Abstract
Subjected to an adequately high magnetic field, Landau levels (LLs) form to alter the electronic transport behavior of a semiconductor. Especially in two-dimensional (2D) limit, quantum Hall effect sheds light on a variety of intrinsic properties of 2D electronic systems. With the raising quality of field effect transistors (FET) based on few-layer black phosphorus (BP), electronic transport in quantum limit (quantum transport) has been extensively studied in literatures. This chapter investigates the electronic transport in few-layer BP, especially in quantum limit. At the beginning of this chapter, a brief introduction to the background of LL, edge state, and quantum Hall effect will be delivered. We then examine the fabrication of high-quality FET based on BP and their electronic performances followed by exploring the magnetoresistances of these high-quality devices which reveal Shubnikov-de Haas (SdH) oscillations and quantum Hall effect in BP. Intrinsic parameters like effective mass, Landé g-factor, and so on are discussed based on quantum transport.
Keywords
- black phosphorus
- field effect transistor
- mobility
- Landau level
- quantum hall effect
1. Introduction
Developing high-quality functional materials such as high-mobility semiconductors is an essential process to explore fundamental condensed matter research. The recent rediscovery of BP as a new member of 2D materials with theoretically predicted high mobility and widely tunable electronic bandgap makes it promising for various electronic devices and for probing interesting physical phenomena [1, 2, 3, 4]. However, there are various obstacles to overcome in order to achieve the theoretically high carrier mobility in few-layer BP. Those obstacles include degradation of flake quality in atmosphere environment, electronic scatterings (against charge impurities, crystal defeats, and so on), high contact resistance, and so on. In devices without high enough mobility, new physical phenomena can be smeared out due to a large charge carrier scattering rate.
Since Landau level (LL) is crucial to understanding the physical phenomena in quantum transport process, we would like to deliver a brief idea about it here. For clarity, we will neglect the spin degeneracy at first and will come to it when necessary. Consider an electronic system without electronic interactions in an applied magnetic field along
In Landau gauge (the wave function is gauge invariant),
where
where
where
Now we take the spin of electron into consideration. Besides the formation of LLs, the energy of electron is also split by Zeeman effect
We conclude now that under applied magnetic field, electrons cycle in quantized traces with quantized energy in one direction. However, when these traces intersect the edge of the electron system, these electrons cannot complete the full cyclotron. Instead, they bounce back and enter another adjacent traces because the cycling direction is fixed. As a result, electrons near the edge are in a skipping motion along one-dimensional boundary moving in a single direction, which is referred to as chiral edge states. Due to the absence of scattering, the edge states carry electronic current with no voltage drop, i.e.,
When an in-plane electric field is applied, electrons are accelerated along one direction. As a result, the balance between the occupation of edge states in two opposite sides is broken, i.e., more electrons are filled in the edge states in one boundary. As we have discussed before, the energy of electrons is quantized. Consequently, the Hall conductance
Alternatively, quantum Hall effect can be understood based on Berry phase [6]. Consider an adiabatic system in which the eigenstate evolves with external parameters slowly. When the external parameters make up a loop and come back to itself, the eigenstate should also come back to itself (let us rule out the possibility of degeneracy here) but with a different phase, because the eigen energy of eigenstates with different phases is invariant. This phase is the so-called Berry phase. This Berry phase manifests itself in an electronic system through weak localization effect, A-B effect, and so on. All these effects come from the dynamics of electrons. For Bloch electron under the perturbation of a weak electric field
where
where the integration is over the entire Brillouin zone and the result of this integration is the Chern number
This reaches the same conclusion with previous analysis, i.e., the Hall conductivity is quantized.
In this chapter, we would introduce the detailed work on BP from the characterization of the electronic transport process and figure out the main limitation to high-quality devices, followed by efforts to conquer the impediment. Once devices with high enough quality can be fabricated, we will go to the electronic transport in quantum limit (cryogenic temperature and high magnetic field). Realization of quantum transport process in BP allows people to determine some critical parameters (effective mass, Landé g-factor, and so on) which are crucial for seeking engineering applications of this material.
2. High-quality FET based on BP
In 2014, several groups first demonstrated the FET based on few-layer BP (Figure 1a) [1, 3, 7, 8], giving rise to the renaissance of BP. As a semiconducting layered material, BP has a high carrier mobility at room temperature, with the value up to 1000 cm
As the most famous elemental layered material beyond graphene, BP provides the needed bandgap for FET applications, especially in infrared region. Unlike semiconducting transition metal dichalcogenides (TMDCs), the bandgap in BP is direct for all number of layers, which makes this material particularly promising for optoelectronic applications. Moreover, the bandgap of BP is widely tunable achieved by layer number, electric field, strain, and alloying. Combined with its strong anisotropic, BP may allow for the exploration of new exotic phenomena and multifunction devices.
2.1 Realization of air-stable devices based on BP
Although bulk BP is considered as the most stable allotrope of phosphorus, it is reported that few-layer BP is unstable in ambient conditions. Significant surface roughening over the time after exfoliation can be observed by AFM [8]. The investigation of water condensation on BP flakes reveals that BP is very hygroscopic and tends to uptake moisture from air [13]. Long-time exposure to air can even etch BP away. The in situ Raman and transmission electron spectroscopy studies of the degradation of BP show that the degradation in air mainly arises from the photo-assisted oxidation reaction with oxygen dissolved in the adsorbed water on its surface [14]. Theoretical work revealed that BP presents a strong dipolar moment out of plane which makes it very hydrophilic [15].
