Photon-Induced Suppression of Interlayer Tunneling in Van Der Waals Heterostructures
Woo-Ram Lee Wang-Kong Tse







Department of Physics and Astronomy, The University of Alabama, Alabama 35487, USA

Center for Materials for Information Technology, The University of Alabama, Alabama 35401, USA




We develop a theory for interlayer tunneling in van der Waals heterostructures driven under a strong electromagnetic field, using graphene/h-BN/graphene as a paradigmatic example. Our theory predicts that strong anti-resonances appear at bias voltage values equal to an integer multiple of the light frequency. These features are found to originate from photon-assisted resonant tunneling transitions between Floquet sidebands of different graphene layers, and are unique to two-band systems due to the interplay of both  intraband and interband tunneling transitions. Our results point to the possibility of tunneling localization in van der Waals heterostructures using strong electromagnetic fields. 








Photon-Induced Suppression of Interlayer Tunneling in Van Der Waals Heterostructures
    Wang-Kong Tse
    December 30, 2023
====================================================================================









When light is incident on a tunneling junction, inelastic tunneling transitions can occur via the exchange of photons between electrons and the electromagnetic field. This photon-assisted tunneling (PAT) phenomenon was first predicted in the classic work by Tien and Gordon  <cit.> in superconductor-insulator-superconductor tunneling junctions <cit.>. PAT has since been studied and observed in many systems <cit.>, including semiconductor quantum dots <cit.>, double quantum wells <cit.> and superlattices <cit.>, and optical lattices <cit.>. 
In particular, tunneling dynamics can be suppressed <cit.> when the light coupling parameter, given by the ratio of the driving field amplitude and frequency, matches a zero of the Bessel function, a celebrated phenomenon called dynamic localization <cit.>. 




Van der Waals heterostructures (vdWHs) <cit.> are an emerging class of nanoscale materials 


that hold great promise as a platform for realizing unconventional electronic properties and desirable functionalities. 


Vertically stacked vdWHs exhibit many distinctive properties not available in conventional semiconductor quantum well systems, including enhanced longitudinal and Hall Coulomb drag <cit.>, tunable metal-insulator transition <cit.> and extraordinary photovoltaic response <cit.>. In addition to in-plane transport, vertical tunneling transport in a field-effect tunneling junction  geometry exhibits superior current-voltage characteristics <cit.>.







































A strong electromagnetic field can provide a heretofore unexplored degree of freedom for tuning the tunneling dynamics in vdWHs. 




In this Letter, we theoretically investigate interlayer tunneling in optically driven vertical vdWHs, using graphene/h-BN/graphene as an archetypical system. Using the Keldysh-Floquet Green's function formalism, we formulate a theory for the non-equilibrium PAT current and elucidate the non-perturbative effects of the driving field on the intraband and interband tunneling transitions. 

Our theory predicts a new type of tunneling localization effect where photon-enabled resonant tunneling processes induce a dramatic suppression of the interlayer tunneling current as a function of the bias voltage, precisely at integer multiples of the photon energy. 


















Model — Our tunneling structure consists of two parallel graphene layers, labeled as top (T) and bottom (B), that are separated by a middle insulating monolayer of hexagonal boron nitride (h-BN). The layers are perfectly aligned and stacked in the ABA (Bernal) configuration, and a bias voltage V is applied across the top and bottom layers. The low-energy excitations around the K and K' points (labeled by ξ = ± 1, respectively) in the Brillouin zone of each graphene layer is governed by the 2D massive Dirac model up to an energy cutoff ℰ_c ≈ 3 eV:


    h_ kξ T,B
    = vħ (ξ k_x σ_x + k_y σ_y) + Δσ_z/2 ±e V𝕀_2/2,

where v is the Dirac velocity, Δ is the band gap induced by the h-BN layer, -e is the electron charge, and {𝕀_2, σ_x, σ_y, σ_z} denote the identity and Pauli matrices in the sublattice-pseudospin (i.e., a and b sites) space. We construct the Hamiltonian of the trilayer system using a  nearest-neighbor hopping approximation, including the coupling between each graphene layer and the h-BN layer and ignoring the negligible direct hopping between the graphene sheets. Due to its large band gap Δ_ BN∼ 5 eV, we can trace out the h-BN layer and obtain an effective double-layer Hamiltonian <cit.> ℋ̃ = ∑_ k∑_ξ∈{+,-}ψ̃_ kξ^† (h̃_ kξ + 𝒲̃)ψ̃_ kξ, where ψ̃_ kξ^† = (ϕ_ kξ T^†, ϕ_ kξ B^†). Tilde symbolizes the layer-pseudospin (i.e., T and B) space. It is convenient to define 𝕀_± = (𝕀_2 ±σ_z)/2, 𝕀̃_± = (𝕀̃_2 ±σ̃_z)/2, σ̃_± = (σ̃_x± i σ̃_y)/2 and use the new basis {𝕀̃_+, 𝕀̃_-, σ̃_+, σ̃_-} for the layer pseudospins to write  the unperturbed Hamiltonian as h̃_ kξ = h_ kξ T⊗𝕀̃_+ + h_ kξ B⊗𝕀̃_- and  the interlayer tunneling Hamiltonian as 𝒲̃ = 𝒲_TB⊗σ̃_+ + 𝒲_BT⊗σ̃_-. The tunneling matrix elements are given by 𝒲_TB = 𝒲_BT^† = 𝒲_0 𝕀_- with 𝒲_0 = - 2Γ^2 Δ_ BN^-1, where Γ is the interlayer hopping energy from a site in graphene to b site in h-BN (or vice versa) and Δ_ BN is the h-BN band gap.




We consider the two graphene layers to be coupled, for simplicity, to the same optical field by imagining two independent but identical laser sources setup symmetrically on both sides of the vdWH, illuminating the two graphene layers at normal incidence. Thus, the surface electric fields on both layers will be in phase with the same amplitude and frequency. 



Choosing the propagation direction along the z axis, the incident light with electric field amplitude E, frequency Ω, and polarization ϑ is described by the vector potential 𝐀(t) = (cE/Ω) [sin(Ω t) x̂ + sin(Ω t + ϑ) ŷ]. 



The Peierls substitution ħ k→ħ k + e A(t)/c in the original Hamiltonian produces the time-dependent Hamiltonian ℋ̃_ kξ(t) = ℋ̃_ kξ + 𝒱̃_ξ(t), with the interaction 𝒱̃_ξ(t) = 𝒱_ξ(t) ⊗𝕀̃_2, where 𝒱_ξ(t) = 𝒱_0 [ξsin(Ω t) σ_x + sin(Ω t + ϑ) σ_y].







The ratio 𝒱_0/Δ = (E/E_0) / (ħΩ/Δ) with E_0 = Δ^2/(evħ) describes the amplitude of the interaction in dimensionless form.  


Theory — Our formalism is developed by treating the interlayer tunneling Hamiltonian as a perturbation while taking into account the optical field non-perturbatively. 

Noting that the total charge is conserved in the system, the interlayer electric current density takes the form J(t) = 2 S^-1⟨Î(t) ⟩, where 2 counts the spin degeneracy, S is the normalization area, ⟨⋯⟩ is the canonical ensemble average, and Î(t) = ∂_t Q̂_T(t) = - ∂_t Q̂_B(t) = (i/ħ) [ℋ̃(t), Q̂_T(t)] is the electric current with Q̂_T,B(t) being the electric charge operator on layer T and B. 

The time evolution of J(t) is determined from the equal-time lesser Green's function, G_ kξ^<(t,t). 

In a non-equilibrium steady state (NESS), the system respects time translational symmetry 
and is thus governed by the Floquet theorem <cit.>. By using the Floquet mode expansion of the Green's function <cit.>, 

we obtain the time-averaged interlayer electric current density J  = - 4 (e/ħ) (1/S) ∑_ k∑_ξ∈{+,-}∫_-ħΩ/2^ħΩ/2 (dħω/2π) Re Tr{ [𝒲_BT⊗𝕀̂_∞] [Ĝ_ kξ^<(ω)]_TB},
where an overhat refers to a quantity in the Floquet space, the subscript “TB" of the Green's function refers to its off-diagonal element in the layer subspace,  

“Tr" is the trace over sublattice pseudospins and Floquet modes, and 𝕀̂_n is the n × n identity matrix in the Floquet space.


