Evidence for Spin Glass Ordering Near the Weak to Strong

Apr 11, 2016 - For each data set shown in Figure 1b, Δρxx/ρO is positive and is .... Conventionally, for spin glasses, remanence measured in the ZF...
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Evidence for Spin Glass Ordering Near the Weak to Strong Localization Transition in Hydrogenated Graphene Bernard R. Matis,*,† Brian H. Houston,† and Jeffrey W. Baldwin† †

Naval Research Laboratory, Code 7130, Washington, DC 20375, United States S Supporting Information *

ABSTRACT: We provide evidence that magnetic moments formed when hydrogen atoms are covalently bound to graphene exhibit spin glass ordering. We observe logarithmic time-dependent relaxations in the remnant magnetoresistance following magnetic field sweeps, as well as strong variances in the remnant magnetoresistance following field-cooled and zero-field-cooled scenarios, which are hallmarks of canonical spin glasses and provide experimental evidence for the hydrogenated graphene spin glass state. Following magnetic field sweeps, and over a relaxation period of several minutes, we measure changes in the resistivity that are more than 3 orders of magnitude larger than what has previously been reported for a twodimensional spin glass. Magnetotransport measurements at the Dirac point, and as a function of hydrogen concentration, demonstrate that the spin glass state is observable as the zero-field resistivity reaches a value close to the quantum unit h/2e2, corresponding to the point at which the system undergoes a transition from weak to strong localization. Our work sheds light on the critical magnetic-dopant density required to observe spin glass formation in two-dimensional systems. These findings have implications to the basic understanding of spin glasses as well the fields of two-dimensional magnetic materials and spintronics. KEYWORDS: hydrogenated graphene, 2D magnetic material, spin glass, transport localization

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transport and results in the emergence of quantum phenomena like strong and weak localization12 and resonant scattering.13 Studies have shown a transition from WL to SL for different atomic species covalently bonded to the graphene, which occurs as a smooth transition from one localization regime to the other.12,14 Chemical modification of graphene has also been proven to introduce point defects that carry magnetic moments, which can influence spin-dependent transport15 and show signs of ferromagnetic ordering.16 In this article, we use magnetotransport measurements to show that hydrogenated graphene is a disordered 2D system that undergoes a transition to a spin glass state as the temperature T is lowered to 4.2 K. At T = 293 K the magnetoresistance (MR) at the film’s Dirac point (DP) is positive and is explained by a classical model. As T is reduced to 4.2 K the MR at the DP becomes negative, and for the lowest and highest hydrogen concentrations used in our experiments a localization model accurately describes the observed low-T MR. However, for a narrow range of hydrogen concentrations, and consequently as the zero-field resistivity approaches h/2e2,

spin glass is a fundamental magnetic system in which the random competition between ferromagnetic and antiferromagnetic exchange interactions results in a frustrated state with no long-range magnetic order. Such a form of magnetism represents a weakly understood and complex problem in condensed matter physics owing to the geometrical frustration, random nature of the exchange interactions, and the lack of a well-defined ground state. Nonmagnetic, threedimensional materials doped with a random distribution of magnetic elements of a few atomic percent have been shown to exhibit a phase transition to a spin glass state.1−3 Recently, experimental evidence suggests spin glass ordering in submonolayer Fe films on cleaved InAs surfaces,4 despite conflicting arguments concerning the possibility of such a magnetic state forming in two dimensions above zero temperature.5−10 However, no studies within two-dimensional (2D) systems have focused on how the ability to observe the spin glass state depends upon the concentration of magnetic impurities, or how other quantum effects such as strong localization (SL) and weak localization (WL) may affect the ability to observe the state via magnetotransport measurements. Introducing chemical impurities into the 2D material graphene distorts the periodic conjugated lattice, thus breaking the translational symmetry of the crystal.11 Such symmetry breaking modifies graphene’s charge transport and magneto© XXXX American Chemical Society

Received: March 22, 2016 Accepted: April 11, 2016

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DOI: 10.1021/acsnano.6b01982 ACS Nano XXXX, XXX, XXX−XXX

