Comparability between 3D FEM Simulation and Experi- Experi-mental I-V Spectra

In order to extract the Schottky barrier height from the spectroscopic data (see sec-tion 4.2), one needs to obtain isolines from the 3D FEM simulasec-tion that are directly com-parable with tunnel current isolines from the 𝐼𝐼-𝑉𝑉 spectra taken along the space charge region. As it will turn out in this subsection, one can extract a quantity from the simula-tion that can be thought of as a measure for the tunnel current. In the following, this quan-tity is denoted as Φ_{𝑉𝑉𝑇𝑇} and represents the energy range in which electrons tunnel solely through the vacuum barrier between tip and sample and not through the additional tip-induced space charge barrier as indicated in Figure 4.4. According to the 1D band model, for the case of a p-type junction Φ_{𝑉𝑉𝑇𝑇} can be written as

12 Here the reader is reminded that the local electrostatic potential energy Φ=−𝑒𝑒𝑉𝑉 differs from the electro-chemical potential 𝜇𝜇_{𝑒𝑒𝑠𝑠} which is given by 𝜇𝜇_{𝑒𝑒𝑠𝑠}=𝜇𝜇𝑠𝑠+Φ where 𝜇𝜇_𝑠𝑠 represents the chemical potential.

4.1 3D Simulation of the Electrostatic Potential of Metal-Semiconductor Interfaces

57 Φ𝑉𝑉𝑇𝑇 =𝑒𝑒𝑉𝑉s− ΦTIBB − ΦSCR−(𝐸𝐸_𝐹𝐹^{𝑆𝑆𝐶𝐶} − 𝐸𝐸𝑉𝑉) (4.11) where 𝑉𝑉s is the applied voltage between tip and sample, ΦTIBB is the tip-induced band bending, ΦSCR is the band bending of the space charge region, 𝐸𝐸_𝐹𝐹^{𝑆𝑆𝐶𝐶} is the Fermi energy in the semiconductor bulk and 𝐸𝐸𝑉𝑉 is the valence band maximum. ΦTIBB can be extracted from the FEM simulation. For any position of the tip along the space charge region of the semiconductor

ΦTIBB =Φsbb− ΦSCR (4.12)

applies. Here Φ_sbb is simply the band bending at the surface of the semiconductor right below the tip. Φ_sbb and Φ_SCR are simulated in dependence of the distance between tip and interface (see also upper panel in Figure 4.5). With relation (4.12) also Φ_TIBB can be ex-tracted from that. However, if one inserts expression (4.12) into equation (4.11) it be-comes apparent that one only needs to read out the surface band bending Φ_sbb below the tip to obtain Φ_{𝑉𝑉𝑇𝑇}. The same considerations can be transferred to the case of an n-type junction, the only difference being that electrons tunnel from the tip into the conduction band of the semiconductor [47].

Figure 4.4: 1D energy band model to explain the derivation of the quantity Φ_{𝑉𝑉𝑇𝑇}. The band schemes depict the case of a negative sample bias voltage 𝑉𝑉_𝑠𝑠 and (a) far away from the metal-semiconductor interface and (b) inside the space charge region (SCR) induced by the metal-semiconductor contact causing a rigid shift of the energy bands by Φ_{𝑆𝑆𝐶𝐶𝑆𝑆}. For more details see text.

Now it will be shown that the quantity Φ_{𝑉𝑉𝑇𝑇} can be seen as a measure for the tunnel cur-rent 𝐼𝐼_𝑇𝑇 or in other words that there is a bijection between the energy interval Φ_{𝑉𝑉𝑇𝑇} and the tunnel current 𝐼𝐼_𝑇𝑇. This is only the case if one can exclude a significant tunnel current from any other energy interval than Φ_{𝑉𝑉𝑇𝑇}. Another energy interval where tunneling can take place is the tip-induced space charge region inside the semiconductor ΦTIBB which

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builts up another finite barrier adjacent to the vacuum gap, see Figure 4.4. Feenstra and Stroscio [82] described the transmission through this tip-induced space charge region in an effective mass scenario and obtained the following expression for the corresponding transmission coefficient

𝑇𝑇TIBB = exp �−𝑤𝑤�2𝑚𝑚^∗Φ_TIBB

ℏ � with 𝑤𝑤 =�2𝜀𝜀_𝑟𝑟𝜀𝜀₀Φ_TIBB

𝑁𝑁𝑒𝑒^{2 (4.13)}

where 𝑤𝑤 is the width of the tip-induced space charge region, 𝑚𝑚^∗ is the effective mass of an electron or hole depending on the type of doping, 𝜀𝜀_𝑟𝑟 is the relative permittivity of the semiconductor, and 𝑁𝑁 is the doping concentration.

