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Graphene possesses a unique Landau level system that is non-equidistantly spaced in energy, as thus a large amount of optical transitions may become possible. Here, by utilizing this unique feature, we propose a novel dual field method which combines both external magnetic field and gate electric field together to control the optical response of the graphene-based devices. The key principle of this method is to selectively allow different optical transitions in graphene among Landau levels via an electric gate tuning of the Fermi level. By applying this method to a graphene based amplitude modulator and through an implementation based on transfer matrix method, we numerically demonstrated the well characteristics of switchable modulation on four individual channels, a huge modulation depth up to 80 dB and an extremely low required energy of tuning Fermi level down to 10 meV. Such excellent frequency tunability and gate controlling ability of this dual field method may open up the potential for applications in active optoelectronics, spin optics, ultrafast optics and etc.
© 2016 Optical Society of America
1. Introduction
Non-electrical-neutral carriers will be quantified into discrete energy levels under an applied external magnetic field. The so-called Landau levels (LLs) is thus brought up to describe such a phenomenon [1]. For the case of carriers with non-zero mass, which is most common in the natural world, the Landau levels are equally spaced by an energy of . Hence, for electrons and holes in conventional materials such as semiconductors and metals, an additional optical transition induced by such Landau quantization process would occur when the photon energy equals to , typically in the order of 0.1 meV by assuming a magnetic field around 1 Tesla (the value of a commercial neodymium magnet). Such a low resonance energy makes this phenomenon almost impossible for potential photonic applications ranging from terahertz (THz) to infrared and even higher frequencies.
Graphene, a unique two-dimensional material with carbon atoms arranged in an iconic honeycomb structure, has shown many intriguing features. One of them is that the effective mass of its carriers is theoretically zero around the Dirac point, which results in relativistic behaviors of carriers [2]. Therefore, Landau quantization in graphene would be different from one of conventional materials and produces a particular Landau level system which is no-longer equally spaced but following relation:
where n denotes the index of each Landau level and vF denotes the Fermi velocity. It is obvious that Landau levels in graphene are highly non-equally spaced and the relative energy of adjacent LLs is much higher than that of a conventional material at lower level indexes. The typical energy difference between 0th and 1st LL at the condition of B = 1 Tesla reaches about 37 meV, which is more than two magnitudes higher than that in conventional materials, and also right within the THz photon’s energy range. For using larger magnetic fields obtained by electrical or optical pumped coils [3–5], higher resonance frequency up to near-IR that could fully cover the frequency band from THz to far-IR is feasible when using graphene.Most importantly, there have many possible optical transitions that can be selectively supported by graphene under a magnetic field. Therefore the selectivity can be embodied on two aspects. On one hand and intrinsically, transitions between LLs are restricted by the selection rule Δn = ± 1 as consequences of both momentum conservation and spatial symmetric configuration of the magneto-electronic wave function in graphene [6–8]; On the other hand, a LL can be controllably kept vacant or occupied according to the energy of Fermi level, so is the corresponding transition either forbidden or not. Each transition results in an optical absorption with different frequencies (energies) in graphene. Hence, by utilizing this controllable selectivity, we propose that the optical response of graphene could be drastically tuned on both amplitude and frequency. Fig. 1 shows a sketch of our concept on allowing different optical transitions in the LLs system via tuning the Fermi level. It also has been widely known that Fermi level in graphene can be efficiently tuned through an electrostatic gate [9–11]. Therefore, through a proper combination of both gate electric field and external magnetic field, graphene can exhibit unprecedented frequency tunability as the magnetic field enables a full-of-transition system to the gate electric field. We refer this method as the dual field control method. As a direct application of the new graphene based devices, whom electrical and optical property of graphene is inherited to, can be thus tailored to modulate electromagnetic waves drastically on both amplitude and frequency. By taking a full advantage of the Fermi level tuning and uniquely using an electrical gate at the presence of a magnetic field, this new method surpasses the previous studies on the gate tunability of graphene [12–21] and on a separate consideration of using a magnetic field as a possible means of tuning the optical response of the graphene based devices [22–25].
Fig. 1 Sketch of our concept on allowing different optical transitions in LLs system via tuning the Fermi level. Blue rings stand for LLs of holes and red ones stand for LLs of electrons. The two-color mixed sphere at the Dirac point indicates the degenerate state of holes and electrons. Solid green arrows and shadowed gray ones indicate the allowed transitions and forbidden transitions respectively. When Fermi level is between L0 and L1, transition L0->L1 is allowed and transition L1->L2 is forbidden; when Fermi level is shifted between L1 and L2, transition L0->L1 is turned off due to the occupation of both L0 and L1, while transition L1->L2 becomes active. Light blue arrows show some interband-like transitions that are free from Fermi level tuning.