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resistivity Δρxx/ρO = {[ρxx(B) − ρxx(B = 0 T)]/ρxx(B = 0 T)} × 100 as a function of B at the DP before and after hydrogenation. For this particular device, Raman spectroscopy measurements following hydrogenation gave ID/IG ∼ 0.6 (for the initial graphene ID/IG ∼ 0.05), where ID and IG are the D and G band intensities, respectively. The appearance of a sharp D band following hydrogenation indicates lattice symmetry breaking due to added defect sites with the bonding of hydrogen.17 Values for ρxx were obtained from ρxx = RxxW/L, where W and L are the sample width and length, respectively, and Rxx is the magnetoresistance obtained by sourcing a current between electrodes 1 and 2, I12, as labeled in Figure 1a and by measuring the resultant voltage drop using electrodes 3 and 4, V34, with B always applied perpendicular to the (hydrogenated) graphene plane. For each data set shown in Figure 1b, Δρxx/ρO is positive and is explained by a classical model that takes into account an inhomogeneous distribution of electrons and holes throughout the conductor with equal mobility and equal concentrations,18 given by

where h is Planck’s constant and e is the fundamental unit of electric charge, and the phase coherence length Lφ becomes comparable to the localization length ξ (indicating a transition from WL to SL), the localization theory fails to accurately describe the MR across the entire range of B used in our experiments. At the point where the localization theory fails we measure long relaxation times for the resistivity following sweeps of B in addition to very different results between fieldcooled (FC) and zero-field-cooled (ZFC) measurements. Measurements of the resistivity following field sweeps show changes of more than 200 Ω over the course of 5 min, which is more than 3 orders of magnitude larger than the change in resistivity previously reported for a 2D spin glass.4 The results indicate a transition to a spin glass state upon cooling to 4.2 K, and the ability to observe the state is highly dependent upon the concentration of hydrogen bonded to the graphene, which affects the resistivity of the system. It is the fact that the zerofield resistivity is approaching h/2e2, when the system transitions from WL to SL, that the effect is realized.

RESULTS AND DISCUSSION An optical image of one of our devices can be seen in Figure 1a. Figure 1b shows the percent change in the longitudinal

⎛ 1 1 1 ρxx (B) = ⎜⎜ + * ρ ρ 1 + (μB)2 xx , O ⎝ xx ,1

⎞−1 ⎟ ⎟ ⎠

(1)

The B-independent ρxx,1 term is a modification to the theory, and it has been suggested that the origin of the term is due to ne ≠ nh and μe ≠ μh near the DP.19 In the data fitting, ρxx,1, ρxx,O and μ served as the free parameters. For the initial graphene the ratio ρxx,O/ρxx,1 changes slightly from T = 293 K to T = 4.2 K (ρxx,O/ρxx,1 ∼ 0.23 and 0.31, respectively), whereas μ increases by roughly 40%, which explains the increase of Δρxx/ρO for the initial graphene as T is reduced. Upon hydrogenation ρxx,O/ρxx,1 increases to ∼ 0.6, whereas μ decreases nearly 95% from the initial graphene case. Covalently bound hydrogen atoms on graphene are known to reduce μ,20 which suggests a reduced contribution from the B-dependent term in eq 1. Hydrogen atoms bound to graphene also distort the crystal lattice11 and induce charge doping,21 which can impact the ne and nh area fractions within the film, where ne and nh are the electron and hole concentrations, respectively, thereby serving to enhance the 1/ρxx,1 term and reduce the magnitude of Δρxx/ρO. However, it is clear that a classical model can account for the observed dependence of Δρxx/ρO on B for all data sets shown in Figure 1b. Furthermore, the T dependence of Δρxx/ρO on B for the hydrogenated graphene is strikingly different from the pristine graphene case. As an example, Figure 1c shows Δρxx/ ρO vs B at the DP and at different T for a device with ID/IG ∼ 1.6. The data in Figure 1c shows a transition from a positive to a negative dependence of Δρxx/ρO on B as T is lowered to 4.2 K; this transition occurred in all of our hydrogenated devices regardless of ID/IG and is similar to the negative MR observed in exfoliated graphene devices that were subsequently hydrogenated.12 We can further quantify the number of hydrogen atoms from the ID/IG ratio, and for ID/IG ∼ 1.6 we estimate the defect density nD ∼ 4.05 × 1011 cm−2.17 This value for nD would correspond to roughly six hydrogen atoms per graphene domain for our device geometry. One characteristic magnetic property of a canonical spin glass is the long relaxation period for the remnant magnetization, which varies approximately linearly as a function of ln(t).22 Figure 2a shows the normalized resistivity ρ/ρO versus logarithmic t for two hydrogenated devices with ID/IG ∼ 0.7