The environmental instability of BP flakes makes it a challenge to fabricate high performance BP-based devices, since most experiments reported on BP relied on mechanically exfoliated flakes. To prevent the degradation, several kinds of passivation were developed. Atomic layer-deposited
2.2 Ohmic contact and ambipolar transport in BP
As that in other semiconducting devices, the electrical performance of devices based on BP is significantly affected by the electrical contacts. Low contact resistance is critical to achieving high mobility, high on-off ratio, and large photo response in BP. The main issue in BP-based FET is the existence of large Schottky barrier in the contacts, which limits the current injection and its potential for applications. Several kinds of contact engineering have been identified toward high-quality electrical contacts. By choosing various metals with different work functions to match BP, different performance of BP transistors can be achieved [26, 27]. The aluminum-contacted BP displays ambipolar characteristic with nearly symmetric electron and hole mobility of 950 cm
Few-layer BP has a direct bandgap with the size around 0.3 eV. The small bandgap of few-layer BP facilitates the injections of both electron and hole, resulting in ambipolar characteristic being easily observed in BP-based transistors. The ambipolar transport in BP can exploit both
2.3 Strong transport anisotropy in BP
BP has a puckered honeycomb lattice, yielding strong in-plane anisotropy. The unusual anisotropic structure of BP results in its strong in-plane anisotropic electrical and optical properties. BP-based devices with electrode probes fabricated at various angles have been used to probe the electrical conductivity and carrier mobility along different direction (Figure 3) [1, 7]. Higher hole mobility and conductivity were found along light effective mass directions (
2.4 Widely tunable electronic bandgap
Due to the strong layer interaction and quantum confinement of the charge carriers in the out-of-plane direction, BP shows stronger thickness-dependent bandgap compared with other 2D semiconductors, such as TMDCs. When the thickness of BP decreases from bulk to monolayer, its direct bandgap was predicted to increase roughly from 0.3 to 2.0 eV [34]. This prediction has been experimentally confirmed by the measurements of optical absorption [35]. The optical bandgaps of monolayer, bilayer, trilayer, and bulk BPs were determined to be 1.73, 1.15, 0.83, and 0.35 eV, respectively. The bandgap of few-layer BP is also widely tunable by electric fields. By using an in situ potassium doping technique, a vertical electric field from dopants modulates the bandgap of few-layer BP and even tunes the system to an anisotropic Dirac semimetal state, owing to the giant Stark effect [36]. For practical applications, the dynamic tuning of bandgap can be realized by the dual-gate FET configuration in BP-based devices [37]. For example, the transport measurement in a 10-nm-thick BP flake shows that its bandgap can be continuously tuned from 0.3 eV to below 0.05 eV, using a moderate displacement field of 1.1 Vnm
3. SdH oscillation and quantum hall effect in BP
Upon subjection to a magnetic field, LLs form in an electronic system. However, high scattering rate of charge carriers leads to a broadening of LLs. The LLs manifest itself only when this broadening is lower than the energy gap between two adjacent LLs. The achievement of high-quality devices (high carrier mobility and low scattering rate) suppresses this broadening and finally allows the observation of LL with laboratory reachable magnetic fields. By the end of 2014, Li et al. and Nathaniel et al. reported the first observation of Shubnikov-de Haas (SdH) oscillations in BP hole system [3, 24] (Figure 4a). Later on Li et al. observed the quantum Hall effect in the same system (Figure 4b) [43]. Figure 4a shows
More quantitatively, according to the Lifshitz-Kosevich model, the temperature dependence of oscillation amplitude follows [45].
where
As device quality further improved, Zeeman splitting can be resolved in laboratory-accessible magnetic field [10], and alternative SdH oscillation amplitudes were observed. By reproducing the oscillation component with high-order spin-resolved LK formula
where
The resolvability of Zeeman splitting makes it possible to determine the Landé g-factor of BP electronic systems via the coincidence method taking advantage that the Zeeman energy
Making use of a magnetic field up to 45 T, fractional quantum state at filling factor
When the electronic band of a semiconductor is occupied by induced electrons (e.g., electrons induced by gate voltage), it will screen the electronic field hence prohibiting the wave function of electrons from penetrating the topmost few layers [52, 53, 54]. While when the band is empty, it will not screen the electronic field, and the wave function of electrons penetrates all the way to the opposite surface. Based on this screening effect, wide quantum wells have been formed in few-layer BP which can be switched between double layers of charge carriers and single layer of charge carriers [55]. The switch in one single device from double quantum wells and single quantum well has never been realized in other systems. Besides, one can tune the distance between the double layers of charge carriers by selecting
4. Conclusions
In this chapter, we mainly focus on the development of high-quality FET and the electronic transport in quantum limitations based on BP. Although few-layer BP degrades in the atmosphere, encapsulation with BN flakes was proven to be a reliable strategy to realize air-stable devices based on BP. The realization of Ohmic contact through selective etching technique lays the foundation for high-quality devices. Thanks to the rather small electronic bandgaps for few-layer BP flakes, by aligning the work function of contact metals with the band edge of BP, ambipolar-conducting channels were realized in single field effect devices. In devices with high carrier mobility, the scattering rate of charge carriers is low leading to limited broadening of LLs. As a consequence, the LLs become resolvable in laboratory reachable magnetic field at cryogenic temperature. Based on the SdH oscillations and quantum Hall effect, a spin-selective quantum scattering process was established in BP 2DHG. Intrinsic parameters such as effective mass and Landé g-factor were measured. The realization of high-quality air-stable BP devices promises its potential in next-generation electronic applications.
Acknowledgments
The authors acknowledge the financial support from the Research Grants Council of Hong Kong (Project No. FB417- UoM-HKUST and C7036-17 W).
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