Introduction

Carrier scattering gives rise to electrical resistance – a measure of frictional losses within semiconductor specimens1. Under microwave/mm-wave/terahertz photoexcitation, high mobility 2D electron systems confined, for example, within GaAs/AlGaAs heterostructures exhibit large amplitude “1/4 cycle shifted” magnetoresistance oscillations with resistance maxima in the vicinity of E = (j + 3/4)ħωc, resistance minima near E = (j + 1/4)ħωc, and nodes in the resistance oscillations in the vicinity of the cyclotron resonance, and integral and half-integral cyclotron resonance harmonics, i.e., E = jħωc and E = (j + 1/2)ħωc. Here, E = energy, ωc = eB/m*, e = electron charge, m* = electron effective mass, ħ = the reduced Planck constant, and B = the magnetic field. Most remarkably, at the lowest temperatures under modest photo-excitation, the deepest resistance minima saturate into zero-resistance states2, about B = 4/5Bf and B = 4/9Bf, of the characteristic field Bf = 2πfm*/e, where f is the electromagnetic-wave frequency2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83. That is, experiments suggest that the diagonal resistance can be switched off in the 2D electron system by photo-excitation in the presence of a small magnetic field. Such experimental reports have motivated the theoretical study of transport in photo-excited 2D semiconductor specimens57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83. Large amplitude magnetoresistance oscillations under photoexcitation also imply strong electrical sensitivity to electromagnetic waves, which could potentially be applied towards the realization of microwave/mm-wave/terahertz photodetectors84,85,86,87. Indeed, since the observations thus far have indicated Bf = 2πfm*/e, the implication is that radiation frequency f = eBf/2πm* can be determined from the characteristic field Bf of the magnetoresistance oscillations. The concurrent observed sensitivity to both the radiation-intensity and frequency has suggested the possibility of a tuned narrow band radiation sensor in the microwave/mm-wave/terahertz bands25.

Above mentioned radiation-induced magnetoresistance oscillations have already served to characterize material systems such as GaAs/AlGaAs heterostructures2,3,5,6,7,9,11,13,15,16,17,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55, strained Si/SiGe88, electrons on the surface of liquid Helium44,89, and the oxide MgZnO/ZnO 2DES system90 for scattering lifetimes, and effective masses. Thus, one expects that photo-excited magnetotransport studies of modern atomic layer 2D systems such as monolayer and bilayer graphene could potentially unveil new science and applications46,91,92,93,94,95,96,97,98,99. However, the expectations for device response of these materials is not known and such phenomena have not been observed thus far in the graphene system. As a consequence, a question of interest is: what will be the oscillatory resistance response of modern atomic layered 2D materials such as, for example, graphene under electromagnetic wave excitation in a magnetic field? Below, we examine the expected characteristics for both the monolayer and bilayer graphene systems. As mentioned above, our results suggest a strong sensitivity of the response in monolayer graphene to both the Fermi energy and the radiation frequency. However, bilayer graphene is expected to show a more GaAs/AlGaAs like behavior, characteristic of parabolically dispersed systems. The results are summarized below.

It is well known that the Landau level dispersion for Dirac fermions in monolayer graphene is given by EN = sgn(N)(2ħeBN)1/2vF97,98. Here, N is the Landau level index, and vF is the carrier velocity. Suppose that the Fermi level lies in the Nth Landau level, EN = EF, and the electromagnetic radiation induces transitions from N’th Landau level to the N + q’th Landau level57,59,60,61, such that EN + q = EF + hf, where h is Planck’s constant, and f is the radiation frequency. From studies of the GaAs/AlGaAs system2, it is known that nodes in the radiation-induced magnetoresistance oscillations appear when the radiation energy spans such an integral number of Landau levels, i.e., EN+q − EN = hf, where q denotes the order of the node. Consider \({E}_{N}^{2}=2\hslash eBN{v}_{F}^{2}\), and note that

$${E}_{N+q}^{2}-{E}_{N}^{2}=2\hslash eBq{v}_{F}^{2}$$
(1)

