Detecting striations via the lateral photovoltage scanning method without screening effect

The lateral photovoltage scanning method (LPS) detects doping inhomogeneities in semiconductors such as Si, Ge and SixGe1−x in a cheap, fast and nondestructive manner. LPS relies on the bulk photovoltaic effect and thus can detect any physical quantity affecting the band profiles of the sample. LPS finite volume simulation using commercial software suffer from long simulation times and convergence instabilities. We present here an open-source finite volume simulation for a 2D Si sample using the ddfermi simulator. For low injection conditions we show that the LPS voltage is proportional to the doping gradient as previous theory suggested under certain conditions. For higher injection conditions we directly show how the LPS voltage and the doping gradient differ and link the physical effect of lower local resolution to the screening effect. Previously, the loss of local resolution was assumed to be only connected to the enlargement of the excess charge carrier distribution.


Introduction
Semiconductor crystals are the very basis for any optoelectronic component such as transistors, LEDs or solar cells. To grow crystals, two well-established techniques dominate the market for single crystalline silicon. On the one hand, Czochralski grown crystals (95% market share) are cheap to produce but systematically introduce oxygen and carbon into the crystal, thus limiting their quality severely [1]. On the other hand, floating zone crystals (5% market share) produce purer crystals, however, significantly increase the production cost [2]. In order to improve the crystal growth design, it is crucial to predict the temperature distribution of the coils which heat up the raw material. Along the solid-liquid interface microscopic variations in the crystal appear, see Figure 1 (left and middle). These so-called striations can be measured even in the cooled-down crystal.
These striations can be measured via the lateral photovoltage scanning method (LPS) [3]. LPS excites the semiconductor crystal with a laser, creating a voltage difference at the sample edges which is proportional to the local doping variation. This opto-electrical measurement procedure detects doping inhomogeneities at wafer-scale and room temperature in a non-destructive fashion, see Figure 1 (right). Interestingly, besides being very cost-effective and fast, this tabletop setup is -unlike other methodsespecially suitable for low doping concentrations (10 12 cm −3 to 10 16 cm −3 ) and thus applies to a large range of doping concentrations.
Tauc predicted already in the 1950ies that the bulk photovoltaic effect, on which LPS relies, could be used to measure any physical quantity which affects the band structure of the material. This shows the potential of the LPS method. Unfortunately Tauc' assumptions (he considers for example a 1D sample) are not adequate for real measurements standards. To overcome these limitations Kayser et al. [4] simulate the LPS method for a given doping profile via a finite volume discretization of the van Roosbroeck system based on the commercial COMSOL Multiphysics toolbox. Eventually, one needs reconstruct the doping from the LPS measurements. However, for now we focus on computing the LPS voltage for a given doping. This step needs to be understood well before tackling the more complicated inverse problem. Unfortunately, the forward COMSOL simulation already requires long computation times to solve for the electron and hole densities, n, p, as well as the electric potential ψ. Additionally, it was not possible to simulate low doping concentrations.
To overcome these obstacles, we rewrite the LPS model in terms of quasi Fermi potentials ϕ n , ϕ p and the electric potential ψ . We introduce the laser profile, the sample inherent parameters of the recombination mechanisms and the charge carrier mobility model by Arora. We also explicitly describe how we implement the nonlinear boundary conditions which models the circuit. We discretize the nonlinear PDE system using a Voronoi finite volume method which we then implement via the opensource software tool ddfermi [5]. Our approach is explained in greater detail in [6].
The aim of this paper is to study the underlying assumption of the LPS method that the LPS voltage is proportional to the doping variation. In particular, it is known that only for low injection conditions this proportionality can be mathematically justified. Here, we investigate for which laser powers this relationship is no longer true [6]. Furthermore, we provide an argument that lower local resolution for higher laser powers is caused by the screening effect of the generated charge carriers.
The rest of this paper is organized as follows: In Section 2 we introduce the model and in Section 3 its discretization. Finally, we present our result in Section 4 before we conclude in Section 5.

