Introduction

The quantum mechanical spin degree of freedom finds remarkable applications in the areas of quantum computing (QC) and spin-based electronics (spintronics)1,2,3,4,5,6,7,8. For example, in QC scenarios, particle spin often serves as a quantum bit or qubit1,2,3,4,5,6. In spintronics, the spin serves to endow electronic devices with new functionality as in the giant magneto-resistive read head or the spin-based transistor7,8. Graphene is a novel two-dimensional system with remarkable properties such as massless Dirac fermions, an anomalous Berry's Phase, a pseudo-spin (valley degeneracy) in addition to spin and half-integral quantum Hall effect9,10,11,12. Graphene is also an appealing material for electron-spin QC and spintronics1,4,5,6,7,8,13,14,15, owing to the expected weak spin–orbit interaction and the scarcity of nuclear spin in natural carbon. Because of QC and spintronics, the microwave control and electrical detection of spin have become topics of interest now in graphene nanostructures1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17, where the small number of spins limits the utility of traditional spin resonance.

Here we report the first observation of resistive detection of spin resonance in epitaxial graphene (EG)9,18,19, provide a measurement of the g-factor and the spin relaxation time, and determine the pseudo-spin (valley degeneracy) splitting at zero magnetic field. Such resistive resonance detection can potentially serve to directly characterize the spin properties of Dirac fermions, and also help to determine and tune the valley-degeneracy splitting for spin-based QC15.

Results

Trilayer graphene

Figure 1a shows the diagonal resistance, Rxx, versus the magnetic field, B, for the trilayer EG specimen, sample 1. The blue curve obtained at T=1.5 K in sample 1 shows a cusp in Rxx near null magnetic field, that is, weak localization (WL)20,21,22, followed by positive magneto-resistance at B>0.2 T. In Fig. 1a, an increase in T, to T=90 K, results in the red curve, which includes a positive displacement of the Rxx versus B trace with respect to the T=1.5 K trace, that is, dRxx/dT>0 at B=0 T, along with the quenching of WL. As WL cannot be observed without inter-valley scattering in monolayer or bilayer graphene22, the observed WL is presumed to be an indicator of a non-zero inter-valley matrix element.

Figure 1: Transport in trilayer graphene.
figure 1

(a) The diagonal resistance, Rxx, is shown versus the magnetic field, B, at temperatures T=90 K, shown in red, and T=1.5 K, shown in blue, for sample 1, in the dark condition, without microwave excitation. The upward displacement of the T=90 K curve with respect to the T=1.5 K curve shows that Rxx increases with the temperature, that is, dRxx/dT≥0. (b) Rxx versus B in the absence of microwave excitation is shown in blue, and under constant F=48 GHz microwave excitation at P=4 mW is shown in red, for sample 1. The photo-excited Rxx trace shown in red exhibits a uniform upward shift with respect to the dark Rxx curve shown in blue for B<1 T. At higher B, resonant reductions in the Rxx are observed in the vicinity of B=±1.4 and ±1.75 T, where the photo-excited Rxx approaches the dark value. Inset: an atomic force microscopy image of the EG/SiC surface with the device superimposed upon it. The size scale bar corresponds to 10 μm. (c) The change in the diagonal resistance, ΔRxx, between the photo-excited and dark conditions in panel b, that is, ΔRxx=Rxx(4 mW)−Rxx(dark), is shown versus B. Note the valleys in ΔRxx in the vicinity of B=±1.40 and ±1.75 T.

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Figure 1b illustrates the influence of microwave excitation on sample 1 at F=48 GHz. Here, for B<1 T, microwave excitation produces a positive displacement of the photo-excited Rxx relative to the blue trace obtained in the absence of photo-excitation, akin to increasing the temperature, cf. Fig. 1a. However, at B>1 T, Rxx shows resistance valleys as the photo-excited curve approaches the dark curve, similar to reducing the temperature. To highlight associated resonances, the change in the diagonal resistance, ΔRxx=Rxx(4 mW)−Rxx(dark), is shown versus B in Fig. 1c. Figure 1c shows two noteworthy features: a high magnetic field resonance at |B|=1.75 T and a low magnetic field feature at |B|=1.4 T. These resonances disappeared upon increasing the bath temperature to T>5 K.

