Research Article 
Corresponding author: Alexander K. Fedotov ( fedotov@bsu.by ) © 2021 Andrei A. Kharchenko, Julia A. Fedotova, Valeryia Yu. Slabukho, Alexander K. Fedotov, Alexey V. Pashkevich, Ivan A Svito, Maxim V. Bushinsky.
This is an open access article distributed under the terms of the Creative Commons Attribution License (CC BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Citation:
Kharchenko AA, Fedotova JuA, Slabukho VYu, Fedotov AK, Pashkevich AV, Svito IA, Bushinsky MV (2021) Electrical and galvanomagnetic properties of black phosphorus single crystals. Modern Electronic Materials 7(4): 127139. https://doi.org/10.3897/j.moem.7.4.78587

Black phosphorus (bP) single crystals having the ntype electrical conductivity produced in a high pressure setup (~1 GPa) with six diamond anvils at 800 °C for 12 h have been studied. The electrical conductivity σ(Т,В) and the Hall constant R_{h}(Т,В) have been analyzed within oneband and twoband models as functions of temperature in the 2 < Т < 300 K range and magnetic field in the 0 < В < 8 T range. Fitting of the experimental σ(Т,В) and R_{h}(Т,В) curves suggests the following key properties of the crystals: (1) intrinsic conductivity type, (2) approximately equal electron and hole concentrations and mobilities, (3) anisotropic behavior of electron and hole conductivities, concentrations and mobilities and (4) combination of negative and positive contributions to magnetoresistance (magnetoresistive effect, MR). In a zero magnetic field the anisotropy coefficient α = [σ_{а} (Т) – σ_{с} (Т)]/σ_{с} (Т) below 50–70 K is positive whereas above 220 K its sign changes to negative due to a specific combination of the temperature dependences of carrier concentration and mobility. It has been shown that the negative sign of relative MR (negative magnetoresistive effect) dominates at T < 25 K and B < 6 T and is presumably caused by the effects of strong localization resulting from structural disorder. The positive MR sign (positive magnetoresistive effect) is associated with the Lorentz mechanism of carrier movement and exhibits itself above 25 K in 6–8 T magnetic fields.
black phosphorus, conductivity anisotropy, magnetoresistance, carrier transport, twoband model.
Black phosphorus (bP) is one of the most atmosphericpressure stable allotropic phosphorus modifications pertaining to the series of materials having a newtype layered 2D structure [
As can be seen from Fig.
Studies of bulk black phosphorus started after P.U. Bridgman first synthesized it in 1914 at high pressures and temperatures [
Despite the obvious success in phosphorene research there are still a number of pending questions in the understanding and explanation of the properties (primarily, electric ones) of bulk bP crystals. Below is a brief overview of literary data on the electrical properties of bulk polycrystalline and single crystalline black phosphorus obtained using various methods.
Works dealing with the properties of black phosphorus can be arbitrarily divided in 2 groups: in the period from 1914 to approx. 1990 [2, 6, 20–24] and after 2014 [1, 3–5, 7–19, 25–33] when interest to bP arose again due to the production of bP single crystals having a higher structural perfection. There are multiple literary data on the electrical resistivity and Hall effect of bulk bP single crystals over a wide range of temperatures and magnetic fields [
As noted above, earlier data [20, 24, 25, 27, 29–33] suggest (see for example Fig.
As noted earlier [
The behavior of anisotropic semiconductors and semimetals (especially those having nontrivial carrier dispersion laws) in magnetic fields plays a key role in the understanding of their carrier transport mechanisms. The first data on the magnetoresistive effect in bP crystals were reported in the earliest works on the topic [
The PMR to NMR transition [
1. With an increase in the magnetic field the longitudinal contribution to the resistivity tensor ρ_{хх} (B) does not undergo saturation in the PMR region whereas in the NMR region its saturation occurs.
2. The Hall resistivity ρ_{хх} (B) grows linearly with an increase in the magnetic field at 3 and 300 K but exhibits a nonlinear behavior in the entire range of intermediate temperatures, i.e., 20–200 K.
The former specific feature of the ρ_{хх} (B) behavior under impact of magnetic field results in necessity to describe it by two models, i.e., the classical resistor array model (caused by disorder) and the magnetic polaron model (due to the high ohmicity of the crystal) [
The latter specific feature of the ρ_{хх} (B) behavior in a magnetic field is accounted for on the basis of twoband conductivity models involving two types of carriers.
