Ultrafast laser-triggered field ion emission from semiconductor tips

In the presence of a high electric field, the surface band structure of a semiconductor–vacuum interface is strongly distorted with the bands bent upward leading to high surface hole density when the applied voltage is positive [14]. However, for low temperature and before the application of the laser pulse, the free-charge density is negligible inside a pure semiconductor. The band structure is of a dielectric type with a linear voltage drop across the sample and a constant internal electric field [22]. The internal field is determined by the surface field in vacuum and the dielectric constant of a semiconductor (ε_r = 16.2 for Ge and ε_r = 11.7 for Si). In our model, we consider the dc field E₀ inside Ge and Si in the range of 1–3 V nm⁻¹, i.e. around E₀ = E_evap/ε_r. The dc electric field allowing evaporation E_evap is 29 and 33 V nm⁻¹ for Ge [23] and Si [24], respectively.

When a femtosecond laser pulse interacts with a semiconductor tip, it creates free carriers (free electrons and holes). The evolution of the carrier density is strongly influenced by the external dc electric field that, in turn, leads to the establishment of strong band bending at the surface. In addition, under photo-generation of electrons and holes, the equilibrium Fermi level is split into two quasi-Fermi levels.

To describe the transient response of the tip to the field during the interaction with the laser pulse, we use the drift-diffusion approach [18, 19]. In the La-APT, the laser pulse is applied perpendicularly to the main tip axis, i.e. the wave vector is perpendicular to the dc field direction of symmetry. The optical absorption depth of Ge at λ = 1030 nm and of Si at λ = 515 nm is >1 μm. Far away from the hemispherical apex of the tip, the tip can be considered as a cylinder with ∼100 nm diameter, which is smaller than the optical absorption depth. Thus, the tip can be approximately described by a 1D model along its axial direction. The model consists of the solution of the transport equations for carrier densities and the Poisson equation for the field inside the semiconductor sample as follows:

$$\begin{equation} \left\{\begin{array}{@{}l} \displaystyle\frac{\partial n_{\mathrm{h}}}{\partial t}-\displaystyle\frac{\partial}{\partial x}\left(D_{\mathrm{h}}\displaystyle\frac{\partial n_{\mathrm{h}}}{\partial x}+\mu_{\mathrm{h}}n_{\mathrm{h}}E\right) = S_{\mathrm{h}}-R_{\mathrm{h}},\\ \displaystyle\frac{\partial n_{\mathrm{e}}}{\partial t}-\displaystyle\frac{\partial}{\partial x}\left(D_{\mathrm{e}}\displaystyle\frac{\partial n_{\mathrm{e}}}{\partial x}-\mu_{\mathrm{e}}n_{\mathrm{e}}E \right) = S_{\mathrm{e}}-R_{\mathrm{e}},\\ \displaystyle\frac{\partial E}{\partial x}=-\displaystyle\frac{e}{\varepsilon \varepsilon_{0}}(n_{\mathrm{h}}-n_{\mathrm{e}}), \end{array} \right. \end{equation} \tag{ 1 }$$

where n_h, n_e and E are the space- and time-dependent hole density, electron density and electric field, respectively, e is the absolute value of the electronic charge, and ε₀ is the electric constant.

The source of carriers, S_c (subscript c stands for either hole or electron), accounts for the interband absorption of the laser energy

$$\begin{equation} S_{\mathrm{c}}=\frac{(1-\Gamma)\alpha^{\mathrm{opt}}I}{\mathcal{E}_{\mathrm{ph}}}, \end{equation} \tag{ 2 }$$

where Γ is the reflectivity of the sample and I is the intensity of the laser pulse. At the given laser intensities and photon energies, multi-photon processes are negligible. For example, in Ge, the two-photon absorption cross section β is less than 10 cm GW⁻¹ at 1030 nm [25]. The given intensity I is smaller than 1 GW cm⁻², which leads to the two-photon absorption coefficient βI < 10 cm⁻¹ being much smaller than the linear absorption coefficient α_opt = 10⁴ cm⁻¹.

