low temperature electron transport in these systems is SOI of the Rashba-type originating from the broken inversion symmetry. The conduction band in doped STO consists of t2g orbital, which under spin–orbit coupling splits into J = 3/2 and J = 1/2 states of energy separation ∼17 meV [46–48]. In the context of the interfacial t2g orbitals in LAO/STO system, the quantum confinement along the z direction pushes the dxy orbitals down in energy compared to the dzx and dyz . Thus dxy can be assigned to J = 1/2 atomic state where as dzx and dyz to J = 3/2. These energy bands are further split by the Rashba SOI, which is expected to be cubic in the wave vector k for J = 3/2 state. It has been shown that the k-linear and k-cubic Rashba splittings leave their distinctive signatures in the MR [49]. The measurements of MR also permit us to differentiate between weak localization, quantum correction to the e–e scattering and the Kondo effect. While WL is an orbital effect and thus a strong negative MR is predicted when the external field is perpendicular to the plane of the 2D gas (H⊥ configuration), the MR due to e–e is 5
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Table 1. Obtained fitting parameters on fitting the R versus T data by using equation (2) in the temperature range of 10–100 K.
x (Cr con.)
R0 ( −1 )
R1 ( −1 K−2 )
R2 ( −1 K−5 )
R3 ( −1 )
0 0.20 0.40
1711.72 3620.05 7630.34
0.094 0.179 0.395
−2.14E−8 −2.65E−8 −7.19E−8
−262.56 −445.62 −1265.25
Figure 5. Magnetoresistance at different temperatures in transverse magnetic field (H⊥ ) for LaAl1−x Cr x O3 /SrTiO3 samples (a, a )x = 0.0 and (b, b )x = 0.40. Inset of figure (a , b ) shows Kohler plot at 15 K.
and 0.4 samples are (6 K, 8 K) and (8 K, 10 K) respectively. The transition from linear to parabolic dependence is however gradual. We also note that the peak magnitude of MR at 2 K increases with x. We first discuss the +ve MR at T > T2 which goes as ∼ H α (α > 1). The +ve sign of the MR seen here is inconsistent with the WL and Kondo scattering pictures, and the T2 is too large for quantum contributions of e–e to MR to be effective. This regime of MR can be attributed to the classical elastic and inelastic scattering events as the charge carriers tend to go in a cyclotron orbit in the out-of-plane field. Further the Kohler rule [62, 63] predicts, R (H ) /R0 ∝ a|µH |2 and its applicability however depends on the validity of the classical Boltzmann transport in this 2DEG system. Based on the Hall data and the fact that ωc τ 1, we believe the Kohler rule can be applied to H⊥ geometry MR. In the inset of figures 5(a ) and (b ) we have plotted the average MR (=1/2(MR(+H) + MR(−H)) at 15 K for x = 0 and 0.4 samples as a function of (H /R0 )2 . For H 0.5 T we notice an
approximate quadratic field dependence which is pronounced at the higher fields. From this MR data, the calculated Kohler mobility is ≈0.30 cm2 V−1 s−1 and 0.07 cm2 V−1 s−1 for x = 0.0 and x = 0.4 samples respectively. The data at T 6 K show a clear deviation from the classical orbital effect. We interpreted these results in the framework of the k-cubic Rashba SOI effect which leads to a conductivity correction of the type expressed in equation (3). Figures 6(a) and (b) shows the fits to equation (3) in the field range of ±6 T. The quality of these fits is excellent. The critical fields, time scales and length scales extracted from such analysis are listed in the table 2. For the intermediate temperature T1 < T < T2 , the data indicate a linear temperature dependence of MR. This is an interesting result in itself as linear MR has been seen in doped and disordered semiconductors and also in some topological insulators [64–68]. A tentative explanation for such dependence is the possibility of spatial fluctuations in carrier concentration and carrier mobility which can distort 6
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P Kumar et al
respectively. The magnetic field dependence of sheet resistance in the Kondo regime can be modelled as [39, 72, 73]. R (H|| ) = R (0) + RK (H|| /HK ).
