Pseudo-magnetic field-induced slow carrier dynamics in periodically strained graphene

The creation of pseudo-magnetic fields in strained graphene has emerged as a promising route to investigate intriguing physical phenomena that would be unattainable with laboratory superconducting magnets. The giant pseudo-magnetic fields observed in highly deformed graphene can substantially alter the optical properties of graphene beyond a level that can be feasible with an external magnetic field, but the experimental signatures of the influence of such pseudo-magnetic fields have yet to be unveiled. Here, using time-resolved infrared pump-probe spectroscopy, we provide unambiguous evidence for slow carrier dynamics enabled by the pseudo-magnetic fields in periodically strained graphene. Strong pseudo-magnetic fields of ~100 T created by non-uniform strain in graphene on nanopillars are found to significantly decelerate the relaxation processes of hot carriers by more than an order of magnitude. Our findings offer alternative opportunities to harness the properties of graphene enabled by pseudo-magnetic fields for optoelectronics and condensed matter physics.


Supplementary Note 1. Fabrication procedure
Fabrication of a nanostructured substrate: Supplementary Figure 1 illustrates the fabrication procedure for a nanostructured substrate. We first spin-coated a polymethyl methacrylate (PMMA) resist (950 PMMA A6, MICROCHEM) on a 300-nm-thick SiO2/Si substrate at 4500 rpm for 30 sec, followed by baking at 180 ºC for 2 min ( Supplementary   Figs. 1a-b). Then, an etching mask (an array with 1-µm diameter holes formed at 1.6-µm intervals) was patterned ( Supplementary Fig. 1c) using Raith e-line e-beam lithography system. The substrate was then immersed in buffered oxide etch (BOE) (12.5% HF, 87.5% NH4F) with 3 min 30 sec at room temperature ( Supplementary Fig. 1d). PMMA resist was removed by acetone (80 ºC, 10 min), isopropyl alcohol (IPA) (room temperature, 2 min), deionized (DI) water (room temperature, 2 min), and O2 plasma treatment (150 W, 3 min), The PMMA-covered graphene layer was floated by 5% diluted hydrofluoric acid (HF) solution by etching away the underlying SiO2 layer. After the PMMA/graphene layer was lifted off in the HF solution, it was transferred to DI water to remove HF residue. The nanostructured substrate was then used to fish the PMMA/graphene layer ( Supplementary   Fig. 2c). The sample was then dried at room temperature while standing at an angle to make a tight adhesion between graphene and the nanostructured substrate by capillary force 1 ( Supplementary Fig. 2d). Afterwards, the PMMA resist was removed by acetone, IPA, and DI water ( Supplementary Fig. 2e).

Supplementary Note 2. Raman analysis on unstrained and strained graphene
The Raman spectra of unstrained and strained graphene fabricated using the previously described procedure are both plotted in Supplementary Fig. 3. Supplementary Figure 3a compares the Raman spectra of the unstrained and strained graphene near the G mode peak, which arises from the doubly degenerate zone center phonons at Γ 2 . Similar to the 2D peak, tensile strain in the nanopillar causes the G peak to red-shift away from the unstrained graphene peak at 1581.9 cm -1 . Furthermore, strain breaks the rotation symmetry of the zone center phonons 3 , creating G-peak splitting with two peaks at 1544.87 cm -1 and 1574.7 cm -1 . The 2D peak shown in Supplementary Fig. 3b, arising from the second order scattering of the zone-boundary phonons, has a peak value of 2691.7 cm -1 in unstrained graphene. By applying tensile strain, the graphene C-C bonds are elongated and become weaker, causing the 2D peak to red-shift. The splitting of the two peaks into 2606.5 cm -1 and 2674.3 cm -1 shows that the C-C bonds are being stretched by varying degrees 4 .

Supplementary Note 3. Calculation of local strain distribution for strained graphene nanopillars
Structural distortion-induced strain in graphene nanopillars can be evaluated using the strain tensors, which are characterized as 5 : (1) where u(r) and h(r) represent the in-plane and out-of-plane deformation fields, respectively. We mainly focus on the out-of-plane deformation induced by nanopillars, since the resulting in-plane lattice distortion is much smaller than the vertical lattice distortion 6 . Consequently, from given atomic structural distortions of graphene, which can be determined by high-resolution atomic force microscopy (AFM), the strain components can be calculated as: where ℎ( ) is the AFM topographic data, and r denotes the position in the x-y plane. The resulting strain distributions are presented in Fig. 2c, which are also used to calculate the spatial distribution of pseudo-magnetic fields as explained in Supplementary Note 4.