The van Roosbroeck model
In this section, we describe first the charge transport model. We model the silicon crystal as a bounded domain Ω ⊂ R 2 . Its doping profile is given by the difference of donor and acceptor concentrations, The current densities for electrons and holes are given by J n (x), J p (x). These variables satisfy the following steady-state system, the so-called van Roosbroeck model, The first equation is a nonlinear Poisson equation. The following two continuity equations describe the charge transport in a semiconductor crystal. Assuming Boltzmann statistics, the relations between the quasi-Fermi potentials and the densities of electrons and holes are given by Here, we have denoted the conduction and valence band densities of states with N c and N v , the Boltzmann constant with k B and the temperature with T . Furthermore, E c and E v refer to the constant conduction and valence band-edge energies, respectively. Using relations (2) the current densities can be written in the following drift-diffusion form The intrinsic carrier density n i is defined via If we neglect both transport equations, and only solve the nonlinear Poisson equation in (1) for the electrostatic potential ψ eq and fixed quasi Fermi potentials ϕ n = ϕ p = 0, we say that the semiconductor is in equilibrium. The corresponding equilibrium charge densities n eq and p eq satisfy n eq p eq = n 2 i .
The stationary van Roosbroeck system is usually supplied with Dirichlet-Neumann type of boundary conditions [7].

Geometries
We will consider a 2D domain of silicon crystal. The sample has to be sufficiently longer than the mean free path of the electrons. Its necessary length depends on the assumed charge carrier life times but it can reach millimeter scale. The active region is highly centralized. For large charge carrier life times (millisecond range) it might be necessary to increase the sample length. The height of the sample is directly related to the penetration depth of the laser. For a laser with wave length λ = 685 nm, the penetration depth is approximately 15 nm. Thus our domain is given by

Generation rate
When a laser hits the crystal, some photons are reflected with constant reflectivity R. The other impinged photons create electron and hole, resulting in a generation rate defined as follows where S(x) is the shape function of the laser (normalized by Ω S(x)dx = 1) and N ph the impinging photon rate on the whole surface of the sample given by N ph = P L λ L hc . Here, P L denotes the laser power, λ L the wave length of the laser, c = 3 × 10 8 m/s the speed of light in vacuum and h = 6.6 × 10 −34 m 2 kg/s = 6.6 × 10 −34 Js the Planck constant.
The shape function in 2D is given by S(x, z) = S x (x) · S z (z) where in z direction the absorption of the laser is assumed to decay exponentially Here d A is the penetration depth, or in other words 1/d A is the absorption coefficient, which heavily depends on the laser wave length. The shape function in x-direction is given by where x 0 denotes the position of the laser along the x axis and σ L the laser spot radius.

Recombination rates
The total recombination rate is given by the sum of the following three recombination rates, i.e. the direct R dir , Auger R Aug and Shockley-Read-Hall R SRH recombination rates.

Arora mobility model
The

Coupling the device to an external circuit
So far we have modeled the silicon crystal but not the LPS setup itself. The electric current j D i flowing through the i-th ohmic contact Γ D i is defined by the surface integral According to the conservation of charge, the currents in (9) satisfy the relation We model now the voltage meter as a simple circuit having a resistance R. The network has two nodes, in which the potentials are respectively u D 1 and u D 2 . Using the formalism of the Modified Nodal Analysis (MNA) we have that the difference between the electric potentials at the nodes is given by where i D is defined in (10). Usually one of the nodes of an electric circuit is assumed to have an electric potential equal to the ground. This means in our case that we can arbitrarily set u D 1 = u ref = 0, and thus (11) reduces to Notice that (12) is an implicit equation for u LP S := u D 2 since i D depends implicitly on u D 2 , via the van Roosbroeck system (1). Once we have found a solution (ψ, ϕ n , ϕ p ) to the van Roosbroeck system where u D 2 enters as a Dirichlet boundary condition, we compute the current i D via (9). More details on this LPS model can be found in [6]. More complicated models were analyzed analytically in [8].

