CHAPTER 3. INTRABAND EXCITATIONS IN MULTILAYER BLACK PHOSPHORUS
3.3 OPTICAL AND ELECTRICAL CHARACTERIZATION OF MULTILAYER BP FIELD
We employed Fourier-transform infrared micro-spectroscopy to measure reflection spectra of multilayer BP structures. A typical field effect heterostructure, schematically illustrated in side view in Fig. 3.1(B), constructed using van der Waals assembly technique is shown in Fig. 3.1(C), where the BP is 18.7 nm, and the top and bottom h-BN are 36.8 nm and 36.4 nm, respectively.
The carrier density in BP was tuned by applying a gate voltage across the bottom hBN and SiO2
(285nm). Such a geometry allowed independent electrical and optical characterization of the induced 2DEG in BP. Polarized Raman spectroscopy was used to identify the AC and ZZ axes of
BP as indicated in the optical image of the device. All optical and electrical measurements were performed in ambient at room temperature.
Figure 3.1. Device schematic and electro-optic characterization. (A) Anisotropic puckered crystal structure of BP (P atoms are in sp3 hybridization). (B) Device schematic and measurement scheme for hBN encapsulated BP devices. (C) Optical microscope image of the device discussed in the main text. (D) Normalized reflection spectrum from the BP device shown in (C). (E) Color-map of source-drain current variation as a function of both gate voltage and source-drain bias. (F) Gate voltage modulated source-drain current at one representative source-drain voltage (100mV). (G) Variation of source-drain current with source-drain voltage showing linear conduction with systematic increase as gate voltage increases on the positive side, the slight dip is due to the fact
that the MCP is not at 0V). (H), (I) Interband optical modulation along the AC and ZZ axis, respectively, showing the anisotropy in the electro-optic effects. (J) Schematic of changes in the AC axis optical conductivity (real part) upon doping.
The reflection spectrum, shown in Fig. 3.1(D), for the same device was measured at the minimal conductance point (MCP), confirmed from two-terminal electrical measurements as shown in Fig.
3.1(E)-(G), with light polarized along the AC direction. This spectrum is normalized to that of optically thick Au (approx. 500 nm) evaporated on the same sample as a reference surface. Three prominent features dominate the spectrum–a narrow hBN phonon around 1370 cm-1, a broad dominant SiO2 phonon around 1100 cm-1 (recent studies42 showmultiple phonon contributions in SiO2) and the beginning of band edge absorption around 3000 cm-1 convoluted with an interference dip coming from the entire stack. Additionally, from our transport measurements, a hole mobility of 1107 cm2/V.s and an electron mobility of 412 cm2/V.s were obtained at low doping levels, corresponding to scattering rates on the order of approximately 5-10 meV. As shown in Figs.
3.1(H) and 3.1(I), these reflection spectra can be heavily modified under positive or negative gate voltages.
Modelling the optical conductivity of the BP electron/hole gas allows us to gain an understanding of the quasiparticle dynamics under applied voltage. Our BP flakes are between 10-20 nm thick and described by a sheet conductivity 𝜎 since the effective modulation is confined to only 2-3nm from the interface of BP/b-hBN. The thickness of this modulated region was estimated from the results of band bending calculations, using a Thomas-Fermi model143. This sheet conductivity has contributions from both interband and intraband processes, given as 𝜎 = 𝜎𝑖𝑛𝑡𝑒𝑟𝑏𝑎𝑛𝑑 + 𝜎𝑖𝑛𝑡𝑟𝑎𝑏𝑎𝑛𝑑 = 𝜎1(𝜔) + 𝑖𝜎2(𝜔). The interband contribution accounts for absorption above the band-edge, including all subbands, while the intraband part accounts for free carrier response. One can explicitly calculate for optical conductivity using the Kubo formalism as follows50:
𝜎𝑖𝑛𝑡𝑒𝑟𝑏𝑎𝑛𝑑 = −𝑖𝑔𝑠ℏ𝑒2
(2𝜋)2∑ ∫ 𝑑𝑘 𝑓(𝐸𝑠𝑗𝑘)−𝑓(𝐸𝑠′𝑗′𝑘′)
𝐸𝑠𝑗𝑘−𝐸𝑠′𝑗′𝑘′ ∗⟨𝜙𝑠𝑗𝑘|𝜈̂𝛼|𝜙𝑠′𝑗′𝑘′⟩⟨𝜙𝑠′𝑗′𝑘′|𝜈̂𝛽|𝜙𝑠𝑗𝑘⟩
𝐸𝑠𝑗𝑘−𝐸𝑠′𝑗′𝑘′+ℏ𝜔+𝑖𝜂
𝑠𝑠′𝑗𝑗′
(3.E1)
𝜎𝑖𝑛𝑡𝑟𝑎𝑏𝑎𝑛𝑑,𝑗 = 𝑖𝐷𝑗
𝜋(𝜔+𝑖𝜂ℏ) , 𝐷𝑗 = 𝜋𝑒2∑ 𝑛𝑖
𝑚𝑖,𝑗∗
𝑁𝑖=1 (3.E2)
Here, 𝜈 ̂𝛼,𝛽 is the velocity operator defined as ℏ−1𝜕𝑘𝛼,𝛽𝐻, 𝑔𝑠 = 2 is used to denote the spin degeneracy, f(E) is the Fermi-Dirac distribution function; the indices s(s′) refer to conduction (valence) bands and the indices j(j′) refer to the subbands. H is the low-energy in-plane Hamiltonian around the Γ point, 𝐸𝑠𝑗𝑘, 𝜙𝑠𝑗𝑘 are the eigen-energies and eigenfunctions of H; 𝑚𝑖,𝑗∗
represents the effective mass of carriers in each subband (i) along a specific crystal orientation j, 𝑛𝑖 represents the charge density in each subband and 𝜂 is a phenomenological damping term.