The carrier losses in the bulk, R_c, are mainly due to the Auger recombination with coefficients of the e–h–h process C_h = 7.8 × 10⁻³² cm⁶ s⁻¹ and the h–e–e process C_e = 2.3 × 10⁻³¹ cm⁶ s⁻¹ for Si [26] and C_h = C_e = 1.1 × 10⁻³¹ cm⁶ s⁻¹ for Ge [27]

$$\begin{eqnarray} &&\begin{array}{l} R_{\mathrm{h}}=R_{\mathrm{e}}=R^{\mathrm{A}}_{\mathrm{h}}+R^{\mathrm{A}}_{\mathrm{e}},\\ R^{\mathrm{A}}_{\mathrm{h}}=C_{\mathrm{h}}n_{\mathrm{h}}^{2}n_{\mathrm{e}},\\ R^{\mathrm{A}}_{\mathrm{e}}=C_{\mathrm{e}}n_{\mathrm{e}}^{2}n_{\mathrm{h}}. \end{array} \end{eqnarray} \tag{ 3 }$$

In the transport equations, the carrier mobilities and diffusion coefficients are considered for Fermi–Dirac statistics in order to take into account degeneracy effects at high carrier densities [28]

$$\begin{equation} \mu_{\mathrm{c}}=\mu_{\mathrm{c}}^{0}\frac{\mathcal{F}_{0}(\eta_{\mathrm{c}})}{\mathcal{F}_{1/2}(\eta_{\mathrm{c}})},\quad D_{\mathrm{c}}=\mu_{\mathrm{c}}\frac{k_{\mathrm{b}}T_{\mathrm{c}}}{e}\frac{\mathcal{F}_{1/2}(\eta_{\mathrm{c}})}{\mathcal{F}_{-1/2}(\eta_{\mathrm{c}})}, \end{equation} \tag{ 4 }$$

where $\mathcal {F}_{\xi }(\eta _{\mathrm { c}})$ denotes the ξth-order Fermi integral, η_c is the reduced quasi-Fermi level [29] and k_B is the Boltzmann constant. Low-density values of electron and hole mobilities at 77 K are given by μ⁰_e = μ⁰_h = 4 m² (Vs)⁻¹ for Ge [30, 31] and μ⁰_h = 0.92 m² (Vs)⁻¹ and μ⁰_e = 2 m² (Vs)⁻¹ for Si [32]. The value of η_c depends on the local carrier density n_c and is calculated by the procedure described in [20]. Both the diffusion coefficient and the reduced Fermi levels depend on the carrier temperature T_c that changes during the interaction with the laser pulse. This is simultaneously calculated from the two-temperature equations described in the next subsection.

Equations (1)–(4) are solved with a finite difference method [17] together with the following initial conditions:

$$\begin{eqnarray} &&n_{\mathrm{h}}(x,t=0)=n_{\mathrm{e}}(x,t=0)=0,\qquad E(x,t=0)=E_0, \end{eqnarray} \tag{ 5 }$$

and boundary conditions

$$\begin{eqnarray} &&\begin{array}{l} E(x=0,t)=E_0,\\ D_{\mathrm{h}}\displaystyle\frac{\partial n_{\mathrm{h}}}{\partial x}+\mu_{\mathrm{h}}n_{\mathrm{h}}E\bigg|_{x=0,t}=v_{\mathrm{sr}}n_{\mathrm{c}},\\ D_{\mathrm{e}}\displaystyle\frac{\partial n_{\mathrm{e}}}{\partial x}-\mu_{\mathrm{e}}n_{\mathrm{e}}E\bigg|_{x=0,t}=v_{\mathrm{sr}}n_{\mathrm{c}},\\ D_{he}\displaystyle\frac{\partial n_{\mathrm{h}}}{\partial x}+\mu_{\mathrm{h}}n_{\mathrm{h}}E\bigg|_{x=L,t}=D_{\mathrm{e}}\displaystyle\frac{\partial n_{\mathrm{e}}}{\partial x}-\mu_{\mathrm{e}}n_{\mathrm{e}}E\bigg|_{x=L,t}=0, \end{array} \end{eqnarray} \tag{ 6 }$$

where L is the length of the considered sample and v_sr is the surface recombination velocity. The value of v_sr depends on the density of surface states and for Ge and Si ranges from 10² to 10⁶ cm s⁻¹ [33–35]. The surface-recombination rate is limited by the surface minority carriers. Thus, in equation (6) n_c = n_e if there are fewer electrons than holes and n_c = n_h in the opposite case.

The boundary conditions (6) for electron and hole currents are chosen to maintain the conservation of particles, i.e. no charge carriers leave either the left (x = 0) or the right boundary (x = L). The left boundary condition for the dc field E (x = 0) is kept constant. In the numerical solution, the sample is divided into layers (numerical cells) with variable thickness. The first numerical cell at the sample surface is 10⁻¹⁰ m, increasing to the bulk. Because of the very high field and, thus, carrier drift velocity the time step is required to be very small, 10⁻²⁰ s, so the calculations are time-consuming. Therefore, we focus our attention only on several picoseconds after the beginning of the laser pulse.