Finite volume discretization
We partion our domain Ω into Voronoï cells ω K such that Ω = N K=1 ω K . Each control volume is associated with a node x K ∈ ω K . Via the divergence theorem we obtain after integration over each control volume a discrete version of the continuity equation in (1). Consistent with the continuous van Roosbroeck system, this finite volume discretization describes the change of the carrier density within a control volume. The corresponding numerical electron flux j n describing the flow between neighboring control volumes can be expressed as a function, depending nonlinearly on the values ψ K , ψ L , η K , η L such that Here a function with subindex, e.g. K, denotes evaluation of the function at the node x K . More details can be found in [7]. We implement this discretization within the open-source software tool ddfermi [5], which solves the nonlinear boundary condition (12) using a secant method [6].

Results
Tauc [9] predicted, that the bulk photovoltaic effect, on which the LPS method relies, is proportional to small resistivity variations under low injection conditions. In particular, recently mathematical investiga-tions [6] showed that The authors concluded that if the integral in the above equation is approximately constant, then under certain assumptions specified in [6] the LPS voltage u LP S is proportional to the doping gradient.
Yet, especially close to the extrema of the doping gradient small discrepancies could be observed which the authors attributed to the approximations and assumptions used in deriving the approximate proportionality (13).
To understand the limits of (13), we investigate the effect of the laser power P on the LPS voltage u LP S . Changing the laser power will change the conductivity variation ∆σ(x − x 0 ) for a constant average doping concentration N D0 . For sufficiently large laser powers, we will eventually violate the low injection conditions. In Figure 3 (a-d) we show the behavior of simulated LPS scans with respect to the local doping gradient (right axis, red line) for various laser powers and the doping profile As can be seen in Figure 3 (a) the simulated LPS voltage perfectly matches the doping variation, which in turn is proportional to the resistivity variations [6]. We present the LPS voltages without a non-physical offset, which as previously stated in [4] can be related to the sample geometry.
However, in Figure 3 (b) it can already be seen that the peak maxima and minima do not perfectly match the doping variation profile anymore. This effect even increases for higher laser powers, see Figure 3 (c-d). The sinusoidal variation is superposed with an envelope function. This function might be explained by the larger diffusion of charge carriers, which are now able to reach the ohmic contacts. Therefore, the scan depends on the distance to the closest ohmic contact. Both contacts are placed at x = ±1.5 mm.
Experimental results have shown that the local resolution of the LPS method is decreasing when increasing the laser power [3]. The behavior was attributed to an enlargement of the charge carrier cloud generated by a larger laser power. But as long as the mean free path for charge carriers remains constant, the convolving effect of this charge carrier cloud stays constant.
In Figure 4, we show the ratio of LPS voltages and corresponding laser powers. For small laser powers (0.2 nW − 2 nW) the profiles are nearly overlapping, which indicates a linear dependency of the LPS voltage on the laser power in this regime. Only slightly reduced is the amplitude of the 20 nW profile (the same power used in Figure 3 (b)). For higher laser powers a large decrease in the amplitude can be observed, clearly showing that in this case the LPS voltage does not depend linearly on the laser power. It was shown analytically that this behavior is logarithmic and due to the screening effect of charge carriers [9,6]. Figure 3 (b) the LPS profile begins to no longer match the doping variations and additionally in Figure 4 the amplitude of the LPS voltage laser power ratio begins to decrease, these underlying physical effects might be connected. Therefore it is likely that the lower local resolution for higher laser powers is caused by the screening effect of the generated charge carriers. This also corroborates similar observations regarding the logarithmic relationship between LPS signal and laser power [9,6].

Conclusion
We presented simulations of the LPS measurement technique used to predict inhomogeneities in semiconductor crystals, using the open source software ddfermi. Due to flexibility of our approach, we were able to overcome efficiency and stability problems that a simulator based on commercial code has. By simulating several LPS profiles using low laser powers we could verify Tauc' assumptions that the bulk photovoltage is proportional to the doping gradient under certain conditions. Furthermore, we showed the limits of this proportionality with respect to higher laser powers. It appears that the loss in local resolution with increased laser power is connected to the screening effect of the generated charge carriers.