Apply the identity \({E}_{N+q}^{2}-{E}_{N}^{2}=({E}_{N+q}-{E}_{N})({E}_{N+q}+{E}_{N})\). Let EN+q − EN = hf and EN+q + EN = 2EF + hf, Then, by substituting into Eq. 1, we obtain

$$(2{E}_{F}+hf)hf=2\hslash eBq{v}_{F}^{2}$$
(2)

or

$$B=(1/q)(\pi f/e)(2{E}_{F}/{v}_{F}^{2}+hf/{v}_{F}^{2})$$
(3)

Thus, Eq. 3 describes the magnetic field values for the q’th node of radiation-induced magnetoresistance oscillations in monolayer graphene and it suggests that such oscillations in monolayer graphene are also dependent upon value of the Fermi energy, EF, unlike in the GaAs/AlGaAs system2,5,7,9,11,13,15,17,25,27,31,35,41. In Fig. 1(a), we plot the nodal positions in magnetic field, B, vs. the inverse of integers, 1/q, for several typical values of the Fermi energy in monolayer graphene. The figure shows a set of straight lines, which indicates that the expected magnetoresistance oscillations are periodic in B−1, just as in the GaAs/AlGaAs system. The characteristic frequency or field, Bf, of the radiation-induced magnetoresistance oscillations that appears in the empirical formula for the oscillatory magnetoresistance lineshape ΔR ≈ −exp(−λ/B) sin (2πBf/B)11, is found from the slopes in Fig. 1(a) of B vs. 1/q, which suggest increases of Bf with EF, at a fixed radiation frequency, f = 100 GHz. The inset of Fig. 1(a) shows the variation of this characteristic frequency Bf of the radiation-induced magnetoresistance oscillation with EF at several radiation frequencies, f = 25, 50 100, 200 and 400 GHz. The inset conveys that Bf increases faster with EF at larger radiation frequencies f. Note that Bf increases linearly with f over this range of f since EF ≫ hf, see eqn. 3. For f = 100 GHz, hf = 4.125 × 10−4 eV is much smaller than a small practical value for EF such as EF = 5 × 10−3 eV.

Figure 1
figure 1

(a) For monolayer graphene, the magnetic field values, B, for nodes in the microwave induced magnetoresistance oscillations are plotted vs. the inverse node index, 1/q, with q = 1, 2, 3…, for different values of the Fermi energy, EF, at a microwave frequency of f = 100 GHz. The inset shows the characteristic field or oscillation frequency Bf of the microwave induced magnetoresistance oscillations as a function of the Fermi energy, EF, for radiation frequencies f = 25, 50, 100, 200, and 400 GHz. (b) This figure illustrates the expected oscillatory magnetoresistance, ΔR, vs. B, in monolayer graphene under microwave photo-excitation at f = 100 GHz for three values of the Fermi energy, EF. Note that the characteristic magnetic field, Bf, of the oscillatory magneto-resistance increases with EF.

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The expectations for the oscillatory resistance, ΔR ≈ −exp(−λ/B) sin (2πBf/B)11, in the monolayer graphene system are illustrated in Fig. 1(b), which exhibits ΔR vs. B at EF = 0.5, 0.25, and 0.166 eV for photoexcitation at f = 100 GHz. Since the characteristic field Bf increases with EF, and the parameters B, f, e, and vF in Eq. 3 can be determined or are well known, the radiation-induced magnetoresistance oscillations in monolayer graphene can be utilized to accurately determine EF.