The lesser Green's function Ĝ_ kξ^< can be calculated within the Keldysh-Floquet Green's function formalism <cit.>. 
The full, tunneling-coupled Green's function is uniquely determined by the uncoupled Green's function of each layer and the interlayer tunneling Hamiltonian 

via the Dyson equation in the Keldysh-Floquet space <cit.>. Expansion of Ĝ_ kξ^< up to first order in 𝒲_0 yields the following photon-assisted tunneling current formula: 

    J_ tun    = - 4 e/ħ1/S∑_ k∑_ξ∈{+,-}∫_-ħΩ/2^ħΩ/2dħω/2π
          × Re{ Tr[ 𝕎̂^†_ kξ TB(ω) 𝒢̂_ kξ T^R(ω) 𝕎̂_ kξ TB(ω) ĝ_ kξ B^<(ω) .
    
       .     + 𝕎̂^†_ kξ BT(ω) 𝒢̂_ kξ B^A(ω) 𝕎̂_ kξ BT(ω) ĝ_ kξ T^<(ω) ]},

where 𝒢̂_ kξ T, B^R(ω) is the retarded Floquet Green's function of the T and B layers, ĝ_ kξ T^<(ω) is the lesser Floquet Green's function in the absence of light. Coupling to the metallic leads provides the energy relaxation mechanism and is captured by a band broadening parameter Γ in the Green's functions <cit.>. We have defined an effective photon-dressed interlayer tunneling Hamiltonian

    𝕎̂_ kξαα̅(ω) = 𝒲_αα̅[ 𝕀_2 ⊗𝕀̂_∞ + 𝒢̂_ kξα̅^R(ω) 𝒱̂_ξ],


where α∈{T, B} and the overbar denotes a complement, e.g., T̅ = B. Eq. (<ref>) together with Eq. (<ref>) is a central result of this Letter. It shows that non-perturbative photon dressing effects are contained not only in the Floquet Green's function  𝒢̂^R, but also in the tunneling Hamiltonian 𝕎̂ through Eq. (<ref>), in which the second term within the brackets represents the modification of the tunneling amplitude by a strong optical field. 


To illustrate the physics contained in Eq. (<ref>) and also to connect with the more familiar case without light, 
let us first consider the weak coupling regime <cit.> defined as 𝒱_0 / ħΩ≪ 1. In this scenario the photon-induced correction ∼𝒢̂^R𝒱̂ to the tunneling Hamiltonian [Eq. (<ref>)] is negligible.   



For the ABA stacking configuration of our system, [𝒲_TB]_cc' = [𝒲_BT]^*_c'c = 𝒲_0 δ_cc'δ_cb where c,c' ∈{a,b} label the sublattice pseudospins, we can reduce Eq. (<ref>) to a compact form 

    J_ tun/J_0
    = (2π vħ)^2/Δ1/S∑_ k∑_ξ∈{+,-}∫_-∞^∞ dħω̃
         ×[ ρ_ kξ B b^( ph)(ω̃) ρ_ kT b^( eq)(ω̃) f_T(ω̃) 
    - ρ_ kξ T b^( ph)(ω̃) ρ_ kB b^( eq)(ω̃) f_B(ω̃) ],

where J_0 = π^-1 e v^-2ħ^-3Δ |𝒲_0|^2 and ω̃≡ω + nΩ is the frequency in the extended zone scheme <cit.>. The two terms of Eq. (<ref>) correspond to a forward and a backward tunneling channel, and 


each channel is modified by non-perturbative light coupling effects through the Floquet mode 

spectral function ρ_ kξα b^ (ph)(ω̃) = - (1/π) Im [𝒢̂^R_ kξα(ω)]_bb;nn. 