In this work, we first considered the effect of a magnetic field together with a gate electric field on graphene-based devices in the view of quantized energy system. With the magneto-optical conductivity of graphene obtained using a full quantum mechanical method, we numerically investigated the optical response of a classic graphene based amplitude modulator under the simultaneous bias of magnetic field and gate electric field. By applying the above proposed dual field control method, we successfully made this single band modulator into a switchable multi-band one, and numerically demonstrated the device with an excellent frequency tunability and a great gate control ability. Potential applications of the device and also the method could be expected on various photonic functional devices, including active control systems, spin optics, nonreciprocal optics and all-optical systems and so on.
2. Methods and structure
We use a full quantum mechanical interpretation to model graphene and subsequently calculate its magneto-optical conductivity σg using the Kubo formula [7,26]. The calculation is performed by taking the contributions from all the LLs into account, thus could adequately reveal the quantized energy structure of graphene under an external magnetic field. Detailed descriptions of the final computer-programmable formula are presented in Appendix I, and also the selection of each parameter.
The graphene based device under studied here is a graphene based amplitude modulator (GAM) [Fig. 2(a)], which is a layered structure that resonantly absorbs radiation at specific frequency, also known as Salisbury screen. Magnetic field B is directly applied perpendicular to the GAM surface, while in order to realize the Fermi level control, an electrostatic gate is introduced here. A thin capacitor layer made by 100nm thick SiO2 together with a thicker layer of p-Si is placed under graphene, acting as the electric gate. Underneath is a layer of gold, who acts as an electrode and also as a back reflector. The incident electromagnetic wave would strongly interfere with the part reflected from the back reflector, resulting in the multi-beam interference phenomenon. Thus resonant absorption would take place in GAM by altering the thickness of p-Si, as constructive interference occurs at the graphene layer [27,28]. Therefore by tuning the surface conductivity of graphene, enhanced levels of absorption and reflection can be achieved in GAM. For the rigorous calculation of GAM’s optical response, transfer matrix method is utilized here [7,29,30], where graphene is treated as an anisotropic conductive surface in the presence of magnetic field.
Fig. 2 (a) Schematic illustration of the graphene amplitude modulator (GAM) under study. Magnetic field B is applied perpendicular to the graphene surface. (b) Relationship between the gate voltage and the Fermi level in GAM, by assuming graphene to be pristine. Inset shows a cross-section illustration of GAM.
The relationship between Fermi level and gated voltage can be calculated from [12,13,16]:
where VG is the gate voltage, fd is the Fermi-Dirac distribution. nc stands for the carrier density and is an intermediate variable. Cgc is the geometrical capacitance, which stands for the ideal capacitance of SiO2. Note that the capacitance of the silicon layer is ignored here, due to its leaky character. V0 is the gate voltage at which Fermi level just reaches the Dirac point. For pristine graphene, V0 equals to zero. However, for arbitrary graphene samples, V0 varies much. In this work, it’s not proper to set it at any value as generality will be lost. Therefore, to maintain generality, we take the place of gate voltage by Fermi level in the following sections. As a reference, Fig. 2(b) plots the relationship between Fermi level and gate voltage for pristine graphene. Once V0 is measured or determined, gate voltage can be immediately calculated from Fermi level using Eq. (2).3. Results and discussion
In order to give a quantitative description on the graphene’s surface conductivity under the influence of both magnetic field and electric field (Fermi level), we first calculate σg versus magnetic field B and versus Fermi level EF respectively [Fig. 3]. Note that the diagonal and off-diagonal elements of σg follow the same law under magnetic field [26,31,32], hence in this section we only show the results of diagonal element σxx for clarity. In Figs. 3(a) and 3(c), the real and imaginary parts of σxx is plotted versus magnetic field B at two individual frequencies which are arbitrarily set at 4 THz and 9 THz. Each curve comprises a train of resonant peaks, which are ascribed to the transitions among LLs directly. For example, the strongest peak at 1 Tesla is formed due to the intraband-like interlevel transition LL0->LL1, which can be confirmed via Eq. (1). In Figs. 3(b) and 3(d), the real and imaginary parts of σxx is plotted versus EF by assuming an external magnetic field of 1 Tesla. Similarly, once the energy of transition is coincident with the photon energy, a sharp rise in the conductivity would appear and thus forms a step as EF sweeps between the two LLs. For example, for EF ranging from 0 to 37 meV, LL0 is occupied and LL1 is vacant, thus transition LL0->LL1 becomes available. Therefore, at the frequency of 9 THz, the photon can excite this transition while at the frequency of 4 THz, the photon doesn’t have enough energy to do so. Subsequently, for EF ranging from 37 to 52.9 meV, both LL0 and LL1 are occupied while LL2 stays vacant, thus transition LL0->LL1 is utterly forbidden and transition LL1->LL2 becomes available instead. Therefore, at the frequency of 4 THz the photon can excite transition LL1->LL2 while at the frequency of 9 THz, the photon’s energy is too high. Consequently, the optical conductivity at 9 THz falls down while it at 4 THz rises and forms two plateaus with the variation of EF. Such a strong dependence of the optical conductivity of graphene on the dual fields forms the basis of using our proposed dual field method to modulate the electromagnetic waves. As a comparison, the dashed lines in Figs. 3(b) and 3(d) are the results under zero magnetic field, where a relatively minor dependence on EF is observed.