From the experimental perspective, measurements can be carried out as a function of the magnetic field, B, or, as is more typical for graphene, vs. the gate voltage VG. In monolayer graphene, the carrier density, nq, varies as the square of the Fermi energy, i.e., nq = (4π/h2)(EF/vF)2. Then,

$${B}_{f}=(\pi f/e)(1/{v}_{F}^{2})(2h{v}_{F}{({n}_{q}/4\pi )}^{1/2}+hf)$$
(4)

Further, for graphene on top of 300 nm SiO2 on doped Si the relation between the gate voltage and the carrier density is nq = sgn(ΔVG)α|ΔVG| with α = 7.2 × 1010 cm2/V97,98. Upon inserting these relations, we obtain from eqn. 4:

$${B}_{f}=(\pi f/e)(1/{v}_{F}^{2})(2h{v}_{F}{(\alpha |{\rm{\Delta }}{V}_{G}|/4\pi )}^{1/2}+hf)$$
(5)

Figure 2(a) exhibits the dependence of the frequency or characteristic field Bf of the magnetoresistance oscillations vs. the electron density, i.e., nq = n, and vs. the difference in the gate voltage with respect to the neutrality voltage, VN, i.e., ΔVG = VG − VN. The square root dependence observed in Eqs 4 and 5 is manifested as a sub-linear variation of BF with respect to these parameters at a fixed radiation frequency, f, i.e., Bf ≈ n1/2 and \({B}_{f}\approx {\rm{\Delta }}{V}_{G}^{1/2}\). Expectations for gate voltage dependence of the radiation-induced magnetoresistance oscillations in monolayer graphene following from Eq. 5 are exhibited in Fig. 2(b) for f = 200 GHz. The Fig. 2(b) shows that oscillations grow in amplitude with increasing magnetic field. Further, it is evident that the number of oscillations over a given span of gate voltage decreases with increasing B. Finally, the spacing between, say, successive oscillatory minima, increases with increasing ΔVG because the magnetoresistance oscillations are actually periodic in \({\rm{\Delta }}{V}_{G}^{1/2}\). From these numerical studies, it is clear that radiation-induced oscillations in monolayer graphene should be substantially different than the oscillations observed in the GaAs/AlGaAs system. The key differences are the dependence of the characteristic field or frequency Bf of the magnetoresistance oscillations upon the Fermi energy when EF ≫ hf even at a fixed radiation frequency in monolayer graphene (Fig. 1(a)), the expected dependence of Bf on the gate voltage (Fig. 2), and the possibility of using such magnetoresistance oscillations to make a measurement of the Fermi energy in monolayer graphene.

Figure 2
figure 2

(a) The characteristic field or oscillation frequency Bf of radiation-induced magneto-resistance oscillations in monolayer graphene is plotted vs. the electron density, n, (bottom abscissa), and the gate voltage, ΔVG, (top abscissa). Bf is proportional to n1/2 and \({\rm{\Delta }}{V}_{G}^{1/2}\). (b) This figure illustrates the expected oscillatory magneto-resistance, ΔR, vs. the gate voltage difference with respect to the neutrality voltage, ΔVG, in monolayer graphene under microwave photo-excitation at f = 200 GHz for three values of the magnetic field, B. Here, since the extrema are periodic in (ΔVG)1/2, they become further apart with increasing ΔVG.

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Unlike in monolayer graphene, charge carriers in bilayer graphene are massive fermions as a consequence of the parabolic band structure. Bilayer graphene is zerogap system in the absence of a transverse electric field and it develops a bandgap under a transverse electric field97,98. Consider the zerogap case: The dispersion of carriers in zerogap bilayer graphene under the influence of a magnetic field is given by EN = (N(N ± 1))1/2ω0B, where ω0 = eħ/m*, and m* = 0.037 m is the nominal effective mass of the carriers97,98. Here, the + and − describe the response of the electron and the hole systems, and N can take on positive and negative integers for electrons and holes, respectively. To determine the expected response for the radiation-induced magnetoresistance oscillations in bilayer graphene with electrons, we suppose that at a magnetic field, B, the Fermi level lies in the Nth Landau level, EN = EF, and the electromagnetic radiation induces transition from Landau level N to the N + q Landau level so that EN+q = EF + hf. Further, assume that nodes in the radiation-induced magnetoresistance oscillations will appear when the radiation energy spans an integral number of Landau levels, i.e., EN+q − Eq = hf, where q denotes the order of the node. To extract the characteristics for radiation-induced magnetoresistance oscillations in such a system, we examine:

$${E}_{N+q}^{2}-{E}_{N}^{2}=(hf)(2{E}_{F}+hf)={\omega }_{0}^{2}{B}^{2}q(q+2N+1)$$
(6)