Monolayer graphene

Figure 2a–c shows the results for sample 2, whereas Fig. 2d–f shows representative data for sample 3. Both sample 2 and sample 3 are monolayer EG specimens. The T dependence of Rxx at B=0 T is shown in Fig. 2a for samples 2 and 3, respectively. Unlike sample 1, samples 2 and 3 show a decrease in Rxx(B=0) with increasing temperature, that is, dRxx/dT≤0. Further, as indicated in Fig. 2b, microwave irradiation of these specimens produces a uniform negative displacement in the Rxx traces with increasing power, P. Yet, the effect of microwave excitation (dRxx/dP≤0 at B=0) is again similar to heating the specimen (dRxx/dT≤0), cf. Fig. 2a. Thus, the Rxx(B=0) regions from Fig. 2b have been marked as coloured discs in Fig. 2a. Apparently, microwave excitation at P=10 mW serves to increase the carrier temperature up to T=32 K in sample 2, and up to T=36 K in sample 3. At such higher P, the radiation helps to manifest, in addition, resonant Rxx peaks in the vicinity of the dashed lines of Fig. 2b, unlike in Fig. 1b, where valleys characterize the resonances in Rxx. Yet, in all three specimens, the photo-excited Rxx moves towards the dark curve at resonance. Fig. 2c shows resonances at nearly the same |B|, at F=18 GHz, in samples 2 and 3.

Figure 2: Transport in single-layer graphene.
figure 2

Panels a–c show some results for sample 2, whereas figs d,f show representative data for sample 3. (a) Sample 2 shows a decrease in the resistance Rxx with increasing temperature T, that is, dRxx/dT≤0, in the absence of a magnetic field, that is, B=0 T. (b) The data traces show a downward shift with increasing microwave power, P, at F=18 GHz, suggestive of microwave-induced carrier heating in the specimen. The Rxx at B=0 T obtained from these traces have been marked as filled circles in panel a above, which shows the T dependence of Rxx. Apparently, P=10 mW serves to increase the carrier temperature up to T=32 K in sample 2. Microwave-induced resonances appear in the vicinity of the dashed lines with increasing P. (c) ΔRxx=Rxx(10 mW)−Rxx (dark) is shown versus the magnetic field for sample 2. Note the change in ΔRxx due to spin resonance in the vicinity of the dashed lines at B=±0.66 and ±0.27 T. (d) Sample 3 also shows a decrease in the resistance Rxx with increasing temperature T, that is, dRxx/dT≤0, in the absence of a magnetic field, that is, B=0 T. (e) At F=18 GHz and T=1.5 K, Rxx is shown versus B for sample 3, at several power levels. The Rxx at B=0 T observed in these data have been marked as filled circles in panel d. (f) This panel shows ΔRxx=Rxx(10 mW)−Rxx(dark) versus the magnetic field for sample 3. Note the change in ΔRxx due to spin resonance in the vicinity of B=±0.66 and ±0.28 T.

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Spin resonance evolution

Figure 3 illustrates the frequency evolution of the ΔRxx resonances for all three specimens. Here, Fig. 3a–e illustrates the results for sample 1, Fig. 3f–h exhibits the data for sample 2 and Fig. 3i shows some results for sample 3. Note the shift of resonances to higher B with increasing F.

Figure 3: Frequency evolution of microwave-induced resonances.
figure 3

(a–e) For sample 1, the photo-induced change in the diagonal resistance, ΔRxx, is plotted versus B for (a) 17 GHz, (b) 22 GHz, (c) 28 GHz, (d) 42 GHz and (e) 48 GHz. Note the uniform shift in the resonances, indicated by dashed lines, to higher B with increasing F. Here, the resonances are characterized by ΔRxx minima. (f–h): For sample 2, the photo-induced change in the diagonal resistance, ΔRxx, is plotted versus B for (f) 17 GHz, (g) 19 GHz and (h) 21 GHz. Note the shift in the resonances, indicated by dashed lines, to higher B with increasing F. Here, the resonances are characterized by resistance maxima. (i,j) For sample 3, the photo-induced change in the diagonal resistance, ΔRxx, is plotted versus B for (i) 17 GHz and (j) 21 GHz. Note the shift in the resonances, indicated by dashed lines, to higher B with increasing F.