Figure 1. (a) Optical image of a graphene Hall bar. Light regions (1−8) are the Cr/Au contacts. The brown background is the SiO2 substrate. (b) Percent change in the magnetoresistivity Δρxx/ρO versus magnetic field B at the DP for graphene at temperature T = 293 K and T = 4.2 K and for hydrogenated graphene at T = 293 K (ID/IG ∼ 0.6). The solid lines are fits to eq 1. (c) Percent change in the magnetoresistivity Δρxx/ρO versus magnetic field B at the DP and for varying temperature T for a hydrogenated graphene device with ID/IG ∼ 1.6. B

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ZFC state, B was raised to a maximum field Bmax (here, the range 1.5 T ≤ Bmax ≤ 2.6 T was used). Upon reaching Bmax the field was ramped back to B = 0 T, after which ρ was immediately recorded. For the FC procedure Bmax was applied while the sample was at 100 K, and the sample was allowed to cool to 4.2 K with Bmax applied. For the FC procedure, once the sample temperature reached 4.2 K, B was set to 0 T and ρ was subsequently recorded (in all cases the device resistance was observed to saturate upon reaching thermal equilibrium, thereby eliminating thermal drift as a cause of the observed device behavior). Figure 2b shows that the measured values of ρ are higher for the ZFC procedure, which is consistent with the isothermal remanent magnetization and thermoremanent magnetization properties of spin glasses22 and is attributed to the spin-glass magnetic ordering. Table SI of the Supporting Information shows a summary of our spin glass results. It is particularly interesting that spin glass properties for our hydrogenated graphene devices have been observed only for 0.7 ≤ ID/IG ≤ 1.6. Rappoport et al. showed using Monte Carlo simulations that for a certain density of hydrogen atoms on graphene the system becomes highly frustrated and that magnetic ordering reminiscent of a spin glass can occur.23 We consider then the question of what hydrogen concentrations meet the critical magnetic-impurity density requirement for observing spin glass properties via magnetotransport. We answer this question by considering the WL to SL transition that occurs within increasingly disordered graphene samples, and we find that the critical density for observing the spin glass behavior is the density that causes ρ (B = 0) to reach a value close to h/2e2. In Figure 3, we plot the T = 4.2 K data from Figure 1c, which corresponds to the hydrogenated device with ID/IG ∼ 1.6 and

Figure 2. (a) Normalized resistivity ρ/ρo as a function of time t at temperature T = 4.2 K and at the DP for the hydrogenated devices with ID/IG ∼ 0.7 and ID/IG ∼ 2.0 following a magnetic field B sweep from 2.6 to 0 T. The solid lines are linear fits to the data. The inset shows the bare resistivity ρ as a function of time t for the device with ID/IG ∼ 0.7. (b) Measured resistivity ρ as a function of maximum set field Bmax for different cooling procedures for the hydrogenated device with ID/IG ∼ 0.8 as detailed in the main text.

and ID/IG ∼ 2.0 where the time t dependence of ρ was recorded at T = 4.2 K immediately following a field sweep when B was ramped from 2.6 to 0 T (the Figure 2a inset shows the relaxation of ρ vs t following the field sweep; here dρ/dt > 0 is consistent with an overall negative MR). The data in Figure 2a shows a long relaxation period for ρ following the sweep to B = 0 T for the device with ID/IG ∼ 0.7, whereas the device with a larger concentration of chemisorbed hydrogen (ID/IG ∼ 2.0) showed no observable sign of remanence following the field sweep; relaxation periods like the one shown in Figure 2a were observed only for 0.7 ≤ ID/IG ≤ 1.6 with the longest relaxation periods on the order of tens of minutes. The solid lines in Figure 2a are fits to aln t + b where the two separate fits across the range of t are suggestive of two separate relaxation periods for the remnant magnetization following the field sweep with an initial rapid relaxation of the magnetic moments (solid red line) followed by a slower relaxation after t ∼ 3 min (solid green line). We point out that, in general, it is the long relaxation time, not a particular functional form of the MR, which distinguishes a spin glass from another magnetic material. For canonical spin glasses the remnant magnetization depends upon whether the system is cooled with or without an applied field. Figure 2b shows the results using the two different cooling procedures. For each data point in Figure 2b, we first raise T to 100 K. For the ZFC procedure the sample was allowed to cool from 100 to 4.2 K with B = 0 T (in all cases the cooling rate was 0.05−0.1 K/sec and the device resistance was monitored while cooling). Upon cooling to 4.2 K in the