We consider the Gaussian temporal profile with the pulse width (full-width at half-maximum (FWHM)) τ_p = 500 fs, the peak intensity of 1 GW cm⁻², laser wavelength λ = 1030 nm for Ge and λ = 515 nm for Si. The size of the sample is taken as L = 20 μm. It is assumed that the lasing starts at t = 0, reaches the maximum intensity at t = 600 fs and ends at t = 1200 fs.

In the presence of a positive external dc field, the photo-generated holes start drifting toward the surface, whereas the electrons drift in the opposite direction. This effect leads to a significant increase in the hole density at the surface with respect to its bulk value (figure 1(a)). The separation of holes from electrons generates space charge that results in the screening of the external field. During the laser pulse, the dc field gradually decreases in the bulk (figure 1(b)). When the carrier distributions reach a quasi-equilibrium governed by the balance between drift and diffusion currents, the field becomes completely screened (t ≃ 1 ps). By this time, the valence band maximum at the surface is significantly higher than the hole quasi-Fermi level in the bulk and the hole density reaches n_h ≃ 2.7 × 10²¹ cm⁻³ for Ge for E₀ = 1.5 V nm⁻¹. For Si, the charge movement and field screening evolve similarly with the maximum hole density n_h ≃ 3.2 × 10²¹ cm⁻³ for E₀ = 2 V nm⁻¹ (due to the similarity, the corresponding figures were not shown). Therefore, these holes will significantly absorb the laser energy at the surface. The free-carrier absorption is related to the change of the complex dielectric function Δε^fcr, which depends on the carrier density and laser wavelength according to the Drude model [36]

$$\begin{eqnarray} &&\begin{array}{c} \Delta\varepsilon^{\mathrm{fcr}}=\Delta\varepsilon_{\mathrm{e}}^{\mathrm{fcr}}+\Delta\varepsilon_{\mathrm{h}}^{\mathrm{fcr}},\\ \Delta\varepsilon_{\mathrm{c}}^{\mathrm{fcr}}=-\displaystyle\frac{n_{\mathrm{c}}e^{2}}{\varepsilon_{0}m_{\mathrm{c}}^{*}\omega^{2}}\displaystyle\frac{1}{1+\mathrm{i}(\omega\tau_{\mathrm{D}})^{-1}}, \end{array} \end{eqnarray} \tag{ 7 }$$

where τ_D is the carrier damping time and ω = 2πc/λ is the laser frequency. For Ge, the carrier effective masses m*_c are taken as m*_h = 0.23m_e and m*_e = 0.12m_e [34] and for Si m*_h = m*_e = 0.5m_e [33]. The free-carrier absorption coefficient is given by

$$\begin{equation} \alpha^{\mathrm{fcr}}_{\mathrm{c}}=\frac{4\pi}{\lambda\sqrt{2}}([\mathrm{Re}^{2}\Delta\varepsilon_{\mathrm{c}}^{\mathrm{fcr}}+\mathrm{Im}^{2}\Delta\varepsilon_{\mathrm{c}}^{\mathrm{fcr}}]^{1/2}-\mathrm{Re}\Delta\varepsilon_{\mathrm{c}}^{\mathrm{fcr}})^{1/2}. \end{equation} \noindent \tag{ 8 }$$

The value of α^fcr_h is about 10⁵ cm⁻¹ for both Ge and Si at the surface when early during the laser pulse the hole density reaches more than 10²¹ cm⁻³ due to the band bending. This value is almost ten times higher than the interband absorption coefficients of Ge and Si at 80 K for infrared and green light, respectively, showing that under a high electric field the optical properties of the surface of the sample are changed following the dynamics of the spatial distribution of charges. Note that we also take into account a change of the sample reflectivity Γ according to the modified dielectric function (equation (7)).

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The carrier damping time τ_D in equation (7) depends on many processes, such as carrier–phonon collisions, carrier–carrier collisions, degeneracy of carrier distribution and surface roughness scattering [37, 38]. It becomes very short at high carrier densities. The value of about 1 fs is reported for optically excited silicon [18, 36]. At low densities, however, it can be 100 times longer. We take τ_D = 1 fs as hole and electron damping time everywhere in the sample, for simplicity. In the regions of low density of holes, i.e. bulk of the sample, the free-carrier absorption is much lower than interband absorption and does not significantly contribute to the total absorption. The density of the electrons and free-electron absorption is low in the bulk, as well as at the surface.