By setting \({E}_{N}^{2}={E}_{F}^{2}=N(N+1){\omega }_{0}^{2}{B}^{2}\), we find \((N+1/2)={(({E}_{F}^{2}/{\omega }_{0}^{2}{B}^{2})+1/4)}^{1/2}\), to obtain

$$q={[hf(2{E}_{F}+hf)/({\omega }_{0}^{2}{B}^{2})+({E}_{F}^{2}/({\omega }_{0}^{2}{B}^{2})+1/4)]}^{1/2}-{({E}_{F}^{2}/({\omega }_{0}^{2}{B}^{2})+1/4)}^{1/2}$$
(7)

Equation 7 serves to determine the B-position of the nodes in the radiation-induced magnetoresistance oscillations as a function of the magnetic field for different values of the Fermi energy, EF. Figure 3(a) presents 1/q vs. B for different values of the EF at f = 100 GHz for bilayer graphene. The remarkable feature in this figure is the relative insensitivity of the results to the value of the Fermi energy for bilayer graphene, unlike in the case of monolayer graphene (see Fig. 1(a)). Thus, in this sense, bilayer graphene looks more similar to the GaAs/AlGaAs system.

Figure 3
figure 3

(a) For radiation-induced magneto-resistance oscillations in bilayer graphene, the inverse node index, 1/q, for q = 1, 2, 3… is plotted vs the magnetic field, B, for various Fermi energies, EF, at a radiation frequency, f = 100 GHz. (b) For radiation-induced magnetoresistance oscillations in bilayer graphene, the inverse node index, 1/q, for q = 1, 2, 3… is plotted vs. the magnetic field, B, for radiation frequencies f = 25, 50, 100, 200, and 400 GHz. The plot implies that increasing the radiation frequency shifts the magnetoresistance oscillations to higher magnetic fields. The inset shows the magnetoresistance oscillation frequency, Bf, vs the radiation frequency, f. The plot shows that the slope of the line is 1.32 mT/GHz which corresponds to an effective mass ratio m*/m = 0.037.

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The expected nodal B-positions, which are insensitive to EF in bilayer graphene, are examined for different f in Fig. 3(b). Figure 3(b) shows that a given node, q = 1, 2, 3…, shifts to higher B as f increases. Further, the frequency or characteristic field Bf of the radiation-induced magneto-resistance oscillations increases linearly with f in the limit where EF ≫ hf, as indicated in the inset of Fig. 3(b). Indeed, the characteristic field or of frequency Bf-field should shift at the rate of 1.37 mT/GHz in bilayer graphene vis-à-vis the ≈2.35 mT/GHz shift observed in the GaAs/AlGaAs system. These calculations have assumed a fixed effective mass of m* = 0.037 m for bilayer graphene, while it is known that non-parabolicity could provide for variation in m* with EF97,98. Thus, studies of radiation induced magnetoresistance oscillations in bilayer graphene as a function of the carrier density could serve to characterize the non-parabolicity and determine the effective mass as a function of the energy.

In comparing the expectations for monolayer and bilayer graphene, the striking feature is the great dissimilarity in the EF dependence of the radiation-induced oscillatory magnetoresistance characteristics. For monolayer graphene, the oscillations depend strongly on EF and therefore also the ΔVG. On the other hand, for bilayer graphene, there is a lack of sensitivity to EF and therefore also on ΔVG. This suggests that the characteristics of the observed magnetoresistance oscillations could also serve to differentiate between these two types of graphene.