In the absence of optical fields, 
ρ_ kξα b^ (ph)(ω̃) 
recovers its equilibrium counterpart ρ_ kα b^ (eq)(ω̃)
= - π^-1 Im [ĝ^R_ kξα(ω)]_bb;nn with Eq. (<ref>) reducing to the well-known 2D-to-2D tunneling current formula <cit.>. 


Fig. <ref>(a) shows the Floquet mode  
spectral function ρ_ kξα b^ (ph)(ω̃) 


as a function of the equilibrium energy dispersion ℰ_ k for different values of the driving field E. At small values of E, the quasienergy dispersions of the Floquet sidebands (i.e., photon-dressed electronic bands)

approximately coincide with copies of the equilibrium bands shifted by integer multiples of ħΩ, and the reduced-zone Floquet quasienergy ℰ̅_ k≈ℰ_ k. The undressed conduction and valence bands each generate their own Floquet sidebands, which for convenience will be called Floquet conduction  and valence sidebands (FCSB and FVSB, respectively). 





As E increases, mixing between Floquet states becomes stronger and quasienergy band gaps appear prominently at the locations of anti-crossing, given by (ℰ_ k, ħω̃ - μ_α) = (m-n, m+n) ħΩ/2  with m,n∈ℤ. 








Single-photon excitations — Before carrying out fully numerical calculations of Eq. (<ref>), we first perform a second-order perturbative analysis in the driving field amplitude. In this work, we focus on low temperatures k_B T ≪Δ and evaluate the tunneling current, 

assuming each graphene layer is at half-filling so that the electrochemical potential under bias is μ_T = - μ_B = eV/2. Treating the light field 𝒱_ξ(t) as a perturbation and expanding the single-layer Green's function into 𝒢̂_ kξα(ω) = ∑_j=0,1,2 (𝒱_0)^j𝒢̂_ kξα^(j)(ω) + 𝒪(𝒱_0^3) <cit.>, Eq. (<ref>) can be written in the form  J_ tun  = ∑_j=0,1,2 (𝒱_0)^j J_ tun^(j) + 𝒪(𝒱_0^3), where the zeroth order term <cit.> yields the dark tunneling current <cit.>. 











The first-order contribution J_ tun^(1) = 0 because J_ tun^(1)(t) ∝sin(Ω t) vanishes under time-averaging. 
Fig. <ref>(a) shows, for different values of frequency, the tunneling current J_ tun  =  J_ tun^(0)+𝒱_0^2 J_ tun^(2) obtained within the present second-order perturbation theory. It is seen that the overall profile remains close to the dark tunneling current J_ tun^(0) (dashed line) except at eV = ħΩ. When the bias voltage is equal to the photon energy, tunneling transitions assisted by a single photon occur, manifesting as a single anti-resonance in the second-order contribution J_ tun^(2).



















































































Photon-assisted tunneling suppression — 
To account for strong field and multiphoton excitation effects,  we now evaluate the tunneling current from Eqs. (<ref>)-(<ref>) non-perturbatively. Fig. <ref>(b) shows our full numerical results 


for different field strengths E (upper set of solid lines) and for different polarization angles ϑ (lower set) with an optical frequency value ħΩ = 1.25Δ slightly larger than the band gap. 

In the absence of light, the dark tunneling current 
remains zero when the bias voltage is smaller than the band gap, as shown by the dashed lines in both Figs. <ref>(a)-(b). 
When light is turned on, we first notice that the PAT current becomes non-zero even when eV < Δ, a feature not captured by the second-order perturbation theory. 

For bias values greater than the band gap eV > Δ, our theory predicts periodic anti-resonance suppression of the tunneling current with a separation ħΩ along the bias voltage axis as seen in Fig. <ref>(b).







Remarkably, the anti-resonance positions are found to be independent of 
the field strength E and polarization angle ϑ, and are precisely given by integer multiples of ħΩ, i.e., eV = NħΩ with N ∈ℤ. 