Fig. 3 (a, c) plot the real and imaginary parts of the diagonal optical conductivity of graphene versus the square root of magnetic field B at 4 THz (blue solid) and 9 THz (red solid) respectively. Fermi level is set to zero here. (b, d) plot the real and imaginary parts of the diagonal optical conductivity versus the Fermi level at 4 THz (blue solid) and 9 THz (red solid) respectively. Magnetic field is set to 1 Tesla here. Dashed lines in (b) and (d) plot the conductivity under zero magnetic field, as a comparison.
3.1 Absorption versus both B and EF
To achieve a bigger picture of GAM’s optical response at the presence of both external magnetic field and gate electric field, we map its absorption versus both B and EF at several typical photon energies. A normal incidence of linearly polarized electromagnetic wave is assumed here as well as in the following sections. Fig. 4(a)-4(c) shows the results at the photon energy of 5 meV, 15meV and 25 meV respectively, where the thickness of p-Si is set to 57.4 μm in order to achieve maximized absorptions at all the three energies. A very clear banded structure is observed in Figs. 4(b) and 4(c), and each band possesses a strong absorption region. This banded structure is formed through the transitions among LLs, as each band corresponds to an individual transition. Subsequently, when the transition’s energy equals to the photon’s energy, a resonant absorption would occur and a maximum in absorption thus shows up within the band. For example, in Fig. 4(c), the lowest three bands correspond to transitions LL0->LL1, LL1->LL2 and LL2->LL3, and each of them possesses a strong absorption regions centered at the magnetic field of 0.5 T, 2.6 T and 4.5 T respectively. For higher bands, the strong absorption regions will be centered at some larger values of B and EF, which can be determined from Eq. (1). It is important to point out that this phenomenon is the unique result of the unequally spaced LLs of graphene. Therefore, when compared with a conventional material like two-dimensional electronic gas (2DEG) who can only match the target photon energy at one fixed magnetic field, graphene could provide much more conveniences in choosing B and EF, and definitely benefit our GAM with great flexibility in design.
Fig. 4 (a, b, c) map the absorption of GAM versus both the magnetic field and the Fermi level at the photon energy of 5 meV, 15 meV and 25 meV respectively. (d, e, f) plot the cut-line view along the three pairs of the dashed white orthogonal lines in (a, b, c) respectively. Note that each pair of the dashed white lines is focused on a local maximum in absorption. (d, e, f) are dual x-axis plots, where the blue solid lines correspond to the down x-axis (a vertical cut) and the red ones correspond to the up x-axis (a horizontal cut). Normal incidence and linearly polarized wave is assumed here as well as in the following figures.
Also, it can be seen from Figs. 4(a)-4(c) that the width of each band along y-axis (EF) decreases with the drop of photon energy. This is because for lower photon energy, the corresponding transition has to be induced by a lower magnetic field, which can be directly seen from Eq. (1). When the width of each band is as small as that of spectral broadening contributed by impurity scattering and thermal excitation [7,11], an impressive fact thus appeals that all the bands overlap with each other and make up a continuous structure [Fig. 4(a)]. The disappearance, or more appropriately, the mergence, of these discrete bands would offer extra freedom in choosing the parameters for GAM when dealing with low frequency situations. Note that here our GAM also shows a near-unity absorption at some special values of B and EF, which indicates it to be a potential perfect absorber. The underlying mechanism is the surface impedance match and the multi-beam interference dominated field enhancement at the graphene layer. For information, a similar graphene based multi-layer structure has already been proposed as a perfect absorber and verified experimentally in Ref [27].