Analysis of literary data on carrier transport in bulk black phosphorus crystals discovers a wide range of models based on which black phosphorus electrical properties are described. It should be noted that there are scarce data on the influence of bP crystal growth technology (primarily, synthesis process pressure, temperature duration and atmosphere), defect state (single crystal, polycrystal, type and quantity of defects, deformation etc.) and measurement mode (number of heating/cooling cycles, storage time and atmosphere) on the temperature dependences of electrical resistivity, magnetoresistive effect, Hall effect, Seebeck effect and other properties of the material. For this reason a number of questions as to the nature of the electrical properties formation in black phosphorus bulk crystals have not been clarified as yet.
Below we present data on the temperature dependences of the electrical conductivity, magnetoresistance and Hall effect for several bP bulk single crystals made by 2D Semiconductors, USA.
(a) Schematic of rifled layered structure, (b) layer top view and (c) layer fragment in bP lattice [4, 5].
The single crystals were synthesized in a high pressure plant (~1 Gpa) with six diamond anvils at 800 °C for 12 h following a technology similar to a earlier described one [
Standard measurements of the electrical resistivity R (T,В) and the Hall constant R_{h}(T,В) of the black phosphorus single crystals were carried out in the 2 < T < 300 K temperature range at 0 ≤ B < 8 T magnetic fields. The test specimens were rectangular, their longer sides (and hence the current vector directions) being parallel to the a crystallographic axis (Specimen 1) or the c axis (Specimen 2) and the B vector being always perpendicular to the ac plane, i.e., it was along the b axis. The R (T,В) and R_{h}(T,В) dependences were recorded in a cryogenfree measurement system of Cryogenics Ltd., UK, on the basis of a closedcycle refrigerator. The current through the specimens during the measurements was set and measured with a Keithley 6430 unit which allows measuring electrical resistance in the 100 mΩ to 10 GΩ range accurate to within 0.1%. The temperature of the specimens was controlled with LakeShore thermodiodes calibrated accurate to 0.0005 K and having a reproducibility of 0.001 K. The temperature was stabilized and measured with a LakeShore 331 controller. The measurement accuracy of the conductivity σ and the Hall constant R_{h} was not worse than 5%, the error being mostly determined by the measurement error in specimen dimensions, electric contact width and contact spacing. Room temperature Hall effect and Seebeck effect sign measurements showed that the test single crystals had the n type conductivity. The stability of the Hall effect sign testified to the predominantly electron conductivity over the whole test temperature range. The quality of the contacts for all the specimens was checked by preliminary measurement of IVs at T = 300 K which proved to be linear at currents of below 1 mA.
Figure
The tests showed that the bP specimens exhibit a strong conductivity anisotropy showing itself primarily in the difference between the behavior of σ_{а} (Т) and σ_{с} (Т) in the measuring temperature range with the current vector being always oriented along the a or c axes, respectively. It can be seen from Fig.
The test showed that R_{h}(В) in fields stronger than 1 T at any temperatures is almost insensitive to magnetic field magnitude. Figure
To correctly estimate the carrier concentrations and adequately describe the carrier transport mechanisms in single crystal bP Specimens 1 and 2 we additionally studied the relative magnetoresistance MR (В,Т) = [R (B) – R (0)]/R (0) as a function of magnetic field and temperature (Fig.
Analysis of the curves showed that in magnetic fields with magnitudes greater than B_{m} the absolute values of NMR effect and in the whole range of PMR contribution, the MR (В) curves are proportional to B^{2} suggesting its Lorentz nature. One should note a number of specific features inherent to the behavior of this contribution with a change in temperature. At 10 K the magnetoresistive effect (even after subtraction of the NMR contribution) is the smallest (almost zero) as compared with its values at higher temperatures although there are indications [
As shown above the experiments revealed a strong anisotropy of the electrical conductivity and magnetoresistance, including a change in the sign of the anisotropy coefficient α = [σ_{а} (Т) –σ_{с} (Т)]/σ_{с} (Т). This anisotropy shows itself not only as the dependence of the conductivity and MR on current vector orientation relative to the crystallographic axes of bP single crystals but also as a change in the course of the σ(Т) (semiconducting or metalliclike) curves and the magnetoresistive effect as a function of magnetic field and temperature MR (В,Т) (NMR or PMR). As follows from the brief literature review presented above the main features of the σ(Т), R_{h}(В,Т) and MR (В,Т) curves for bP bulk crystals are explained in different sources within four most widely used approaches.
