# Electron Transport in \(n\)-Type InSe van der Waals Crystals with Co Impurities
## 3 Results and Discussion
### Electrical Properties along InSe Layers  
We show the recorded temperature dependences of the in-plane electrical conductivity \(\sigma_{\perp\mathrm{C}}\) (Figure 4), the in-plane electron Hall mobility \(\mu_{\perp\mathrm{C}}\), i.e., along the layers, (Figure 5), and the Hall coefficient \(R_{\mathrm{H}}\) (Figure 6) for the pristine as-grown, intercalated and doped with cobalt InSe van der Waals samples. Table 2 shows the corresponding numerical values. It was revealed that in the case of the initial sample at low temperatures the measured Hall coefficient decreases with an increase in temperature. Such behavior is caused by a shallow, mostly ionized even at \(T=80\) K, donor level (18.5 meV) arising from interstitial In\({}_{\mathrm{i}}\) atoms [30]. High values of the mobility indicate high quality of the samples, and the decrease in \(\mu_{\perp\mathrm{C}}\) with increasing temperature takes place because the electron-phonon interaction is the dominant scattering mechanism. This behavior of Hall coefficient \(R_{\mathrm{H}}\) and electron Hall mobility \(\mu_{\perp\mathrm{C}}\) determines a metallic temperature dependence of the in-plane electrical conductivity component \(\sigma_{\perp\mathrm{C}}\) at \(T<380\) K. However, at higher temperatures the electron density strongly increases due the activation of electrons from deeper donor states leading to a typically semiconductor \(\sigma_{\perp\mathrm{C}}\) (\(T\)) dependence. Figure 2: Photoluminescence spectra of bulk InSe samples measured with the excitation laser wavelength of 532 nm (\(p=100\) mW) at \(T=300\) K for: as-grown pristine, Co-intercalated and Co-doped InSe crystals. Figure 4: The in-plane electrical conductivity versus temperature for as-grown pristine (1), Co-intercalated (2) and Co-doped (3) \(n\)-type InSe crystals. Figure 5: The Hall coefficient versus temperature for as-grown pristine (1), Co-intercalated (2) and Co-doped (3) \(n\)-type InSe crystals. Figure 3: Scanning electron microscope micrographs showing cleaved surfaces of (**a**) as-grown pristine, (**b**) Co-intercalated and (**c**) Co-doped InSe crystals. According to the previous theoretical studies on electron-lattice interaction in layered semiconductors reported in the literature [31], the scattering of charge carriers with homopolar optical phonons, which are polarized along the \(C\) crystallographic axis (i.e., out-of-plane) and which modulate the thickness of separate layers, is prevailing in layered van der Waals crystals. In this case, the relaxation time can be expressed as following:  
\[1/\tau_{FS}=2g^{2}\omega\Big{[}((\varepsilon/\hbar\omega)+1)^{1/2}n_{ph}+\Big{(} n_{ph}+1\Big{)}\Theta(\varepsilon-\hbar\omega)((\varepsilon/\hbar\omega)-1)^{1/2} \Big{]} \tag{1}\]  
Here \(g^{2}=\varepsilon_{0}^{2}m^{*\frac{3}{2}}\)\(M\hbar\hbar(\hbar\omega)^{3/2}\) defines the electron-phonon coupling constant, \(\hbar\omega\) defines the energy of the phonon, \(\varepsilon_{0}\) defines the deformation potential per unit displacement with respect to the normal coordinate of the phonon mode, \(N\) defines the number of cells per unit volume, \(M\) defines the reduced ionic mass of the phonon mode, \(\varepsilon\) defines the electron energy, \(n_{ph}=[\exp(\hbar\omega/kT)-1]^{-1}\) defines the phonon occupation number, \(m^{*}=\left(m_{\perp C}^{*}\,m_{\parallel C}^{*}\right)^{1/3}\) defines the density-of-states effective mass and \(\theta\) (\(x\)) is equal to unity at \(x\geq 0\) and is 0 at \(x<0\). In the 3D approach, taking into account the scattering with homopolar optical phonon \(A_{1g}^{\prime}\) (\(\hbar\omega\) = 14.3 meV) and \(g_{2}\) = 0.051 the mobility temperature dependence \(\mu_{\rm FS}(T)\) has a form shown by the curve 4 in Figure 6. In this case, the slight deviation of the apparent mobility from the phonon one at low temperature for the pristine InSe sample can be interpreted taking into account additional scattering with the ionized impurities.