Close to the band gap eV ≳Δ in particular, we find a dramatic suppression of the PAT current to almost zero from its dark value due to the first anti-resonance located at eV = ħΩ = 1.25Δ.  
Fig. <ref>(b) also shows that the strength of the anti-resonances increases with the field strength,  

resulting in a progressive suppression of the tunneling current and indicating a tendency towards complete localization at stronger fields.  






Note here that the nature of our predicted 
tunneling suppression under optical illumination 
is fundamentally different from dynamic localization <cit.>, which occurs only when the coupling parameter 𝒱_0/ħΩ is equal to a zero of the Bessel function. 







The periodic occurrence of the tunneling suppression uncovered by our calculations stems from resonant intraband tunneling assisted by multiphoton excitations.


Let us first consider 

the case without illumination. Due to conservation of in-plane momentum, conventional 2D-to-2D intraband tunneling in coupled semiconductor quantum wells occurs when the layers' Fermi levels are closely matched, and resonance tunneling happens at exact matching whereby J_tun∼τ becomes sharply peaked (here τ is the electron's lifetime)  <cit.>. 






Graphene allows for interband tunneling when the bias voltage exceeds the band gap. 

If illumination is absent in our system, only interband (but not intraband) tunneling can occur 

since the layers' Fermi levels are assumed to be inside the band gap. However, an optical driving field opens up many additional channels for intraband tunneling via photon-assisted transitions between Floquet sidebands (see Fig. <ref>). 

When one layer's undressed valence band (VB) edge is aligned with one of the many FVSB edges of the other layer, electron momentum and energy conservation are simultaneously satisfied and resonant tunneling can occur. This Floquet band edge alignment happens when the bias voltage is tuned to an integer multiple of the driving frequency. 

To understand why a suppression instead of an enhancement occurs, it is useful to refer to Eq. (<ref>) under a forward bias condition eV > 0. 

Physically, the forward tunneling contribution involves tunneling from the undressed VB in the top layer to the n > 0 FVSBs in the bottom layer (Fig. <ref>), while the backward tunneling contribution involves tunneling from the undressed VB in the bottom layer to the n < 0 FVSBs in the top layer. 










Because they are further removed from the n < 0 FCSBs, the n < 0 FVSBs carry a higher spectral weight than their n > 0 counterparts <cit.>, 
as confirmed by our numerical calculations of integrated spectral weights of the Floquet mode spectral function [Fig. <ref>(b)].  
















As a result, under the resonant condition eV = NħΩ, the backward tunneling contribution is dramatically enhanced causing a suppression of J_ tun. 
























































Proposed experimental setup — We close by commenting on the conditions for observability. The assumption that the electromagnetic field on the two graphene layers are the same is not a stringent requirement and using a single light source to illuminate the system does not preclude the resonant tunneling suppression from occurring. Taking account of light reflections, the graphene layer further removed from the light source will experience a driving field with a weaker amplitude but nonetheless the same frequency. The resonant tunneling suppression is therefore expected to occur at the same values of bias voltage eV = NħΩ albeit with less pronounced anti-resonances. To realize the condition with the same electric field ampitudes on both layers using a single laser, two alternative scenarios can be devised. First, this condition can be approximately achieved when the laser wavelength is long compared with the thickness of the vdWH.

Second, an optical cavity can be used (see Fig. <ref>). When the cavity mode of the standing wave is odd and the vdWH is placed at the cavity's center, both graphene layers will experience the same magnitude of electromagnetic field. 

The resonant intraband tunneling mechanism we discovered is a generalization of the usual dark resonant tunneling to the scenario with Floquet sidebands under strong optical illumination. 
This phenomenon should apply not only to graphene layers, but also to trilayer vdWHs with other 2D materials such as bilayer graphene and transition-metal dichalcogenides. The fact that the tunneling current can be turned on and off by illumination at frequency values equal to an integral fraction of the bias voltage suggests a time-dependent control scheme for switching applications, opening the door to dynamical tuning of tunneling dynamics using periodic drives.   










