Transport of electron-hole pairs

After an electron-hole (e -h) pair has been created, the charge carriers start to drift in the electromagnetic field and to move due to the thermal diffusion.

Figure 2.2: The energy loss spectrum for2 GeVprotons traversing a Si absorber of thickness1µm as calculated in [72] (solid line). The functionf(∆)extends to a maximum value∆M = 9 MeV. The separate peaks at17,34,51...eVcorrespond to1,2,3… plasmon excitations. The Landau function for such a absorber is shown as dashed line.

2.1.2.1 Description of the electric field

Electric field inside the sensor has a complex shape. There are two approaches to model the motion of e-h pairs in the electric field:

1. To calculate a detailed map of the electric potential (taking into account strips, metallisation layers, doping, etc.) and then to solve the equation of motion for each hole and electron⃗v =µ ⃗E, whereµdenotes the mobility of the charge carrier [64–66,74,75]. The first order differential equation of motion is solved numerically with Runge-Kutta method. At each time step δt, the step

δ⃗r=⃗r(t+δt)−⃗r(t) (2.1) is evaluated.

2. To approximate the electric field with an analytic expression for a field in a planar abrupt p-n junction in overdepleted regime [61,63,66,68–70] (fig. 2.3):

|E(z)|= V_bias+V_dep

d −2z

d²V_dep, (2.2)

where dis the thickness of the sensor,V_depthe depletion voltage, and V_bias the bias voltage. Then the produced charge is projected on the readout plane.

Figure 2.3: Distribution of the electric field forVbias< Vdep(left) andVbias> Vdep(right) (taken from [61] and modified).

The current induced at time t on the kth electrode by a moving charge carrier is given by the Shockley-Ramo theorem:

i_k(t) =−q⃗v·E⃗_wk, (2.3) where q is the charge of the carrier, ⃗v its velocity, and E⃗_wk the weighting field associated to thekth electrode, which is determined by setting thekth electrode to a unit potential and all others — to zero potential [64–66,74, 75] (fig. 2.4).

Figure 2.4: Drift (left) and weighting (right) potentials of a two-dimensional strip detector with: the lateral extent of1000µm, the thickness of300µm, the strip pitch of 50µm and the strip width of 20µm[64].

If the integration time of a pre-amplifier is larger than the collection time of all the charge carriers, the integrated charge is the full deposited charge. In this case, the measured electrode gets the current induced only by those carriers, which moving terminates on this very electrode [61]. The other electrodes get zero net current.

2.1.2.2 Diffusion

There are several methods to model diffusion from the most detailed to the simplest ones:

1. Additionally to the drift in electric field, a small shift of each charge carrier in random direction due to diffusion can be modelled. There are two methods to evaluate it:

(a) following [64, 76], the velocity of charge carrier is assumed to obey the Boltzmann distribution: with the mean thermal velocity:

v =

(3k_bT m_eff

)1/2

,

where k_b is Boltzmann’s constant, T the temperature, m_eff the effective mass. To get v, a uniformly distributed random number ξ is converted to random number distributed according eq. 2.4:

v_abs =

A random direction for the movement is generated by a random unit vector.

(b) The random walk can be described with the Gaussian law [74]:

dN

wheredN/N is the fraction of particles in the line element drat distance r from the production point and after time t. The standard deviation is given by:

σ =√

2Dt, D= kT

e µ, (2.6)

D is called the diffusion coefficient, T the temperature, ethe elementary charge, and µthe mobility of the charge carrier. The diffusion effect can be taken into account during the solution of the equation of motion: the stepδ⃗rfrom eq. (2.1) can be decomposed as: δ⃗r =δ⃗r_E+δ⃗r_D, whereδ⃗r_E is the step due to the electric field (that is evaluated with the Runge-Kutta

method) and δ⃗rD is induced by the diffusion. The δ⃗rD term is simulated with 3 numbers (σ_i) that represent the motion into 3 directions:

δ⃗r_D =

where σ_i are three random values distributed normally around zero with the width parameter given by eq. (2.6). Here, θ and ϕ are the angles of the direction of motion with respect to the electric field lines.

The additional random velocity calculated with the method from [64] or from [74]

is used during induced current evaluation in eq. (2.3).

2. If the integration time is larger than the collection time, one can simplify the estimation of diffusion influence to the charge collection. Gaussian distribu-tion (2.5) can be used to estimate the cloud increasing size during the whole drift time. In the electric field given by (2.2), the drift time for a charge carrier can be estimated as [61]

t_e = d²

where V_depl is the depletion voltage, V_bias the bias voltage, µ the mobility, d the sensor thickness, z the initial position of the charge carrier along the axis perpendicular to the sensor plane. Since the collection time t∝1/µ, the resultingσin (2.6) doesn’t depend on the type of the charge carrier. A fraction of the charge package collected in strip i is

f_i = 1

where ∆_i is a distance from the charge package to the strip centre,pthe strip pitch [70].

3. The simplest method to model the diffusion is to assign the Gaussian profile to the charge in the direction perpendicular to the incident particle track. The Gaussian width corresponds to the characteristic range of charge diffusion while the charge is collected by electrodes (2µmin [65]). The evolution of the diffusion during the collection time is ignored.

2.1.2.3 Magnetic field

In presence of the magnetic field orthogonal to the electric field in the sensor, the charge carriers do not longer drift straight to the electrodes but are deflected due to the Lorentz force (see fig. 2.5, [63, 64, 75]). The direction of the movement is then changed by the Lorentz angle θ_L. Since the velocities of electrons and holes are different, the Lorentz forces acting on them will also differ:

tanθ_L,i = ∆x_i

d =µⁱ_HB =µⁱr_HB,

∆xi =µⁱ(E, T)rHBd. (2.9) Here, µ_H is the Hall mobility, µthe drift mobility,r_H the Hall scattering factor, ∆x the Lorentz shift,i denotes the carrier type (e – electrons or h – holes) [77].

Figure 2.5: An illustration of the Lorentz shift. Electrons and holes travel at different angles. This angle also depends on the electric field.

The drift mobility depends on the electric field and temperature

µ=µ_low (

1 +

(µ_lowE v_sat

)β)₋1/β

,

whereµ_low,β, and v_sat are temperature-dependent parameters and are different for electrons and holes (see [77]), the electric field E can be determined at current z-coordinate using eq. (2.2). Thus, the value of the Lorentz shift depends not only on the type of the charge carrier but also on the initial coordinate.

In order to take a magnetic field into account, the following approximations can be involved:

1. Solving the equation of motion of the charge carries (2.1) one can include a shift at each step. In this case, a proper value of the electric field is taken into account at each step [65].

2. Calculate the mean electric field along the trajectory of each charge carrier and use this value to calculate the Lorentz shift.

3. Assume that the shift angle is the same for all charge carriers of one type and calculate the shift according to this angle [66].