Supplementary Note 4. Calculation of pseudo-magnetic fields for strained graphene nanopillars
The Hamiltonian modified by pseudo-gauge fields induced by strain in wave-vector space can be written as 7 : (5) where v 0 = 3a 0 t 0 2 is the Fermi velocity, and the typical value is about 10 6 ms -1 , a 0 = 0.14 nm is the length of the carbon-carbon bond, t 0 = 2.7 is the hopping amplitude in the tight-binding model. σ is the 2 × 2 Pauli matrix for the sublattice freedom. The pseudogauge field or vector potential A arising from the strain effect is expressed as 5,7 : Where β ≈ 3 is a constant. Since strain does not break the time reversal symmetry, the vector potential A will possess opposite signs for the two inequivalent K and K ' points, making the net magnetic field to be zero. The pseudo-magnetic field perpendicular to the surface is given by B = ∇ × A and can be written as a function of the strain tensors 7 : One can see that the pseudo-magnetic field, B S , vanishes for a uniform strain and that only non-uniform strain contributes to the finite value of the induced field strength. When the pseudo-magnetic field is relatively uniform under a specific strain field (e.g., a welldesigned triaxial strain), the Dirac bands can be rearranged into non-equidistant pseudo-Landau levels having energies of E n , which can be approximated as: where n is the Landau level index and e 0 is the electron charge. One can insert the values of all constants, and obtain a simplified one:

Supplementary Note 5. Calculation of local density of states (LDOS) of strained graphene nanopillars
LDOS with different pseudo-magnetic fields can be calculated by the following expression: where γ is the broadening factor of Landau level. Supplementary Figure 4 shows the calculated LDOS for the strained graphene with different pseudo-magnetic fields (10, 40, and 80 T).

Classic light-matter interaction and pumping pulse
We consider a classic light-carrier coupling for pumping process. The excitation pulse can be written as 8 : where A env is the envelope function that has the following form: where e pf is the pump fluence, ω is the frequency of pump light, ϵ 0 is the dielectric constant, and σ FWHM is the full width at half maximum (FWHM) of the pump light.

Quantum mechanical description of many-particle system
The full many-particle Hamiltonian with electron, phonon and photon parts in our system can be expressed as: Each term has the following expressions: where a i † , b νq † , c µ † are the electron, phonon, and photon creation operators, ϵ i is the single-particle energy for electron, v kl ij is the Coulomb matrix element, M ij is the optical matrix element, A is the vector potential of the external pump light, Ω νq is the frequency of phonon, G ij νq (g ij µ ) is the electron-phonon (electron-photon) coupling matrix element, and ω µ is the frequency of photon 9 .

Optical Bloch equation
We study the carrier dynamics using the optical Bloch equations (OBE). With the Heisenberg equation of motion, the time evolution of the electron population can be described by 8 : proportional to the measured reflection change (ΔR/R) presented in our study 10 . The electron-electron Coulomb scattering is the main factor that dominates the non-equilibrium carrier dynamics and is considered for the many-particle interactions in the OBE. The electron-electron Coulomb interaction is written as 8 : The many-particle scattering rate by the electron-electron interactions can be written by: where V bc ia is the Coulomb matrix element. It can be written as: where F F b i (q) is the combined form factor, which considers the shape of different Landau levels: is the general form factor 11 , and the coefficient α n b n c n i n a is written as: α n b n c n i n a = >√2? δ n i ,0 +δ n a ,0 +δ n b ,0 +δ n c ,0 , (24) and c bc ia is a constant. Lorentzian L γ >ΔE bc ia ? for the conservation of energy is defined as: Due to the presence of many electrons and the surrounding material, the Coulomb potential will be screened and renormalized as 8 : The dielectric function ϵ r (q,ω) can be calculated as: where ϵ b is the background dielectric constant, Π(q,ω) is the polarizability calculated in the random phase approximation (RPA) according to Goerbig et al. 12 Thus, the Coulomb interaction-modified OBEs now read as 8 : We point out that the main many-body interactions considered in the rise process are electron-electron scattering, in which the photoexcited carrier distribution rapidly broadens as pairs of electrons scatter to lower and higher energy states. Since the pump energy is larger than the probe energy, we assume that the dynamical picture of the rise process is that the excited carriers are first pumped to the initial Landau level, H , and then inject to a lower energy level that is the final Landau level, I , through electron-electron scattering channels. The experimentally measured probe signals indicate the time of carrier filling in the target I level. In Fig. 4, the calculated probe rise times for the target I level show a clear slow-down of carrier relaxation when the pseudo-magnetic field intensity is increased from 20 to 80 T (4 cases: 20 T, 40 T, 60 T, 80 T). Based on the pump (1030 nm) and probe energies (1450 nm), the target levels are chosen differently for different pseudo-magnetic fields. In principle, the dominated electron-electron scattering requires the energy and momentum conservation, but the non-equidistant energetic separation feature of the pseudo-Landau levels in strained graphene can effectively suppress the carrier scattering because energy conservation is not fulfilled 13 . This is in fact in contrast to the data for unstrained graphene having a short rise time of a few tens of femtoseconds owing to the existence of ample electron-electron scattering channels within the continuous Dirac (28) bands. A larger pseudo-magnetic field and a larger separation of pseudo-Landau levels lead to a longer time for carriers to relax from the pump energy level to the probe energy level, which is highly consistent with our experimental observation (Fig. 3).