Figure 3. Change in the magnetoresistivity Δρxx versus magnetic field B at the DP and at T = 4.2 K for ID/IG ∼ 1.6. The solid line is a fit to eq 2. The dashed line is a fit to Δρxx = a + bBc, where a, b, and c are constants (see the main text). The arrow marks the saturation field BS for WL.

for which a long relaxation period for ρ was observed following a sweep of B to 0 T, where Δρxx = [ρxx(B) − ρxx(B = 0 T)]. The solid line in Figure 3 is a fit to a localization theory for graphene,24 where we have calculated Δρxx from the theoretical expression for the semiclassical Drude conductivity given by δσ =

⎡ −1 ⎛ ⎞ τ −1 e 2 ⎢ ⎛⎜ τB ⎞⎟ ⎟ F ⎜ −1 ⎟ − F ⎜⎜ −1 B −1 ⎟ πh ⎢⎣ ⎝ τφ ⎠ ⎝ τφ + 2τinter ⎠ ⎛ ⎞⎤ τB−1 ⎟⎥ − 2F ⎜⎜ −1 −1 −1 ⎟ ⎝ τφ + τinter + τintra ⎠⎥⎦

C

(2)

DOI: 10.1021/acsnano.6b01982 ACS Nano XXXX, XXX, XXX−XXX

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ACS Nano Here, F(x) = ln(x) + ψ(0.5 + x−1), where ψ is the digamma function, τB−1 = 4eDB/ℏ, τφ = Lφ2/D the phase coherence time, and ℏ is Planck’s constant divided by 2π. The diffusion constant D is expressed as D = 0.5vF2(τinter−1 + τintra−1)−1 where vF ∼ 1 × 106 m/s is the graphene Fermi velocity, τinter = σsh/ [2e2vF(πn)1/2] is the intervalley scattering time due to weakpoint disorder, and τintra = h/(2evFπ1/2)μn1/2 the intravalley scattering time due to charged-impurity disorder (see the Supporting Information for estimates of the short-range conductivity σs and charge carrier density n). From fitting to eq 2, we find Lφ ∼ 45 nm. We also determine 2

the localization length ξ from ξ = Lee(σD/e /h), where σD is the Drude conductivity and Le the elastic mean free path determined from Le = σDh/[2e2(πn)1/2] (see the Supporting Information for the determination of σD). Here, we find ξ ∼ 59.3 nm, which indicates the system is on the WL side of the WL to SL transition (because ξ > Lφ); the WL to SL transition occurs when ξ and Lφ become comparable. WL should saturate around the saturation field Bs = ϕo/ξ2 where ϕo = h/e is the magnetic flux quantum. Here, the value of ξ ∼ 59.3 nm gives Bs ∼ 1.1 T. This value for Bs coincides with the value of B, where the data in Figure 3 deviates from the theory described by eq 2. This transition at B ∼ 1 T is not accompanied by any transition in Le (comparable to La), the magnetic length LB (less than ξ and greater than Le and La), the curved arc trajectory length rc (less than Le and La), the thermal length LT (independent of B), or the cyclotron radius Lc (greater than Le and La). Therefore, it would seem that the transition near B ∼ 1.0 T in the Δρxx vs B data set is not governed by any of these quantities, whereas for B < 1.0 T, the negative MR is attributed to WL (see the Supporting Information for a detailed discussion regarding the calculated length scales). Furthermore, the dotted red line in Figure 3 represents a fit of the data for B > 1.0 T to Δρxx = a + bBc, where a, b, and c are constants. From the fit we find c ∼ 1.04, suggesting a linear relationship between Δρxx and B. The linear relationship of Δρxx on B is in agreement with measurements of the MR of canonical spinglasses in higher dimensional materials.1 We thus suspect that the low-T, linear negative MR observed for the hydrogenated samples with 0.7 ≤ ID/IG ≤ 1.6 and for fields stronger than Bs, is due to spin-dependent scattering between the charge carriers and the local magnetic moments. The application of B aligns the spins of the magnetic moments thereby modifying the spinscattering cross section, which for a negative MR suggests less spin-dependent scattering with increasing B. The alignment of the spins as T is cooled is dependent upon the cooling procedure: if the system is cooled in the FC state, then the application of Bmax serves to align the spins and modify the remnant magnetization. Conventionally, for spin glasses, remanence measured in the ZFC state is generally less than that for the FC state, indicating that the FC state, with a higher degree of remanence would have a lower resistance value, which is the behavior observed in Figure 2b. The negative MR is reduced at higher T where thermal fluctuations dominate and modify the magnetic moment up and down populations. Figure 4 shows Δρxx/ρO vs B for increasing ID/IG (i.e., increasing hydrogen concentration) at T = 4.2 K and at the DP. For the lowest ID/IG used in our experiments (ID/IG ∼ 0.6, solid black circles), eq 2 accurately describes the data below B ∼ 0.8 T, whereas above B ∼ 0.8 T Δρxx/ρO saturates for all B (the dotted black line in Figure 4 represents a fit of this ID/IG ∼ 0.6 data set to eq 2); for this data set we found ξ > 2Lφ and an