Electrostatic doping of BP primarily brings about two fundamental changes in the optical response:
the emergence of a strong intraband component in the mid to far infrared and a shift of the optical gap (interband transitions), shown schematically in Fig. 3.1(J). A combination of multiple electro- optical effects at the band-edge has been shown to explain the observed modulation, discussed next, also summarized in Fig. 3.2.
1. Pauli blocking/Burstein Moss shift
The fermionic nature of electrons and holes in a semiconductor dictates that optical transitions between occupied states in the valence band and unoccupied states in the conduction band are blocked if the electron states at the same energy and momentum are already filled, leading to reduced absorption. This effect is known as Pauli blocking/Burstein Moss shift. Additionally, since BP has a quantum well electronic band structure, characteristic absorption dips are seen for different subbands as they are filled with increasing doping.
2. Quantum confined Stark effect
When a quantum well is subjected to an external electric field, the electron states shift to lower energies and the hole states to higher energies thereby reducing the effective optical bandgap.
Additionally, electrons and holes shift to the opposite sides of the well reducing the overlap integral which reduces the oscillator strength of each transition.
3. Forbidden transitions / ‘mixed’ transition
In an unperturbed quantum well system, certain transitions have allowed dipole transitions and optical matrix elements which do not vanish. From symmetry arguments those transitions happen to be between subbands of equal principal quantum number index (j=1 VB to j=1 CB, etc.).
However, upon the application of an external electric field, modification of the overlap integral between electrons and holes causes the previously vanishing optical transitions to be allowed and they appear as mixed transitions.
4. Band bending
In multilayer systems for a typical field effect heterostructure geometry, a degenerate charge gas is induced at the interface of the active material and the gate dielectric; in our case, the BP/b-hBN interface. However, the charge is not distributed equally in the out of plane direction because the first layer of charge screens the remaining charges. This gives rise to a thickness-dependent charge profile approximated by the Thomas-Fermi screening model. For BP it can be seen from calculations for a charge density of about 5-7x1012/cm2 the effective channel thickness is about 2.9 nm- meaning the induced electron/hole gas is two-dimensional in nature and not three- dimensional. In all our fitting routines, it is assumed this is the case, and a sheet conductivity for the 2DEG is used (with a static dielectric constant as a background for the whole BP).
To summarize, we explain our observation in Figs. 3.1(H) and (I) as follows. As we dope the system with electrons, we see a suppression of absorption along the AC direction (appearing as a dip) due to Pauli blocking which increases with applied voltage. Higher lying features such as subbands show very weak modulation. Additionally, a mild red shift of the band edge is seen at the highest positive voltages indicative of a Stark shift. However, on the hole side, not only do we see a strong dip at the onset of band-edge transition and at the subband energies we also see a stronger red-shift of the band gap due to a more dominant Stark shift. The band-edge shifts to about 2800 cm-1 which is ~30meV below the pristine gap on the hole side, and to about 2900 cm-
1 which is ~15meV below the optical gap on the electron side. This asymmetry in the Stark shift might be from impurities/residual doping in the system causing an additional field which cancels out in the electron doped case but adds up in the hole doped case. These impurities could also be
causing the reduction of prominent subband oscillations on the electron side. Further nanoscale studies would be needed to elucidate more about the underlying mechanism. The noisy weak modulation for ZZ around 3000 cm-1 arises from the fact that the interference dip of the entire stack in our device (which also happens serendipitously to be around the same energy as the band- edge) results in low signal.
Figure 3.2. Quantum Well electro-optic effects. Schematic of different electro-optic effects occurring at energies near and above the band-edge of a multilayer BP thin film.
All of the observed reflection modulation spectra exhibit strong anisotropy with respect to the BP crystal axes under AC and ZZ polarized illumination. This strong anisotropy is predicted by theory, and results from the puckered honeycomb lattice crystal structure of phosphorene150. Our results in Figs. 3.1(H) and 3.1(I) indicate significant optical modulation in the 2DEG and are in excellent agreement with results from previous studies23,141–143.