Supplementary Note 7. Theoretical modelling for the carrier dynamics under pseudomagnetic fields with carrier-carrier and carrier-optical phonon scatterings
The main carrier dynamics mechanism we present in this study can be described by the Boltzmann-like scattering equation 14 : where the scattering rate Γ i in/out = Γ i cc-in/out + Γ i ph-in/out dominates the whole thermalization process, which contains two parts: pure carrier-carrier scattering Γ i cc-in/out and carrieroptical phonon scattering Γ i ph-in/out . The in-scattering rate Γ i in and out-scattering rate Γ i out contain all the contributions from and into other allowed Landau levels, respectively. The explicit forms of Coulomb interaction-induced scattering rates can be written as 8,14 : where V bc ia is the Coulomb matrix element calculated by the graphene Landau level basis, In pristine graphene, it is possible to achieve thermalized carrier distribution within 200 fs by efficient Auger processes (i.e., Auger recombination and impact ionization) enabled by carrier-carrier scattering 14 . However, in Landau quantized graphene system with pseudomagnetic fields, the Auger process will not be sufficient for the whole thermalization process 13 . Particularly for the higher Landau levels, the energy conservation for carrier scattering will not always be matched, leading to a very long thermalization time when only carrier-carrier interaction is considered. We therefore emphasize that optical phononassisted carrier relaxation is also critical to establish full thermalization. (30) We consider all the optical phonon modes that can influence phonon-assisted carrier depopulation, including ΓTO, ΓLO, KTO, and KLO, which have the energies of 192 meV, 198 meV, 162 meV, and 151 meV, respectively 8 . We also considered the out-of-plane optical phonon ZO mode that is relevant to the thermalization process in the higher Landau levels with small energy spacing 15 . In the calculations, the electron-optical phonon coupling matrix element for the above-mentioned phonon modes can be expressed as 8 : where V p o (p) is the electron-optical phonon coupling potential with only off-diagonal elements, and ψ f,m f ξ f is the wave function of the Landau-quantized state. Using coupling matrix elements, we can calculate the hot carrier thermalization assisted by optical phonon modes. The optical phonon-induced scattering rates can be written as 8 : where ρ j is all the possible allowed states for the carrier-optical phonon scattering process, n pµ is the population of the µ-mode phonon with wave vector p, which is determined by Bose-Einstein distribution, and L γ >ΔE ijµ em ? is the Lorentzian for the energy conservation.
In the carrier-optical phonon scattering process, electrons can raise their energy levels by absorbing phonons (dominated by the term ρ j >n pµ + 1?L γ >ΔE ijµ em ?), or lower their energy levels by emitting phonons (dominated by the term ρ j n pµ L γ >ΔE ijµ ab ? ) Considering the ambient temperatures of our study (4 K and 300 K), the population n pµ of the optical phonon is very small, which makes the phonon emission as the main process. layer. The PMMA/graphene layer was then transferred to DI water. The nanostructured substrate was used to fish the PMMA/graphene layer. d, The sample was then dried with standing at an angle to make a tight adhesion between graphene and the substrate by capillary force. e, PMMA was removed using acetone, IPA and DI water.

Supplementary Figure 3 | Raman analysis of unstrained and strained graphene a,
Raman spectra of G mode for unstrained (black) and strained (red) graphene. b, Raman spectra of 2D mode for unstrained (black) and strained (red) graphene. By applying high strain on graphene using nanopillars, 2D and G peaks split into 2D + / 2Dand G + / G -, respectively. Corresponding 2D and G peaks have been fitted with two Lorentzian peaks in each spectrum (dashed line).