The temperature of the surface of the tip is determined by the processes of tip heating and heat dissipation in the bulk. The heating of the semiconductor material by the laser pulse is caused by the transfer of energy from the carriers to the lattice through carrier–phonon coupling and carrier recombination.

The excess energy $\mathcal {E}_{\mathrm { c}}$ of a created electron–hole pair depends on the photon energy and band gap of the material. It gives to carriers the initial kinetic energy and leads to the elevated initial carrier temperature. The carriers thermalize then into Fermi–Dirac distribution via carrier–carrier collisions on a femtosecond time scale. Hence, the total kinetic energy density in the carrier system is given as [19]

$$\begin{equation} \mathcal{E}_{\mathrm{c}}n_{\mathrm{c}}=\frac{3}{2}n_{\mathrm{c}}k_{\mathrm{B}}T_{\mathrm{c}}\frac{\mathcal{F}_{3/2}(\eta_{\mathrm{c}})}{\mathcal{F}_{1/2}(\eta_{\mathrm{c}})}. \end{equation} \noindent \tag{ 9 }$$

The excess energy is divided between a hole and an electron according to their effective masses. For Ge at infrared excitation and band gap $\mathcal {E}_{\mathrm {g}}=0.73$ eV at 80 K the excess hole energy is equal to $\mathcal {E}_{\mathrm {h}}\frac {m_{\mathrm {h}}^*}{m_{\mathrm {h}}^*+m_{\mathrm {e}}^*}(\mathcal {E}_{\mathrm { ph}}-\mathcal {E}_{\mathrm {g}})\sim 0.31$ eV. From equation (9), it follows that the initial hot-hole temperature T_h0 ≃ 2420 K. Similarly, the initial hot-electron temperature is T_e0 ≃ 1250 K. For Si at green light excitation ($\mathcal {E}_{\mathrm {g}}=1.15\,\mathrm {eV}$) the excess energy is $\mathcal {E}_{\mathrm {h}}=\mathcal {E}_{\mathrm {h}0}\simeq 0.6\,\mathrm {eV}$ and the hot-hole and hot-electron temperature is T_h0 = T_e0 ≃ 4640 K. The additional kinetic energy is provided through free-carrier absorption to electrons and holes. During the Auger recombination process, two carriers recombine and a third carrier takes a part of the total energy, corresponding to the sum of kinetic energies and band gap. Thus, Auger recombination reduces the carrier number, while increasing the carrier temperature.

Several effects can lead to different energies of electron and hole subsystems and, thus, energy exchange between them. However, as shown in the previous subsection, in the bulk of the sample the dc field is screened, thus an electron–hole pair behaves like one particle and only its total energy is important. At the surface, the holes and electrons are separated and cannot interact. This allows us to neglect the electron–hole energy exchange in the calculations.

We note that because of surface defects, the band gap of a semiconductor can be lower at the surface than in the bulk [39], which can lead in turn to higher excess energy and higher generation rate of the carriers at the surface. However, as shown above, the value of the hole density at the surface is determined by the applied dc field and average density of the holes inside the sample (figure 1(a)). The additional increase in carrier number in a small near-surface region would not change this process significantly. Thus, we fix the band gap at the constant along the sample.

Furthermore, we neglect the effect of carrier heating by the dc field. To acquire some energy from the dc field, the carriers need to travel a sufficient distance. However, they are just slightly and rapidly redistributed under the dc field here. When quasi-equilibrium is reached, the carriers do not drift anymore. This fact also means that we can neglect the possible carrier energy loss through the impact ionization process. For this process to have a higher rate than energy loss through carrier–phonon coupling, the energy of a carrier has to be in excess of $1.5\mathcal {E}_{\mathrm {g}}$ [40]. In the considered case, the excess energy of a carrier is below $1.5\mathcal {E}_{\mathrm {g}}$. Also, the impact ionization rate is diminished at the surface, where only holes are present.