To illustrate the above properties of GAM more clearly, Figs. 4(d)-4(f) show x-cut view and y-cut view of each 2D map along a pair of orthogonal white dashed lines plotted in Figs. 4(a)-4(c). Y-cut view is for varying Fermi level under constant magnetic field and is plotted as blue curves in Figs. 4(d)-4(f). It shows an iconic step-like line shape where each step strictly corresponds to an individual transition. X-cut view is for varying magnetic field under constant Fermi level and is plotted as red curves in Figs. 4(d)-4(f). However, it shows an absorption peak (indicating a reflection valley) with a line shape that seems to be ‘twisted’. In fact, this twisted line shape is the result of switching among different LLs transitions. When magnetic field B sweeps under a static EF, the allowed transition will be changed among different LLs. At the joint point of two allowed transitions, the slope of the curve will be changed abruptly and result in an unusually ‘twisted’ line shape. It must be noted that, no matter the curves are step like or twisted like, there is no doubt that through a proper combination of magnetic field and gate electric field, a drastic modulation on the THz radiation can be achieved on our GAM.
Moreover, since this field tunable characteristic is effective on a wide range of photon energy, it is completely feasible to turn GAM from a traditional single channel device into a multi-channel one using our proposed dual field control method.
3.2 Four-channel modulation via tuning magnetic field under a static EF
In this section, we consider using magnetic field B to realize the multi-channel modulation of GAM. For clarity and simplicity, Fermi level is set constant here at an arbitrarily value of 60 meV. Firstly we calculate the reflection spectrum of GAM over 0.1 to 25 THz, versus magnetic field B ranged from 0 to 8 Tesla, which is obtainable by commercial magnets. And the thickness of p-Si is kept at 45 μm here. Fig. 5(a) shows the result and it can be seen that a series of deep reflection valleys are orderly formed on frequency spectrum with the variation of both B and f. As previously discussed, this phenomenon is due to the interlevel transitions among LLs in graphene with intrinsic dependence on external magnetic field [Eq. (1)]. Only photons with energy matching the energy of allowed transitions in graphene can be resonantly and strongly absorbed by GAM. We mark the first five transitions using dashed lines in Fig. 5(a), and it’s no coincidence that all the reflection valleys fall precisely on them. Note that transitions LL2->LL3 and LL3->LL4 possess no reflection valleys, as they just miss all structural resonances originating from the multi-beam interference. Another notable phenomenon in Fig. 5(a) is there exists a train of shallow horizontal lines that are equidistantly spaced. They are also the result of the multi-beam interference, however, the optical conductivity of graphene at those structural resonance frequencies is relatively very low since no optical transitions are active. As a consequence, only very shallow dips can be formed at those structural resonant positions.
Fig. 5 (a) maps the reflection spectrum of GAM versus magnetic field B at a Fermi level of 60 meV. Five dashed lines indicate transitions (from up to bottom): L0->L1, L1->L2, L2->L3, L3->L4, and L4->L5. Small local minimums on the up left corner correspond to a series of interband-like transitions, which are much weaker than the intraband-like ones indicated by dashed lines. (b) plots the reflection spectrum of designed four-channel modulation of GAM. The four channels are represented by the four deep valleys which correspond to magnetic fields (from left to right): 0.5 T, 1.4 T, 3.2 T, and 4.5 T. (d) plots the corresponding relative modulation depths of the four channels. (c) plots the relationship between the square root of magnetic field and the reflection of GAM at frequency 16.2 THz (red solid) and 27.9 THz (blue solid) respectively.
Proceeding from the 2D mapping of reflection spectrum in Fig. 5(a), we carefully select four different values of magnetic field B to accomplish a four-channel modulation on the THz waves by GAM. Results are shown in Figs. 5(b) and 5(d), where a four-channel modulation is successfully demonstrated. By tuning B at 0.5 T, 1.4 T, 3.2 T and 4.5 T, GAM is able to modulate waves at four frequency channels centered at 1.48 THz, 4.38 THz, 16.2 THz and 18.95 THz respectively. The modulation depth of those channels, which is defined as −20log10(RB/Rmax), reaches great values ranged from 35 dB to 75 dB [Fig. 5(d)]. Moreover, each channel has a uniform full width at half maximum (FWHM) of 0.95 THz, which indicates a very good isolation among each other to avoid potential crosstalk in practice.