1. Movement of one type of carriers (holes) in intrinsic bP crystals (singleband model) [
2. Movement of two differentsign carriers with different effective masses in two bands [
3. Movement of carriers in largescale potential pattern relief causing the socalled mobility fluctuations [
4. Movement of carriers under strong localization conditions (hopping conductivity of carriers or polarons) [
We will now justify the criteria of choosing one of the above listed approaches that we will use hereinafter for analyzing the experimental σ(Т), R_{h}(В,Т) and MR (В,Т) dependences and discussing the measurement results.
The use of the singleband model in the former case for analysis of the movement of one type carriers is most likely incorrect for the following reasons. First, this model is only applicable to undoped bP which is an intrinsic semiconductor and should have predominantly hole conductivity type since the effective mass of holes is lower than that of electrons [
The model allowing for carrier movement in the largescale potential relief region is typically used for analysis of Hall mobility in heavily inhomogeneous semiconductors containing largescaled structural defects. In bP crystals these defects can be grain boundaries (in polycrystals), dislocations, interlayer cracks (caused by the extremely high brittleness of these crystals), stacking faults (violation of the layering rule of phosphorus atoms) as well as point defect clusters. The presence of largescale potential relief in inhomogeneous semiconductor crystals shows itself at low temperatures when and if the temperature dependence of μ_{h}(Т) in double logarithmic coordinates is linear and has extra large positive slopes (the exponent values k >> 3/2) [
Attempts to describe carrier movement under strong localization conditions showing itself as hopping or polaron conductivity at temperatures below 40–50 K also failed because lowtemperature σ(Т) curves in Mott’s coordinates [
Analysis of the possibility to use the four abovementioned most widely used approaches to the description of the σ(Т), R_{h}(В,Т) and MR (В,Т) dependences showed that correct description of the magnetoresistive properties of bP requires the twoband model be used. Note also that when discussing the properties of bP single crystals we ruled out Drude’s theory of quantum corrections to the conductivity due to a carriers phase breaking under weak localization conditions which was formerly observed only in single or multilayered specimens of splitted phosphorene, graphene and transition metal chalcogenides [
Relative magnetoresistance MR (В,Т) for black phosphorus specimens (a) 1 and (b) 2 as a function of magnetic field B at different temperatures T, K: (1) 2; (2) 5; (3) 10; (4) 25; (5) 50; (6) 100; (7) 150; (8) 200; (9) 250; (10) 275; (11) 300.
(a) Hall mobility vs temperature dependences µ_{h}(T) in linear coordinates and (b) Hall concentration vs temperature dependences n_{о}(T) in double logarithmic coordinates for Specimens (1) 1 and (2) 2. Inset: µ_{h}(Т) dependences in double logarithmic coordinates.
For a quantitative evaluation of the parameters describing the conductivity and galvanomagnetic properties of bP single crystal Specimens 1 and 2 on the basis of the twoband model [
where μ_{i}, n_{i} are the mobility and concentration of ith type carriers (i = p, n), respectively, B is the magnetic field induction in which the specimen is located, e is the electron charge and r_{h} is the Hall factor. Since the Hall coefficient does not depend on magnetic field at В > 1 T then assuming electron and hole mobilities to be equal (μ_{i} = μ) we can rewrite Eqs. (1a)–(1c) as follows:
where
is the sum of the carrier concentrations. Then Eq. (1a) reduces to the following relation:
Introducing the relationship
we obtain the formula for relative magnetoresistance tensor:
where μ_{0} is the mobility at B = 0. This implies the equality
For the two carrier types having different signs (electrons and holes) and mobilities in classically weak magnetic fields μ^{2}В^{2} << 1 for which r_{h} depends on the scattering mechanism [
whereas for μ_{p} = μ_{e} = μ it transforms as follows:
Therefore, the experimental value 1/eR_{h} will be related to the actual carrier concentrations (n and p) through the following formula:
In this case the conductivities for equal electron and hole mobilities (μ_{p} = μ_{e} = μ) and for coexistence of two differentsign carrier types will be as follows:
σ = σ_{h} + σ_{e} = enμ + epμ = eμ (n + p) = eμn_{2}, (11a)
σ = σ_{h} + σ_{e} = enμ_{e} + epμ_{h}, (11b)
whence
n _{2} = (n + p). (12)
Therefore, the n_{1} parameter in Eq. (10) can be obtained by measuring the Hall coefficient (1/eR_{h}), the n_{2} parameter can be borrowed from the conductivity formula (Eq. (11a)) and the mobility can be calculated using Eq. (6) for the MR_{xx} parameter:
Simple transformations of Eqs. (1) and (12) yield the following set of equations for n and p calculation:
The relationships obtained above (Method 1) will be hereinafter used for the calculations of carrier concentrations and mobilities (provided the latter are equal).