The temperature of the holes T_h, electrons T_e and lattice T_L are then calculated separately by using the following equations [20]:

$$\begin{equation} \fl \left\{\begin{array}{@{}l} \displaystyle\frac{\partial}{\partial t}(C_{\mathrm{h}}T_{\mathrm{h}})= S_{\mathrm{h}}\displaystyle\frac{m^*_{\mathrm{h}}}{m^*_{\mathrm{h}}+m^*_{\mathrm{e}}}(\mathcal{E}_{\mathrm{ph}}-\mathcal{E}_{\mathrm{g}})+\mathcal{E}_{\mathrm{g}}R^{\mathrm{A}}_{\mathrm{h}}+(1-\Gamma)\alpha^{\mathrm{fcr}}_{\mathrm{h}}I -\displaystyle\frac{C_{\mathrm{h}}}{\tau_{\mathrm{ph}}}(T_{\mathrm{h}}-T_{\mathrm{L}})+\displaystyle\frac{\partial}{\partial x}K_{\mathrm{h}}\displaystyle\frac{\partial T_{\mathrm{h}}}{\partial x},\\ \displaystyle\frac{\partial}{\partial t}(C_{\mathrm{e}}T_{\mathrm{e}})= S_{\mathrm{e}}\displaystyle\frac{m^*_{\mathrm{e}}}{m^*_{\mathrm{h}}+m^*_{\mathrm{e}}}(\mathcal{E}_{\mathrm{ph}}-\mathcal{E}_{\mathrm{g}})+\mathcal{E}_{\mathrm{g}}R^{\mathrm{A}}_{\mathrm{e}}+(1-\Gamma)\alpha^{\mathrm{fcr}}_{\mathrm{e}}I -\displaystyle\frac{C_{\mathrm{e}}}{\tau_{\mathrm{ph}}}(T_{\mathrm{e}}-T_{\mathrm{L}})+\displaystyle\frac{\partial}{\partial x}K_{\mathrm{e}}\displaystyle\frac{\partial T_{\mathrm{e}}}{\partial x},\\ \displaystyle\frac{\partial}{\partial t}(C_{\mathrm{L}}T_{\mathrm{L}})= \displaystyle\frac{C_{\mathrm{h}}}{\tau_{\mathrm{ph}}}(T_{\mathrm{h}}-T_{\mathrm{L}})+\displaystyle\frac{C_{\mathrm{e}}}{\tau_{\mathrm{ph}}}(T_{\mathrm{e}}-T_{\mathrm{L}})+\displaystyle\frac{\partial}{\partial x}K_{\mathrm{L}}\frac{\partial T_{\mathrm{L}}}{\partial x}.\\ \end{array} \right. \end{equation} \noindent \tag{ 10 }$$

The coefficient C_c describes the heat capacity of the carrier system and is given as [20]

$$\begin{equation} C_{\mathrm{c}}=\frac{3}{2}n_{\mathrm{c}}k_{\mathrm{B}}\left\{\frac{\mathcal{F}_{3/2}(\eta_{\mathrm{c}})}{\mathcal{F}_{1/2}(\eta_{\mathrm{c}})}-\eta_{\mathrm{c}}\left[1-\frac{\mathcal{F}_{3/2}(\eta_{\mathrm{c}})\mathcal{F}_{-1/2}(\eta_{\mathrm{c}})}{\mathcal{F}_{1/2}(\eta_{\mathrm{c}})^2}\right]\right\}. \end{equation} \tag{ 11 }$$

The thermal conductivity of carriers is taken as [20]

$$\begin{equation} K_{\mathrm{c}}=\frac{k^2_{\mathrm{B}}n_{\mathrm{c}}\mu_{\mathrm{c}}T_{\mathrm{c}}}{e}\left(6\frac{\mathcal{F}_2(\eta_{\mathrm{c}})}{\mathcal{F}_0(\eta_{\mathrm{c}})}-4\frac{\mathcal{F}_1(\eta_{\mathrm{c}})^2}{\mathcal{F}_0(\eta_{\mathrm{c}})^2}\right). \end{equation} \tag{ 12 }$$

The lattice heat capacity C_L strongly depends on lattice temperature below 300 K [41], as well as thermal conductivity K_L that is reported in the literature for Ge and Si nanowires, whose geometry is close to the APT tip geometry [42]. These dependences are taken into account in the model.

The electron and hole energy coupling to the lattice is described by the carrier–phonon energy relaxation time τ_ph, determined by optical- and acoustic-phonon scattering. This process is known to be different for electrons and holes in Ge and Si [43] and depends in a complicated way on carrier density, carrier temperature and lattice temperature [44, 45]. Also, the peculiarity of the given configuration with the electron–hole separation and a very thin layer of the high-density hole gas can additionally influence the process of energy coupling. Because of such an uncertainty in the exact value of τ_ph, we approximate it to be constant and the same for the electrons and holes, following the reported empirical values of 300 fs for Ge and 240 fs for Si [18, 19, 34].