It should be noted that for graphene, its optical conductivity is associated the strength of transition dipole moment. The strength can be scaled by the spectral weight of optical conductivity, and it is found to be varied with the square root of B [7,32]. This is a unique property originating from the massless fermions in graphene. Thus for instance the channel opened by B = 4.5 T possesses larger transition dipole moment than the ones opened by B = 3.2 T, 1.4 T and 0.5 T. As previously mentioned, our GAM not only follows the law in graphene, but also follows the law of multi-beam interference. Only a moderate strength of transition dipole momentum would result in a near-unity absorption [28]. Therefore in Fig. 5(d) we can see that the modulation depth of channel (B = 4.5 T) is actually lower than it of channel (B = 3.2 T).
We also plot the relation between reflection and magnetic field B at two representative frequencies [Fig. 5(c)]. At frequency of 16.2 THz indicated by red solid line, a curve similar to that in Fig. 3(a) is shown. However, at another frequency of 27.9 THz indicated by blue solid line, a train of small dips rises and cannot be ignored. This phenomenon is raised from the interband-like interlevel transition among LLs, e.g., L-3->L4, L-2->L3, etc., which corresponds to a much higher photon energy but a much smaller strength than the intraband-like interlevel transition does. Although these interband-like interlevel transitions are relatively small, from Fig. 5(c) we can see that it’s not proper to ignore them at the situation of high photon energies. Therefore, it’s very important to take them into consideration when using our proposed method in high frequency applications.
In this section we have demonstrated a four-channel modulation of GAM using solely magnetic field tuning under a static gate electric field. Compared with former works utilizing magnetic field alone to control the device’s optical response [22,23], our method exhibits unprecedented frequency tunability. Moreover, this four-channel modulation can be fully expandable to even more modulation channels and to satisfy wider requirements in practice.
3.3 Multi-channel modulation via tuning Fermi level under static B
Another practical case is electrostatic gate tuning under a constant external magnetic field. In this section we also firstly simulate the reflection spectrum of GAM over 0.1 to 15 THz, versus EF ranged from 0 to 120 meV. Magnetic field is kept 1 Tesla here. And the thickness of silicon is altered to 63.7 μm, in order to minimize the mismatch we discuss below. Figure. 6(a) shows the result and it can be seen that instead of valleys, a series of reflection plateaus or steps are formed on frequency spectrum with the variation of both EF and f. These plateaus are the same as the ones that shown in Fig. 3(b) and Figs. 4(d)-4(f), and each of them corresponds to an interlevel transition that dominates the absorption of photons. We mark the transitions who are responsible for the four steps in Fig. 6(a) using dashed lines, and it is found that there existed some small deviations between the transitions and the steps. This is due to the mismatch between the transition frequency and the structural resonance frequency. Detailed speaking, to make GAM work at its best status, the structural resonance frequency has to be carefully adjusted to meet the transitions in graphene; however, the structural resonance frequencies are equally distributed on the frequency spectrum, while the transitions in LLs system are highly non-equally distributed. Therefore the mismatch arises and results in the deviations between the transitions and the steps.
Fig. 6 (a) maps the reflection spectrum of GAM versus Fermi level at B = 1 Tesla. Four dashed lines indicate transitions (from up to bottom): L0->L1, L1->L2, L2->L3, and L7->L8. Note that there exist small deviations between the transitions and the minimums in three of them, except for the transition L1->L2. (b) plots the reflection spectrum of designed four-channel modulation of GAM. The four channels are represented by the four deep valleys which correspond to Fermi levels (from left to right): 78 meV, 58 meV, 48 meV and 28 meV. (d) plots the corresponding relative modulation depths of the four channels. (c) plots an optimized dual channel modulation of GAM. Solid lines correspond to the modulation depths (left axis) and dashed ones correspond to the reflection spectrum (right axis). The two channels of GAM are at frequencies of 2.88 THz (red solid and red dashed) and 4.06 THz (blue solid and blue dashed) respectively.