(5)
To extract HK , a magnetic field scale, related to Kondo temperature and g factor of the impurity spin [72], we apply equation (5) to the H field data shown in figures 7(a) and (b) for 10 K. The best fitting obtained is HK = 80.5 T and 85.5 T for x = 0.20 and 0.40 respectively. A generalized expression of the type equation (5) can also be written for the Kondo contribution to the temperature dependence of zero field resistivity RK (T /TK ). Following Goldhaber Gorden [72] we have; s
T TK2 RK = RK (T = 0) , (6) TK T 2 + TK2 where
TK =
(21/s
TK − 1)1/2
assuming only the Kondo contribution to resistance equation (6) dominated, a fit to the data shown in figure 3(d) to a equation of the type R = R0 + R1 T 2 + R2 T 5 + RK (T /TK ) .
Over a temperature range of 2–100 K yield a TK of ≈47.6 K and ≈58.5 K for x = 0 and x = 0.40 respectively. The TK and HK thus derived are in order of magnitude agreement to equation (4). The fitting parameters are listed in table 3. At this stage, we revisit the data given in table 1 and compare the critical fields and length scales for different samples deduced in the frame work of Rashba SO contribution to MR. Caviglia et al [74] has studied the behaviour of Hϕ , HSO , Lϕ , τϕ and τSO as a function of gate voltage in LAO/STO interface. The ratio of HSO /Hϕ in their case is ≈50 at 1.5 K and zero bias, and increases with the increase in carrier concentration. For our undoped samples this ratio is ≈4 at 2 K, and decreases further with the Cr doping. This behaviour is consistent with the results of Caviglia et al as they also see a drop with the decreasing carrier concentration (increase negative gate voltage). The trend of τϕ and τSO with increasing Cr concentration is also consistent with the effect of gating, although absolute numbers differ. It is worthwhile to compare our results with the data on 2D electron gas produced at the top gated (0 0 1) SrTiO3 surface [49]. The absolute values of critical fields in this case are lower by an order of magnitude but their dependence on carrier concentration (gate field) is similar. It needs to be pointed out here that the carrier density in gated (0 0 1) STO is lower by a factor of ≈15 as compared to the sheet carrier concentration at the LAO/STO interface. Much lower critical fields have been reported in high mobility 2D gas in Gax In1−x As quantum well structures [75].
Figure 6. Variation of magneto-conductance [σ (B) =
σ (B) − σ (0)], σ being the sheet conductance, measured in a magnetic field applied perpendicular to the LaAl1−x Cr x O3 /SrTiO3 interface at different temperatures (a) x = 0.0, (b) x = 0.40. Black line is the fit to SOI theory equation (3).
the carrier path as seen earlier in epitaxial layers of InAs and InAsN [69]. Such inhomogeneities can arise from the two different sources of carriers in LAO/STO and LTO/STO interface [70], in addition, since the three temperature regimes are closely spaced, one has to consider the effect of competition between the contributions of SOI and the classical orbital scattering. A rigorous analysis of the data in this temperature window should allow separation of these two contributions. We see a striking difference in the field dependence of MR when the magnetic field is aligned parallel to the plane of the film. The results of these measurements for x = 0.2 and 0.4 samples performed at 2 K, 6 K, 10 K and 20 K are shown in figures 7(a) and (b) respectively. While the MR curve at 2 K for H has the same shape as in the case of H⊥ , the overall value of MR is smaller by a factor of two. There is a remarkable change in the sign and shape of the MR curve for T > 8 K. Now it appears quadratic in field, but with a negative sign which is not consistent with the Kohler type MR emanating from the classical orbital effect. We attributed it to the Kondo scattering where the magnetic field acts in the same manner as the temperature, the identity between the Kondo temperature (TK ) and Kondo field (HK ) goes as [71]. SgµB HK = kB TK ,
(7)
4. Conclusions
(4)
where S, g, µB and kB are impurity spin quantum number, Lande ‘g’ factor, Bohr magnetron and Boltzmann constant
In summary, we have successfully grown epitaxial LaAl1−x Cr x O3 films on TiO2 terminated SrTiO3 . The sheet 7
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Table 2. Parameters extracted from fitting the magnetoconductance data of figure 6 to equation (3). Hφ and HSO are the characteristic magnetic field, τφ and τSO the relevant relaxation times and D the diffusion coefficient. The carrier effective mass used in these calculation is 1.5me , me is the mass of electron.