Figure 4.5: (upper panel) Plot of the surface band bending Φsbb, the band bending of the space charge region ΦSCR in the semiconductor bulk, and the tip-induced band bending

Φ_TIBB along the space charge region of a p-type interface (interface is located at 𝑥𝑥= 0 nm) [120]. For the simulation the acceptor concentration is taken to be 𝑁𝑁_𝐴𝐴= 2.75 × 10¹⁸ cm^-3 and Φ𝐹𝐹𝑆𝑆=−0.776 eV (Φ_{𝑆𝑆𝑆𝑆}= 0.78 eV) is assumed. (lower panel) Logarithmic plot of the transmission coefficient along the space charge region calculated according to Ref. [82]. The dashed vertical line indicates up to which position the data was fitted (see also section 4.2).

In the lower panel of Figure 4.5 𝑇𝑇_TIBB is plotted logarithmically in dependence of the dis-tance to the interface. The transmission coefficient is smaller than 2 × 10⁻⁶ for the rele-vant data (that are fitted up to 1.5 nm to the interface, see also chapter 4.2). If one as-sumes a maximal tunnel current in the order of 100 pA, the contribution from the

tip-4.1 3D Simulation of the Electrostatic Potential of Metal-Semiconductor Interfaces

59 induced space charge region to the tunnel current would be in the sub-fA range. This is well below the resolution limit of the experiment. Therefore, any contribution to the tun-nel current from the tip-induced space charge region can be neglected in the analysis of the data. Thus, one finds a bijection between Φ_{𝑉𝑉𝑇𝑇} and the tunnel current 𝐼𝐼_𝑇𝑇: in general one can say that the larger Φ_{𝑉𝑉𝑇𝑇} is, the larger also 𝐼𝐼_𝑇𝑇 will be. This relation applies at any distance of the tip to the interface. Hence, the bias voltages at which the same tunnel cur-rent 𝐼𝐼_𝑇𝑇,ref was measured (a tunnel current isoline in the 𝐼𝐼-𝑉𝑉 spectra along the space charge region) correspond to the bias voltages with a constant Φ_{𝑉𝑉𝑇𝑇,ref} [47]:

𝑉𝑉(𝑥𝑥,𝐼𝐼𝑇𝑇,ref) ≅ 𝑉𝑉(𝑥𝑥,Φ𝑉𝑉𝑇𝑇,ref) (4.14) This equivalence allows the direct comparison of the experimental and simulated data in a way that is decribed in more detail in chapter 4.2.

Figure 4.6: Electrostatic potential energy Φ(𝑥𝑥) along the space charge region of a p-type interface with 𝑁𝑁_𝐴𝐴= 2.75 × 10¹⁸ cm^-3 and Φ𝐹𝐹𝑆𝑆 =−0.776 eV. The black solid, the red dashed, and the blue dashed-dotted lines represent the potential energy at the surface without an STM tip, the potential energy 50 nm below the surface, and the potential ener-gy at the surface with a scanning tip, respectively.

At the end of this section the significance of electrostatic effects due to the surface geom-etry of this approach is discussed. Therefore, the metal-semiconductor contact is simulat-ed without the STM tip. The corresponding electrostatic potential energy Φ(𝑥𝑥) is plottsimulat-ed in Figure 4.6. The solid black and the red dashed lines represent the electrostatic potential energy Φ(𝑥𝑥) simulated without an STM tip at the surface and 50 nm below the surface, respectively. The two curves almost perfectly lie on top of each other. The blue dashed-dotted line demonstrates that the tip has a much stronger impact on the electrostatic po-tential at the surface (due to the tip-induced band bending) than the cross-sectional geom-etry itself. As a conclusion one can say that a purely electrostatic effect resulting from the

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surface geometry is small and will not have a significant impact on the quantitative analy-sis of the Schottky contact.

4.2 Extracting the Schottky Barrier Height from the