Despite the mismatch, we still demonstrate a four-channel modulation of GAM using the gate electric field tuning under a constant external magnetic field [Figs. 6(b) and 6(d)]. By tuning the EF at 28 meV, 48 meV, 58 meV and 78 meV, GAM is able to modulate the waves at frequency channels centered at 1.59 THz, 2.67 THz, 3.76 THz and 9.09 THz respectively. Due to the mismatch discussed above, the modulation depth of the four channels are ranged from 9.2 dB to 13.3 dB, which are much smaller than it in section 3.2, but still can be regarded as an effective modulation [Fig. 6(d)]. The uniformity of the FWHM is maintained and each channel has a FWHM of 0.21 THz. Moreover, it’s important to point out that, the required energy of tuning EF to switch GAM’s working channel reaches down to only 10 meV, which is extremely small compared with former studies using the electrostatic gate alone [12,33–35]. They typically require the EF to be tuned for more than 200 meV, which is more than one magnitude larger than our method. By assuming graphene to be pristine, we can calculate that only 0.3 V gate voltage is needed to be applied on GAM to make it works at the first channel. And by tuning the gate voltage from 0.3 V to 2.5 V, GAM can be controlled to switch its working channel among the four, which clearly indicates an ultra-high modulation efficiency.
In addition, since the modulation depth of this four-channel modulation isn’t fully satisfying, we re-select the parameters for GAM in order to obtain better modulation depth, at the cost of less modulation channels. This is reasonable because eliminating the mismatch between two channels might be easier than doing that among four. By increasing magnetic field B to 2.8 Tesla and altering the thickness of p-Si to 38.1 μm, we successfully realize a dual channel modulation in GAM with modulation depths both exceeding 40 dB [Fig. 6(c)]. The new frequency channels are centered at 2.88 THz and 4.06 THz, requiring EF to be tuned at 120 meV and 160 meV respectively. Although the number of working channels is cut from four to two, a much better modulation depth is achieved. In practice, such trade-off between the modulation depth and the amount of working channels should be carefully managed. Note that for a static magnetic field, the strength of transition dipole is proportional to EF, as the density of states in graphene is monotonic increasing with |EF| at transitions [11,26]. A larger EF would bring a larger conductivity at the transition, but not definitely a larger modulation depth. For our GAM only a moderate strength of transition dipole would result in a maximum modulation depth, as previously discussed.
In this section we have demonstrated multi-channel modulations of our GAM using the gate electric field tuning under a static magnetic field. Besides the excellent frequency tunability that has been confirmed previously, our GAM also exhibits great qualities such as extremely low gate voltage and ultra-high modulation efficiency.
4. Conclusion
In summary, we have studied a graphene based device (GAM) under the control of electric field and magnetic field simultaneously. Both fields can result in a significant modulation on the properties of graphene, and their combination, referred to as the dual field control, has been proposed as a novel method of tuning graphene based device’s performance. The crucial principle of this method is using the Fermi level to selectively allow different transitions in the LLs system formed through external magnetic field. The unprecedented characteristics of our prototype device, such as great frequency tunability, huge modulation depth and ultra-low required modulation energy, can be also realized in other realms like active optoelectronics, spin optics, all-optical systems and etc. A bright future of our proposed dual field control method is to expect.
Appendix I Magneto-optical conductivity of graphene
Graphene is described by its conductivity σg, which should be written in the form of a second rank tensor:
To determine their values at given parameters, the final computer-programmable equations can be written as [26]:
where Ω is the photon energy (also radio frequency f), EF is the Fermi level, B is the magnetic field, nF is referred to as the Fermi-Dirac distribution. Other variables are: n denotes the index of Landau levels, En denotes the relativistic energy of each Landau level. Other parameters are: e is the elementary charge, vF is the Fermi velocity whose typical value is 1.06 × 106 m/s, Γ is the Landau level scattering rate, which is set to be 1 meV here according to the experiment data [36]. Note that Γ is actually a phenomenological parameter, which is used to describe the spectral broadening effect. Γ is also assumed to be constant throughout all LLs, which is consistent with both experimental and theoretical results [26,37,38]. The summations from zero to infinity in Eqs. (4) and (5) include the contributions from all LLs, hence to numerically solve it, one must make a proper truncation at an enough high index of n, typically at the order of 6 to 10. A too low truncated index would lose the significant contributions from high LLs, and result in an inaccurate spectrum of conductivity. Note that the energy unit should be unified, e.g., all in Joule (J) or Kelvin (K).Acknowledgments
This work was supported by the National Basic Research Program of China (2012CB922000). Dr. Lu thanks the support from US Air Force Office of Scientific Research (AFOSR) and the support from DTRA (DTRAA 122221).
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