Figure 4. Percent change in the magnetoresistivity Δρxx/ρO versus magnetic field B at the DP and at T = 4.2 K for increasing ID/IG. The dashed lines are fits to eq 2 for the lowest and highest ID/IG (solid black circles and open green diamonds, respectively). Solid lines are linear fits to the data for 0.7 ≤ ID/IG ≤ 1.6.

unsatisfied Ioffe−Regel criterion kLe ≤ 1, where k = πn is the magnitude of the particle wavevector, indicating the WL regime dominates. Additionally, for the highest ID/IG (ID/IG ∼ 2.0, open green diamonds) eq 2 can be used to describe the data across the full range of B, which is shown by the dotted green line; for this data set k Le ∼ 1.13 indicating the onset of a strongly localized state and a transition to the SL regime. However, for intermediate ID/IG (0.7 ≤ ID/IG ≤ 1.6) eq 2 fails to describe the dependence of Δρxx/ρO on B above a critical field as exemplified by the fitting in Figure 3 and by the linear fits (solid lines) shown in Figure 4. It is for this intermediate range of ID/IG that we have measured spin glass properties and conclude that the negative MR for B > Bs is the result of spindependent transport becoming the dominant scattering process as the system transitions from the WL regime to that of SL. The ability to observe the spin glass state within our devices therefore is highly dependent upon both the amount of hydrogen covalently bonded to the graphene as well as the arrangement of the hydrogen atoms throughout the graphene (because both quantities can affect ID/IG), which is consistent with prior calculations.23 We find that the samples for which spin glass properties were measured all had a zero-field resistivity of approximately the quantum unit of resistivity at T = 4.2 K and all displayed spin glass properties as the system transitioned from the WL regime to that of SL. Thus, all of the devices for which we have measured spin glass behavior showed ρ(B = 0) ∼ h/2e2, where ρ(B = 0) was measured using the ZFC procedure and before the application of B. Samples with a MR fully explained by eq 2 were found to have ρ(B = 0) either above or below h/2e2 depending upon whether the system was in the WL or SL regime. The transition from WL to SL has been shown to occur for chemically modified and disordered graphene in the vicinity of ρ(B = 0) ∼ h/2e2.11,14,25 This leads us to conclude that the critical magnetic-dopant density required to observe spin glass formation in two-dimensions via magnetotransport is the density that results in ρ(B = 0) ∼ h/2e2. Although the 4-fold degeneracy is expected for graphene (double valley and double spin degeneracy),26 it has been suggested that doping the graphene sheet with localized spins can create a magnetic texture resulting in valley symmetry breaking.27 The lifting of the valley degeneracy while preserving the spin degeneracy can account for this factor of 2 and explain why the spin glass formation was observed when ρ(B = 0) ∼ h/ 2e2. D

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CONCLUSION We have used magnetotransport measurements to observe spin glass ordering in hydrogenated graphene. The spin glass state is observable when the zero-field resistivity approaches the quantum unit h/2e2 as the system undergoes a transition from weak to strong localization. This work provides an understanding of the critical magnetic-dopant density needed to observe spin glass formation in 2D systems. The results presented here have implications to the basic understanding of spin glass formation in addition to the emergent fields of 2D magnetic materials and 2D spintronics.

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METHODS Our devices consist of polycrystalline graphene grown by chemical vapor deposition (CVD) on Cu foil that has been transferred to a SiO2 (285 nm)/Si (doped) substrate. An oxygen plasma is used to etch the graphene into a Hall bar geometry, and electron-beam lithography is used to define the Cr (10 nm)/Au (50 nm) contacts. The graphene devices before hydrogenation are spatially inhomogeneous such that the average grain size La is several orders of magnitude less than the sample length between the contacts used to measure a voltage drop (electrodes 3 and 4 in Figure 1a) along the current path. For the device shown in Figure 1a, we find La ∼ 56 nm before hydrogenation (see the Supporting Information for the Raman spectra and estimation of La). All of our transport measurements were carried out in vacuum (