 

















We thank Patrick Kung, Takashi Oka, and Godfrey Gumbs for useful discussions. This work was supported by startup funds from the University of Alabama and the U.S. Department of Energy, Office of Science, Basic Energy Sciences under Early Career Award #DE-SC0019326.











99

Tien63 P. K. Tien and J. P. Gordon, Multiphoton Process Observed in the Interaction of Microwave Fields with the Tunneling between Superconductor Films, Phys. Rev. 129, 647 (1963).

SIS_exp A. H. Dayem and R. J. Martin, Quantum Interaction of Microwave Radiation with Tunneling Between Superconductors, Phys. Rev. Lett. 8, 246 (1962).

PAT_Rev1 G. Platero and R. Aguado, Photon-assisted transport in semiconductor nanostructures, Phys. Rep. 395, 1 (2004).



PAT_QD1 L. P. Kouwenhoven, S. Jauhar, K. McCormick, D. Dixon, P. L. McEuen, Yu. V. Nazarov, N. C. van der Vaart, and C. T. Foxon, Photon-assisted tunneling through a quantum dot, Phys. Rev. B 50, 2019(R) (1994).

PAT_QD2 L. P. Kouwenhoven, S. Jauhar, J. Orenstein, P. L. McEuen, Y. Nagamune, J. Motohisa, and H. Sakaki, Observation of Photon-Assisted Tunneling through a Quantum Dot, Phys. Rev. Lett. 73, 3443 (1994).

PAT_QD3 C. A. Stafford and N. S. Wingreen, Resonant Photon-Assisted Tunneling through a Double Quantum Dot: An Electron Pump from Spatial Rabi Oscillations, Phys. Rev. Lett. 76, 1916 (1996).



PAT_QW H. Drexler, J. S. Scott, S. J. Allen, K. L. Campman, and A. C. Gossard, Photon-assisted tunneling in a resonant tunneling diode: Stimulated emission and absorption in the THz range, Appl. Phys. Lett. 67, 2816 (1995).

PAT_SL B. J. Keay, S. J. Allen Jr., J. Galán, J. P. Kaminski, K. L. Campman, A. C. Gossard, U. Bhattacharya, and M. J. W. Rodwell, Photon-Assisted Electric Field Domains and Multiphoton-Assisted Tunneling in Semiconductor Superlattices, Phys. Rev. Lett. 75, 4098 (1995).

PAT_SL2 G. S. Vieira, S. J. Allen, P. S. S. Guimarāes, K. L. Campman, and A. C. Gossard, Resonantly enhanced photon-assisted tunneling in a multiple-quantum-well superlattice, Phys. Rev. B 58, 7136 (1998).

PAT_OL C. Sias, H. Lignier, Y. P. Singh, A. Zenesini, D. Ciampini, O. Morsch, and E. Arimondo, Observation of Photon-Assisted Tunneling in Optical Lattices, Phys. Rev. Lett. 100, 040404 (2008).

PAT_OL2 R. Ma, M. E. Tai, P. M. Preiss, W. S. Bakr, J. Simon, and M. Greiner, Photon-Assisted Tunneling in a Biased Strongly Correlated Bose Gas, Phys. Rev. Lett. 107, 095301 (2011).

DL_2 B. J. Keay, S. Zeuner, S. J. Allen Jr, K. D. Maranowski, A. C. Gossard, U. Bhattacharya, and M. J. W. Rodwell, Dynamic localization, absolute negative conductance, and stimulated, multiphoton emission in sequential resonant tunneling semiconductor superlattices, Phys. Rev. Lett. 75, 4102 (1995).

DL_3 K. W. Madison, M. C. Fischer, R. B. Diener, Qian Niu, and Mark G. Raizen, Dynamical Bloch band suppression in an optical lattice, Phys. Rev. Lett. 81, 5093 (1998).

DL_4 S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, Observation of dynamic localization in periodically curved waveguide arrays, Phys. Rev. Lett. 96, 243901 (2006).

DL D. H. Dunlap, and V. M. Kenkre, Dynamic localization of a charged particle moving under the influence of an electric field, Phys. Rev. B 34 6, 3625 (1986).