To estimate these parameters in the assumption of the electrical neutrality condition (if the bP crystals are accepted to be intrinsic semiconductors) we will consider the electron and hole concentrations to be equal:
n = p. (15)
Then the conductivity, the Hall constant and the magnetoresistance will be respectively described by the following relationships [
σ = σ_{h} + σ_{e} = enμ_{e} + epμ_{h} = eN (μ_{e} + μ_{h}); (16a)
Introducing the notations
we obtain the following set of equations for the calculation of mobilities:
D = C^{2} – 4A,
which has two solutions:
Calculations using Eqs. (15)–(18) are the principle of Method 2.
We will now estimate some parameters of the test bP single crystal specimens using Methods 1 and 2 for the fitting of the galvanomagnetic properties as described above. It should be noted that for calculations based on either of the Methods using the experimentally observed magnetoresistive effect MR (T,B) at temperatures below 25 K we took into account the NMR contribution to the magnetoresistive effect in the region of a magnetic field by subtracting its maximum absolute value at В = B_{m} (see insets in Fig.
Results of fitting using both Methods described above for derivation of the electron and hole concentration and mobility temperature dependences yield close values, see Figs
(a) Electron and hole concentrations (n = p) vs temperature and (b) electrontohole concentration ratio (n/p) for Specimens (1) 1 and (2) 2. Inset: same functions in double logarithmic coordinates.
(a) Electron and hole mobilities µ(T) vs temperature and (b) electrontohole mobility ratio (µ_{n}/µ_{р}) for Specimens (1) 1 and (2) 2. Inset: mobility vs temperature in double logarithmic coordinates.
Specimen  Concentration (10^{19} m^{3})  Mobility (m^{2}/(V⋅s))  
n  p  n = p  μ_{n}  μ_{p}  μ_{p} = μ_{n} = μ  
Method 1  Method 2  Method 2  Method 1  
1  7.878  7.860  7.869  0.02212  0.02210  0.02211 
2  3.328  3.332  3.330  0.02148  0.02147  0.02147 
Before discussing the dependences shown in Figs
Analysis of the dependences shown in Fig.
The behavior of the n (Т) and p (T) dependences described above suggests that the experimental bP single crystals are intrinsic semiconductors with predominantly electron conductivity type.
As can be seen from Fig.
There is an obvious correlation between the curve behavior shown in Fig.
On the other hand, since the carrier concentration for Specimen 1 shown in Fig.
It follows from the above that the conductivity anisotropy is mainly determined by the carrier concentration anisotropy. Since the carrier concentration and mobility in Specimen 1 see a saturation plateau at high temperatures whereas the carrier concentration in Specimen 2 always grows with temperature, the sign of the conductivity anisotropy in Specimen 2 changes. Thus the correlation between the temperature behavior of the conductivity shown in Fig.
We will now compare the presence and behavior of the NMR effect in the test bP specimens at low temperatures (Fig.
The NMR effect is often observed in strongly disordered semiconductors (i.e., under strong localization conditions) when hopping electron transport over localized states occurs [
Study of the galvanomagnetic properties of black phosphorus (nP) single crystals showed that these crystals are intrinsic semiconductors with two carrier types (electrons and holes) having almost equal concentrations and mobilities. The temperature dependence of the electrical conductivity σ(Т) is determined by the orientation of the current vector relative to the a and c crystallographic axes and depends primarily on the carrier concentration anisotropy. At below 50–70 K the anisotropy coefficient α = [σ_{а} (Т) – σ_{с} (Т)]/σ_{с} (Т) is positive whereas at above 220 K its sign changes to negative. We showed that the resistivity vs magnetic field dependences for both specimens incorporate two competing contributions, i.e., the negative (NMR) and positive (PMR). The NMR contribution seems to originate from structural disorder and is observed at Т < 25 K and В < 6 T while the PMR one is associated with the Lorentz mechanism and shows itself at above 25 K in 6–8 T magnetic fields.