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Figure 4a presents a plot of the microwave frequency, F, versus the resonance magnetic fields, B, extracted from Fig. 3. Figure 4a shows that the resonance B values for the three specimens collapse onto two lines: a gold-coloured line in Fig. 4a, which represents the high B-field resonances of Fig. 3, shows a linear increase as F(GHz)=27.2B(T), with the ordinate intercept at the origin. Another line shown in magenta in Fig. 4a, which represents the low-B resonances of Fig. 3, shows a linear increase as F(GHz)=10.76+26.9B(T), with a non-zero intercept, F0=10.76 GHz. In such a plot, spin resonance for an electron with g-factor ge=2.0023 would follow: F(GHz)=28.01B(T). Thus, the observed slopes, dF/dB=26.9±0.4 GHz T−1 (dF/dB=27.2±0.2 GHz T−1) for the low (high) field resonance correspond to spin resonances with g//=1.92±0.028 (g//=1.94±0.014).

Figure 4: Relation between the resonance magnetic fields and the microwave frequency or energy.
figure 4

(a) This plot shows that the high B-field resonance found in Fig. 3 follows a linear fit, indicated by the gold line, as F(GHz)=27.2B(T) with an intercept at the origin. The low B-field resonance found in Fig. 3, indicated by the magenta line, follows a linear fit as F(GHz)=10.76+26.9B(T) with a non-zero intercept, F0=10.8 GHz. Here, the lines indicate the fits, while the symbols show the data points. The circles correspond to the data for sample 1, the squares correspond to the data for sample 2 and the triangles correspond to the data for sample 3. (b) The observed experimental results appear to be consistent with spin resonance and zero-field pseudo-spin (valley-degeneracy) splitting enhanced spin resonance. A fourfold degeneracy is lifted in the absence of a magnetic field to produce a pair of spin degenerate levels (doublets) separated by E/h=F0. Zeeman splitting then lifts the spin degeneracy of the upper and lower doublets. Microwave photo-excitation induces spin-flip transitions between the spin levels of the lower- or upper doublet, as shown by the gold lines. Such transitions require vanishing photon energy in the B→0 limit. On the other hand, the transition shown in magenta requires non-vanishing photon energy in the limit of B→0.

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Discussion

The g-factors measured here are comparable to the g-values obtained from traditional ESR studies of graphite, which have indicated that the g-factor for c-axis, , increases from 2.05 at 300 K to 2.15 at 77 K, while, at T=300 K, the g-factor for c-axis, =2.003 (refs 23, 24). In graphite, the g-factor depends upon the orientation of the B field, the temperature, the location of the Fermi level and the sign of the charge carriers, with opposite g-factor shifts, Δg, from ge, for electrons and holes23,24,25. The negative Δg and reduced g// observed here relative to ge are consistent with expectation for holes.

From these data, we also estimate a resonance half-width ΔB≈0.05 T, which corresponds to a spin relaxation time τs=h/(4πΔE)=6 × 10−11 s. The spin relaxation time has been a topic of great interest in graphene. Spin relaxation26 has been experimentally studied in monolayer and bilayer exfoliated graphene on SiO2/Si (refs 27, 28, 29, 30, 31, 32, 33, 34) and, more recently, on EG35,36 using spin valve devices, and possible mechanisms involved in spin relaxation have been examined by theory37,38,39. For monolayer exfoliated graphene, observed spin relaxation times generally fall in the range of 40–150 ps (refs 28, 30, 31, 34), while exfoliated bilayer graphene shows spin relaxation times as long as 2–6 ns (refs 32, 33). The observed shorter-than-expected spin lifetime in exfoliated monolayer graphene has been attributed to extrinsic mechanisms based on impurity adatoms37, charged impurities and phonons from the substrate38, spin–orbit coupling due to ripples in graphene39, and so on. Note that the τs reported here for C-face EG is comparable to previous reports of the spin relaxation time for monolayer exfoliated graphene on SiO2/Si. The spin-diffusion length here is λs=(Dτs)1/2=1.4 μm (ref. 13), while the Hall bar width, w=4 μm. Here, D is the diffusion constant. In such a situation, edges could be having a role in spin relaxation, given that edges in graphene can be magnetically active12, and the electrical contacts include gold, a heavy element. Thus, there could be additional avenues for spin relaxation in the small specimen, in addition to the other above-mentioned mechanisms37,38,39. Finally, inhomogeneities could serve to broaden the resonance linewidth and help to produce an apparently reduced spin relaxation time.