Antonova16 I. V. Antonova, Vertical Heterostructures Based on Graphene and Other 2D Materials, Semiconductors 50, 66 (2016).



Drag R. V. Gorbachev, A. K. Geim, M. I. Katsnelson, K. S. Novoselov, T. Tudorovskiy, I. V. Grigorieva, A. H. MacDonald et al., Strong Coulomb drag and broken symmetry in double-layer graphene, Nature Phys. 8, 896 (2012).

MIT L. A. Ponomarenko, A. K. Geim, A. A. Zhukov, R. Jalil, S. V. Morozov, K. S. Novoselov, I. V. Grigorieva et al., Tunable metal-–insulator transition in double-layer graphene heterostructures, Nature Phys. 7, 958 (2011).

Geim L. Britnell, R. M. Ribeiro, A. Eckmann, R. Jalil, B. D. Belle, A. Mishchenko, Y.-J. Kim, R. V. Gorbachev, T. Georgiou, S. V. Morozov, A. N. Grigorenko, A. K. Geim, C. Casiraghi, A. H. Castro Neto, K. S. Novoselov, Strong Light-Matter Interactions in Heterostructures of Atomically Thin Films, Science 340, 1311 (2013).

TunExp1 L. Britnell, R. V. Gorbachev, R. Jalil, B. D. Belle, F. Schedin, M. I. Katsnelson, L. Eaves, S. V. Morozov, A. S. Mayorov, N. M. R. Peres, A. H. Castro Neto, J. Leist, A. K. Geim, L. A. Ponomarenko, and K. S. Novoselov, Electron Tunneling through Ultrathin Boron Nitride Crystalline Barriers, Nano Lett. 12, 1707 (2012).

TunExp2 L. Britnell, R. V. Gorbachev, R. Jalil, B. D. Belle, F. Schedin, A. Mishchenko, T. Georgiou, M. I. Katsnelson, L. Eaves, S. V. Morozov, N. M. R. Peres, J. Leist, A. K. Geim, K. S. Novoselov, and L. A. Ponomarenko, Field-effect tunneling transistor based on vertical graphene heterostructures, Science 335, 947 (2012).

TunExp3 L. Britnell, R. V. Gorbachev, A. K. Geim, L. A. Ponomarenko, A. Mishchenko, M. T. Greenaway, T. M. Fromhold, K. S. Novoselov, and L. Eaves, Resonant tunnelling and negative differential conductance in graphene transistors, Nat. Commun. 4, 1794 (2013).

TunExp4 A. Mishchenko, J. S. Tu, Y. Cao, R. V. Gorbachev, J. R. Wallbank, M. T. Greenaway, V. E. Morozov, S. V. Morozov, M. J. Zhu, S. L. Wong, F. Withers, C. R. Woods, Y.-J. Kim, K. Watanabe, T. Taniguchi, E. E. Vdovin, O. Makarovsky, T. M. Fromhold, V. I. Fal'ko, A. K. Geim, L. Eaves, and K. S.
Novoselov, Twist-controlled resonant tunnelling in graphene/boron nitride/graphene heterostructures, Nat. Nanotechnol. 9, 808 (2014).

TunExp5  T. Georgiou, R. Jalil, B. D. Belle, L. Britnell, R. V. Gorbachev, S. V. Morozov, Y.-J. Kim, A. Gholinia, S. J. Haigh, O. Makarovsky, L. Eaves, L. A. Ponomarenko, A. K. Geim, K. S. Novoselov, and A. Mishchenko, Vertical field-effect transistor based on graphene-WS_2 heterostructures for flexible and transparent electronics, Nat. Nanotechnol. 8, 100 (2013).













Comment1 See Supplementary Material for detailed derivation.

Grifoni98 M. Grifoni and P. Hänggi, Driven quantum tunneling, Phys. Rep. 304, 229 (1998).

Oka_Review H. Aoki, N. Tsuji, M. Eckstein, M. Kollar, T. Oka, and P. Werner, Nonequilibrium dynamical mean-field theory and its applications, Rev. Mod. Phys. 86, 779 (2014).