Sample Hϕ LaAl1−x Cr x O3 /STO Temperature (T) x = 0.0
2K 4K 6K 2K 4K 6K
x = 0.40
0.17 0.30 0.43 0.41 0.50 0.51
Lϕ (nm) HSO Phase Spin–orbit field coherence D = vF2 τ/2 (T) length (m2 s−1 ) × 10−4
τϕ (s) × 10−12 De-phasing time
τSO (s) × 10−13 Spin–orbit scattering time
0.72 1.79 2.08 1.06 1.34 1.88
1.18 0.66 0.44 2.06 1.66 1.58
2.79 1.10 0.92 8.00 6.19 4.28
31.25 23.33 19.60 20.15 18.10 17.91
8.22 8.38 8.64 1.95 1.99 2.05
Figure 7. Magnetoresistance at different temperatures in longitudinal field (H ) from 6 to −6 T field for (a) x = 0.20 and (b) x = 0.40. Inset of figure 7(a) shows MR for LaAlO3 /SrTiO3 in longitudinal mode. Table 3. Obtained fitting parameters on fitting the R versus T data by using equation (7) in the temperature range of 2–100 K.
x (Cr con.)
R0 ( −1 )
R1 ( −1 K−2 )
R2 ( −1 K−5 )
RK ( −1 )
Kondo temperature (K)
0 0.20 0.40
186.80 846.01 45.00
0.088 0.184 0.371
−2.33E−8 −3.40E−8 −6.23E−8
1049.3 2041.4 5175.3
47.6 49.6 58.5
resistance in the temperature range 2–300 K increased significantly with substitution of Cr at the Al sites. For x 0.6, the sheet resistance goes beyond the 2D quantum limit (R ≈ h/2e2 ). The sheet carrier density is also suppressed monotonically with the increasing Cr substitution. The magnetoresistance data at T 20 K has been analyzed in the detail to extract various classical and quantum contributions to electronic transport. T 10 K out-of-plane field MR suggests strong Rashba spin–orbit coupling dominated transport. The in plane field MR and characteristic minimum in R (T ) suggest Kondo-like scattering of charge carrier by localized spins associated with Ti3+ and Cr3+ ions. The effect of Cr substitution on magneto-transport characteristics mimics the behaviour of LAO/STO interface on negative gate bias, which reduces the carrier concentration at the interface.
the Indo-French Centre for Promotion Advanced Research. (Project # IFCPAR 4704-I), and PK thanks CSIR for a senior research fellowship. RCB has also been supported by the JC Bose National Fellowship of the department of Science and Technology. References [1] Hwang H Y, Iwasa Y, Kawasaki M, Keimer B, Nagaosa N and Tokura Y 2012 Nat. Mater. 11 103 [2] Ohtomo A and Hwang H Y 2004 Nature 427 423 [3] Ohtomo A and Hwang H Y 2006 Nature 441 120 [4] Ohtomo A, Muller D A, Grazul J L and Hwang H Y 2002 Nature 419 378 [5] Biscaras J, Bergeal N, Kushwaha A, Wolf T, Rastogi A, Budhani R C and Lesueur J 2010 Nat. Commun. 1 89 [6] Reyren N et al 2007 Science 317 1196 [7] Li L, Richter C, Mannhart J and Ashoori R C 2011 Nat. Phys. 7 762 [8] Rastogi A, Kushwaha A K, Shiyani T, Gangawar A and Budhani R C 2010 Adv. Mater. 22 4448 [9] Rastogi A and Budhani R C 2012 Opt. Lett. 37 317
Acknowledgments
We acknowledge CSIR-India for financial support through AQuaRIUS Project. AD and RCB acknowledge funding from 8
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