The initial and boundary conditions for the temperatures are as follows:

$$\begin{eqnarray} &&\begin{array}{l} T_{\mathrm{h}}(x,t=0)=T_{\mathrm{h}0},\quad T_{\mathrm{e}}(x,t=0)=T_{\mathrm{e}0},\\ T_{\mathrm{L}}(x,t=0)=80~{\rm K}, \end{array} \end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray} &&\begin{array}{l} \displaystyle\frac{\partial T_{\mathrm{h}}}{\partial x}\Big|_{x=0,t}=\displaystyle\frac{\partial T_{\mathrm{e}}}{\partial x}\Big|_{x=0,t}=0,\\ K_{\mathrm{L}}\displaystyle\frac{\partial T_{\mathrm{L}}}{\partial x}\Big|_{x=0,t}=-v_{\mathrm{sr}}n_{\mathrm{c}}\mathcal{E}_{\mathrm{g}},\\ T_{\mathrm{h}}=T_{\mathrm{e}}=T_{\mathrm{L}}=80~{\rm K}~~{\rm at}~x=L. \end{array} \end{eqnarray} \tag{ 14 }$$

Because there are no carriers before the application of the laser pulse, the initial condition for the carrier temperature means that when the first carriers are created they have temperature equal to T_c0. The boundary conditions for the carrier temperature are adiabatic at x = 0. The value of T_L at x = 0 depends on the surface-recombination rate. In this process, the energy $\mathcal {E}_{\mathrm {g}}$ per electron–hole pair is absorbed directly by the lattice via multi-phonon emission [33]. At x = L the carrier and lattice temperature is kept at 80 K.

The self-consistent numerical solution of equations (10)–(12) together with equations (1)–(4) and boundary conditions (5), (6) (13) and(14) gives the change of the lattice temperature in space and time. The surface recombination becomes negligible and temporal evolution of the temperature is determined by the thermalization process between the hot holes and the lattice, because of the field-induced charge separation and very low density of electrons at the surface [46]. The lattice temperature is determined mostly by the density of holes, their initial temperature and temperature increase due to the free-carrier absorption effect. Thus, the tip temperature at the surface is higher than in the bulk due to the substantially higher surface hole density (figure 2(a)). At 1 GW cm⁻² laser intensity, the surface temperature reaches T_L ≃ 285 K for Ge at E₀ = 1.5 V nm⁻¹ and T_L ≃ 270 K for Si at E₀ = 2 V nm⁻¹ (figure 3(a)).

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For comparison, we performed the same calculations for Ge at zero dc field. In this case, the thermalization of the hot carriers with lattice leads to a small increase in average temperature along the whole sample because of the relatively low carrier density. The charge separation does not take place in the absence of the field and surface heating is determined solely by the surface recombination process (figure 2(b)). At 1 GW cm⁻² laser pulse intensity and for the lowest reported value of surface recombination velocity, the temperature rise at the surface is 3 K, whereas for the highest it is 63 K (figure 3(b)). In both cases the surface temperature rise is substantially lower than that in the presence of the high dc field E₀ = 1.5 V nm⁻¹ and holes accumulation (205 K). We also note that the tip heating and cooling is a much longer process in the case of zero field.

Moreover, the surface peak temperature is calculated as a function of laser intensity for different dc electric fields. The results are presented in figure 4. The surface temperature increases with laser intensity due to the increase in both the absorbed energy by holes and the hole density at the surface during the laser pulse. However, the hole density is limited by the maximum band bending at the given field, determined by the stationary solution of the Poisson equation [14]. When this maximum possible hole density is reached at the surface early during the laser pulse, the additional increase of intensity does not lead to a significant increase of temperature. In addition, the heat capacity is higher at higher temperature; thus much more energy is needed to additionally increase the sample temperature. This leads to the much slower increase of surface temperature at higher laser intensities, showing a tendency towards saturation. At lower dc field the maximum hole density at the surface is lower, leading to a lower surface temperature. One can note that for the same dc field applied to Ge and Si samples the maximum temperature of Si is lower. This can be attributed to the lower hole mobility and higher thermal conductivity of Si with respect to Ge.

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