Lee17 W.-R. Lee and W.-K. Tse, Dynamical quantum anomalous Hall effect in strong optical fields, Phys. Rev. B 95, 201411(R) (2017).

Lee14 W.-R. Lee and K. Park, Dielectric breakdown via emergent nonequilibrium steady states of the electric-field-driven Mott insulator, Phys. Rev. B 89, 205126 (2014).

Tsuji09 N. Tsuji, T. Oka, and H. Aoki, Nonequilibrium Steady State of Photoexcited Correlated Electrons in the Presence of Dissipation, Phys. Rev. Lett. 103, 047403 (2009).

InCond In our approach, electron thermalization is achieved in two steps: (i) Before switching on the optical field, each graphene layer is tunnel-coupled to a metallic lead on each side to achieve thermal equilibrium resulting in a finite damping rate Γ; (ii) The optical field is adiabatically switched on with a switch-on time that is long compared with other time scales in the system <cit.>. In this condition, the initial NESS can be adiabatically connected to the thermal equilibrium state <cit.>. Readers are referred to the Supplementary Material for a more detailed discussion. 

Adia V. M. Galitsky, S. P. Goreslavsky, and V. F. Elesin, Electric and magnetic properties of a semiconductor in the field of a strong electromagnetic wave, Sov. Phys. JETP 30, 117 (1970).

Adia2  S. Schmitt-Rink, D. S. Chemla, and H. Haug, Nonequilibrium theory of the optical Stark effect and spectral hole burning in semiconductors, Phys. Rev. B 37, 941 (1988).











weak_coupling E. K. Irish, Generalized Rotating-Wave Approximation for Arbitrarily Large Coupling, Phys. Rev. Lett. 99, 173601 (2007).

2D_Tunneling1 L. Zheng and A. H. MacDonald, Tunneling conductance between parallel two-dimensional electron systems, Phys. Rev. B 47, 10619 (1993).

2D_Tunneling2 N. Turner, J. T. Nicholls, E. H. Linfield, K. M. Brown, G. A. C. Jones, and D. A. Ritchie, Tunneling between parallel two-dimensional electron gases,  Phys. Rev. B 54, 10614 (1996).

















J_0_expression A closed-form analytic expression can be obtained, see <cit.>. 

Vasko13 F. T. Vasko, Resonant and nondissipative tunneling in independently contacted graphene structures, Phys. Rev. B 87, 075424 (2013).

Brey L. Brey, Coherent Tunneling and Negative Differential Conductivity in a Graphene/h-BN/Graphene Heterostructure, Phys.
Rev. Appl. 2, 014003 (2014).

Polini K. A. Guerrero-Becerra, A. Tomadin, and M. Polini, Resonant tunneling and the quasiparticle lifetime in graphene/boron nitride/graphene heterostructures, Phys. Rev. B 93, 125417 (2016). 

Comment3 We have checked that ℰ_c / Δ = 4 is sufficient for the tunneling current to achieve convergence within the range of bias voltage eV/Δ≤ 4. Details are provided in Sec. S-3.4 of the Supplementary Material.  


Kohler S. Kohler, J. Lehmann, and P. Hänggi, Driven quantum transport on the nanoscale, Phys. Rep. 406, 379 (2005).

Eisenstein  J. P. Eisenstein, New transport phenomena in coupled quantum wells, Superlattices Microstruct. 12, 107 (1992).




































specfun 
If one imagines the formation of Floquet sidebands in two steps: (1) generation of “Floquet copies” via displacement of the  undressed conduction and valence bands each by nħΩ (where | n | = 0, 1, 2, …) and (2) band-mixing between the Floquet copies, then one can readily understand that the n < 0  FVSBs have a higher spectral weight  compared to the n > 0 FVSBs, because the n < 0 Floquet copies of the valence band and  n > 0 Floquet copies of the conduction band have less overlap  and hence less band-mixing that would result in spectral gaps in the quasienergy dispersions.    