The observation of similar double resonances in monolayer and trilayer graphene can be viewed as a consequence of rotational (non-AB) layer stacking in EG, which makes it possible even for multilayer EG to show the same electronic properties as isolated graphene18. Note also that sub-lattice or pseudo-spin degeneracy lifting is known to occur at high B in graphene40,41,42. For example, the progression of quantum Hall effect from the Rxy=[4(N+1/2)]−1h/e2 sequence10,11,43, to observations of σxy increases in steps of e2/h (ref. 40), reflects the lifting of both the spin- and pseudo-spin degeneracy. In addition, a non-linear interaction-enhanced valley-degeneracy splitting has been reported from a scanning tunnelling spectroscopy study42. Finally, the manifestation of WL, which is observable in Figs 1 and 2, is an indicator of inter-valley coupling in these specimens22. As sub-lattice degeneracy splitting is not unexpected owing to the above, the observed F0=10.76 GHz is attributed to a zero magnetic field pseudo-spin (sub-lattice degeneracy) splitting of Δ0=hF0=44.4 μeV.

A provisonal interpretation of the F versus B plot of Fig. 4a is provided in Fig. 4b. Chiral eigenstates and linear energy-wavevector dispersion characterize carriers in graphene. The application of a B field nominally produces fourfold, valley- and spin-degenerate Landau levels characterized by EN=±νF(2eħBN)1/2, where N=0, 1, 2…, e is the electron charge, νF is the Fermi velocity and ħ is the reduced Planck's constant.

We imagine the fourfold degeneracy being lifted by hF0 even at B=0, to produce energy doublets as EN′=EN±hF0/2. Then, owing to the Zeeman effect, associated Landau levels show a further splitting of the spin degeneracy as EN″=EN′±gμB/2. Observed microwave-induced transitions occur within the highest occupied Landau level in the vicinity of the Fermi level. As EN>>hF0/2 and gμB/2, we remove the EN term and plot E/h=(EN″−EN)/h in Fig. 4b.

Here, microwave photo-excitation induces spin-flip transitions, shown in gold, of unpaired carriers between the spin levels of the lower or the upper doublet. Such transitions require vanishing photon energy in the limit of vanishing B. In contrast, a transition between the lower spin ('up') level of the lower doublet and the higher spin ('down') level of the upper doublet requires additional energy hF0, and such a transition, shown in magenta, exhibits non-vanishing photon energy in the B→0 limit. Thus, the F versus B plot appears consistent with spin resonance and a zero-field pseudo-spin (valley degeneracy) splitting enhanced spin resonance.

In summary, we have realized the resistive detection of spin resonance in EG, provided a measurement of the g-factor and the spin relaxation time, and identified—and measured—a pseudo-spin (valley degeneracy) splitting in the absence of a magnetic field. Such resistive resonance detection can potentially serve to directly characterize the spin properties of Dirac fermions, and also help to determine—and tune—the valley-degeneracy splitting for spin-based QC.

Methods

Graphene samples

EG was realized by the thermal decomposition of insulating 4H silicon carbide (SiC)18. The EG specimens were characterized by ellipsometry and the extracted layer thickness was converted to the number of layers at the rate of 0.335 nm per layer. The C-face of the EG/SiC chip was processed by e-beam lithography into micron-sized Hall bars with Pd/Au contacts. Measurements are reported here for three Hall bar specimens labelled 1, 2 and 3. Sample 1 is nominally trilayer graphene, while samples 2 and 3 are monolayer graphene. The samples are p-type, with a hole concentration, p≈1013 cm−2, and a carrier mobility μ≈103 cm2 V−1 s−1.

Measurement configuration

Typically, an EG Hall bar specimen was mounted at the end of a long straight section of WR-62 rectangular microwave waveguide. The waveguide with sample was inserted into the bore of a superconducting solenoid, immersed in pumped liquid Helium and irradiated with microwaves over the frequency range 10≤F≤50 GHz, at a source power 0.1≤P≤10 mW, as in the usual microwave-irradiated transport experiment44. Here, the applied external magnetic field was oriented along the solenoid and waveguide axis, as a probe-coupled antenna launcher excited the Transverse Electric (TE-10) mode in the waveguide. Thus, the microwave electric field was oriented perpendicular to the applied external magnetic field. The microwave magnetic field lines formed closed loops, with components in the transverse and axial directions of the waveguide.

Additional information

How to cite this article: Mani, R.G. et al. Observation of resistively detected hole spin resonance and zero-field pseudo-spin splitting in graphene. Nat. Commun. 3:996 doi: 10.1038/ncomms1986 (2012).