Metal-Semiconductor contacts

For almost all semiconductor devices understanding and tailoring of the metal - semiconductor contact are most important issues. Therefore, metal - semiconductor contacts have been subject to many investigations. Schottky [113,114] developed a first detailed model. Today, metal -semiconductor contacts are still called “Schottky contacts”. A good review of this model and its extensions is given for example by Schroder and Meier [115] and by Rhoderick and Williams [116]. In this work, only the aspects relevant for thick film contacts to silicon are briefly re-sumed.

For the subsequent calculations the following definition of the contact resistivity ρ_C is used [117]. with the current densityJand the applied voltageV.

Following the Schottky model (Figure 6.1), the barrier heightφ_B is given by

φ_B=φ_M−χ (6.2)

withφ_M, metal work function, andχ, the semiconductor electron affinity. φ_B is independent of the position of the Fermi level in the bandgap of the semiconductor. A more detailed description

(a) (b)

Figure 6.1.: A rectifying metal - semiconductor contact in the Schottky model. (a) Metal and semiconductor separated (b) Metal and semiconductor in intimate contact withW: Depletion layer width

includes barrier lowering due to the electric field in the semiconductor at the contact interface.

The image-force lowering ∆φ is proportional to the square root of the electric field E, which depends on the doping concentration (equation 6.3) [32,115].

∆φ =

q³N_D(V_bi−kT/q) 8π²K_s³ε₀³

^1/4

(6.3) withV_bi being build-in voltage andK_s being dielectric constant of silicon. The effective barrier heightφB,e f f is

φ_{B,e f f} =φ_B−∆φ (6.4)

To obtain ohmic contacts, the metal work function should be equal or smaller than the semicon-ductor electron affinity, i. e. φ_B would be zero or negative. The majority carries in contact near regions are accumulated or unchanged, respectively, compared to their density in the neutral substrate. Those contacts are frequently named “accumulation” or “neutral contacts” [59]. If φ_M is greater,φ_Bis positive and a rectifying contact is formed. The majority carries in the con-tact near regions are depleted. However, the predicted linear dependency of the barrier height on the metal work function was not confirmed experimentally (see e.g. [118]). Frequently, the Fermi level at the metal - semiconductor interface is supposed to be pinned by surface states leading to rectifying contacts independent of the metal work function, especially for n-type sil-icon substrates [115,116,118]. Ohmic contacts have to be fabricated with other means. In the following the current transport in a metal - semiconductor contact is therefore briefly discussed.

Thermionic emission

The current in contacts to lowly doped material (N_D<1×10¹⁷cm⁻³) is transported via thermionic emission. Only those electrons with an energy greater than the barrier contribute to the current transport. The current - voltage relation is

J=A^∗T²e^−qφB/kT×

e^qV/kT−1

(6.5) with A^∗ = A×m^∗_R/m, A: Richardson constant, m^∗_R: effective mass in effective Richardson constant, V: applied voltage (positive in forward bias). The effective mass used to calculate

the effective Richardson constant depends on the type of dopant in silicon and the crystal orientation. The underlying theory is well described in [32]. For the following calculations A^∗=112 Am⁻²K⁻²for n-type andA^∗=32 Am⁻²K⁻²for p-type, given by Andrews and Lep-selter [119], are used. Contacts to lowly doped material with the thermionic emission transport mechanism are rectifying. The contact resistivity for thermionic emission is then calculated to

ρ_C(T E) = k

qA^∗Texpq(φ_B−∆φ)

kT (6.6)

Thermionic field emission (TFE)

With increasing doping concentration the depletion layer width of the metal - semiconductor contact decreases. Electrons thermally excited to an energy less than the barrier height are in that case able to tunnel through the thin barrier. The contact resistivity in this doping region can be calculated using the WKB (Wentzel-Kramers-Brillouin) approximation [117]. The functional dependency ofρ_Con the semiconductor doping levelN_dis given by equation 6.7 [117].

ρ_C(T FE) = k

E₀₀ is a characteristic energy given by E₀₀= qh¯

m^∗_tunnel is the tunnelling effective mass. In general, the tunnelling effective mass is not equal to the effective mass used to calculate the effective Richardson constant. In literature differ-ent values are used. Following Ng [120] and Schroder [59], for the following calculations m^∗_tunnel/m_e=0.3 was used for metal contacts on n-type silicon².

Field emission

Very high doping surface concentrations, N_D≥1×10²⁰cm⁻³, lead tokT/E₀₀ <<1 at room temperature. Tunnelling is possible near the bottom of the conduction band because the de-pletion layer width is now thin enough. Following Yu [117], the dependency of the contact resistivity on the doping concentration including image force lowering writes

2The used value for m^∗_tunnel is close to the density of states effective mass for <111> oriented n-type silicon (m^∗_DOS/m_e=0.33).

ρ_C(FE) = k

The ratio kT/E₀₀ indicates the relative importance of the thermionic process compared to the tunnelling process. For KT/E₀₀ 1 thermionic emission, for KT/E₀₀ 1 field emission is dominant. Thermionic-field emission connects the two regimes (KT/E₀₀ ∼=1).

In Figure 6.2 the contact resistivity for an silver - silicon contact is plotted versus the dop-ing concentration for the three contact resistivity regimes includdop-ing image force lowerdop-ing. For N_D>1×10¹⁷cm⁻³thermionic-field emission is dominant, forN_D>8×10¹⁹cm⁻³the current transport mechanism is field emission. The calculated values slightly differ from the calcu-lations presented in [115]. The difference mainly occurs due to the different effective mass used for the calculations. In [115]m^∗_tunnel/m=1 was used. In this work the more actual ratio m^∗_tunnel/m=0.3 given by Ng and Liu [120] is used.

Figure 6.2.: Contact resistivity versus doping concentration for the three transport mechanism.

Parameters: n-type <111> oriented silicon, φB =0.78 eV [32], m^∗_tunnel/me = 0.3 [59], A^∗ = 112 Am⁻²K⁻²[119],T =300 K, image force lowering