1.7 Metal-semiconductor contact
1.7.2 Current transport and contact resistance of a Schottky contact . 25
⎝q3N
Vbi− kTq 8π2(r0)3
⎞
⎠
14
(1.15) with the built-in voltage for a non-biased n-type and p-type semiconductor Vbi,n and Vbi,p respectively
Vbi,n=φB− 1
q (EC −EF) Vbi,p =φB− 1
q (EF −EV). (1.16) N is the doping density, 0 the vacuum permittivity and r the dielectric constant of the semiconductor.
Considering image force lowering results in an effective barrier height of
φB,eff =φB−ΔφB. (1.17)
An overview of different methods to experimentally obtain the barrier heights as well as measured values for φB for several metals on n- and p-type silicon can be found in [98, 99]. Measured barrier heights for silver and aluminum, that present the metal components of the pastes investigated in this work, are shown in table 1.1.
Table 1.1: Measured barrier heights for Ag and Al on n- and p-type silicon [99, 100].
barrier height (eV) Ag Al n-type Si 0.56 0.50 p-type Si 0.54 0.58
1.7.2 Current transport and contact resistance of a Schottky contact
The current transport over a Schottky barrier depends on temperature and doping den-sity of the semiconductor. Three different mechanisms can be distinguished: thermionic emission (TE) over the barrier, field emission (FE) by tunneling through the barrier at energies near the bottom of the conduction band and thermionic field emission (TFE) occurring at energies in between. The different mechanisms are depicted in figure 1.11.
With increasing doping density the Fermi level shifts towards conduction (n-type) or valence band (p-type), respectively. Additionally, the barrier widthWB is reduced [98].
With this, the tunneling probability through the thin layer increases and thermionic field emission as well as field emission get more important. In the following, the
differ-Thermionic Emission Thermionic Field Emission
Field Emission
Metal Metal Metal
Metal Metal Metal
n-type semi-conductor
n-type semi-conductor
n-type semi-conductor
p-type semi-conductor
p-type semi-conductor
p-type semi-conductor
Figure 1.11: Schematic band diagrams illustrating the current transport mechanisms over Schottky barrier for n- (upper images, after [74]) and p-type (lower images) semiconductor/
metal interfaces.
ent current transport mechanisms are discussed and equations for the specific contact resistance in the different regimes are introduced.
1. Thermionic emissionFor lowly doped semiconductors, TE presents the preva-lent current path (figure 1.11 left images). Only those charge carriers thermally excited to energies high enough to overcome the potential barrier contribute to the current transport. With the definition of the contact resistance
ρc = ∂J
∂V −1
V→0 (1.18)
and the expression for the current density in the case of thermionic emission, derived, e.g., in [99], the specific contact resistance is given by
ρc(T E) = k qA∗T exp
qφB,eff
kT
. (1.19)
A∗ is the effective Richardson constant. Andrews and Lepselter give values forA∗ for n-type silicon of 112 Acm−2K−2 and p-type silicon of 32 Acm−2K−2 [101].
2. Thermionic field emission The reduction of the barrier thickness WB with
1.7 Metal-semiconductor contact
rising doping density leads to an increased tunneling probability: charge carriers thermally exited to energies smaller than the barrier height are able to tunnel through the barrier (see figure 1.11 center). The current-voltage characteristics for the current transport via tunneling can be approximated by the Wentzel-Kramers-Brillouin (WKB) approximation [102]. With equation 1.18 this leads to the specific contact resistance for thermionic field emission:
ρc(T F E) =c1 k and the characteristic energyE00
E00= q 2
N
m∗tr0, (1.22)
m∗t being the tunneling effective mass of the regarded charge carrier. E00 is sometimes referred to as tunneling probability [103].
3. Field emissionFor very high doping densities the barrier gets thin enough that tunneling can take place near the bottom of the conduction band. According to Yu [102] the specific contact resistance is given by
ρc(F E) =c2 k
In addition to the reduction of the barrier thickness at high doping densities the height of the barrier is decreased by an enhanced image force lowering according to equation 1.15. Therefore, at high doping densities the reduced contact resistance
through the thin barrier by field emission is further lowered due to a reduced barrier height by image force lowering.
To evaluate which current transport mechanism is dominating at a given temperature T and doping density N the characteristic energy E00 is compared with kT [91]. For kT E00 TE is the dominating mechanism, for kT ≈ E00 TFE is prevailing and for kT E00 FE dominates. For a metal-semiconductor contact to p-type silicon with m∗t/m0 ≈ 0.5 [103] andT = 300 K the doping ranges of the different transport regimes are:
• N <1·1018cm−3: thermionic emission
• 1·1018cm−3 < N <1·1020cm−3: thermionic field emission
• N >1·1020cm−3: field emission.
A model combining TFE and TE was proposed by Varahramyan [104]
ρc(T F E+F E) = ¯c k qA∗T exp
qφB,eff
E00· cothE
kT00
(1.25) with c1 and c2 replaced by one constant ¯c= 0.425 for n-type and ¯c= 0.355 for p-type silicon.
For the charge carrier tunneling effective mass different values are specified in literature.
In addition to deviations that can be attributed to different measuring conditions and techniques, the tunneling effective mass depends on doping density, temperature and crystallographic orientation of the interface [103, 105]. For contacts to n-type silicon often the valuem∗t/m0 = 0.3 is used [97, 103]. Considering holes in p-type silicon, Nget al. give values of m∗t/m0 between≈0.37−0.51 for doping densities between 1018cm−3 and 1021cm−3 on (100) surfaces. Due to the lack of exact values for m∗t, especially in the case of p-type silicon, this quantity introduces a large error for the determination of the specific contact resistance in the occurrence of tunneling.
2 Characterization Methods
In this chapter, the characterization methods most frequently used within this work are introduced. In the first section, electrical methods are presented. The following section deals with the methods applied for the microstructural investigation of the screen-printed contacts.
2.1 Electrical characterization
In the following, the measurement procedures for the characterization of solar cell char-acteristics commonly used in this work are introduced. Then the determination of the specific contact resistance by transfer length method and the emitter characterization by means of electrochemical capacitance-voltage measurement are discussed.