The Band Model of Solids

The fundamental theory of solid state detectors is the band model of solids. The main feature of this theory is that the energy levels of the atoms within a solid are merged to form collective energy levels called bands. Due to the Pauli exclusion principle each band can host a finite number of electrons, the lowest energy levels will be fully filled while over a certain value the occupancy will be null.

The highest fully filled band is named valence band and the lowest empty or partially filled band is named conduction band. When the conduction band is partially filled the solid is a conductor, otherwise the energy gap,E_g=V_C−V_V, between these bands is used tu distinguish semiconductors and insulators, see figure 2.1. Insulators’

energy gap is usually larger than 3 eV while all the materials with a lower energy gap are called semiconductors [4]. Silicon, the most common semiconductor used in radiation detectors has aE_g ∼1.2 eV.

Figure 2.1: Scheme of the band structure of solid state materials. V_C and V_V are potentials of the conduction and valence levels, VA and VD the acceptors and donors ones [4].

The most common semiconductors are silicon and germanium, both have four va-lence electrons so that four covalent bonds are formed around each atom. Electrons can be excited from the valence band to the conduction band becoming free to move in the crystal lattice and leaving vacancies in their original positions. These vacan-cies, called holes, are positively charged and can then be filled by other electrons of the valence band leaving holes in their original positions. The holes therefore move through the crystal lattice as well, behaving as carriers of positive charge.

Electron-hole pairs are continuously generated in a semiconductor by thermal

2.1. SOLID STATE DETECTOR WORKING PRINCIPLE 7 excitation, as well as destroyed since a certain number of free electrons will recombine with holes. The concentration of electrons in the conduction band can be calculated by

n= Z ∞

EC

g_e(E)f(E) dE, (2.1)

where g_e(E) is the density of states, f(E) is the Fermi-Dirac distribution and the integral is calculated from the minimum energy level of the conduction band E_C to infinity. In an analogous way the concentration of holespcan be calculated integrating from zero to the maximum energy level in the valence band E_V:

p= Z EV

0

gh(E)f(E) dE. (2.2)

Electrons and holes are fermions obeying to the Fermi-Dirac distribution

f(E) = 1

1 +e^(E−EF)/kT, (2.3)

withEF being the Fermi energy, kthe Botlzmann constant and T the absolute tem-perature. The density of states for electrons and holes can be calculated assuming they are free to move within the semiconductor but cannot escape it. This is repre-sented by a box potential that is zero inside the boundaries of the semiconductor and infinity outside of it and can be written as

ge(E) = 1

where mn and mh are the effective masses of electrons and holes, and ~the reduced Planck constant. Integrating equation 2.1 and 2.2 the electron and hole concentrations result to be where N_C and N_V are the effective density of states in the conduction and valence band. The product of the density for electrons and holes reads

np=n²_i =N_CN_Ve^−Eg/kT (2.6) wheren_i indicates the electron and hole density in an intrinsic semiconductor, where

they are assumed to be equal, and E_g is the energy gap. Typical values for n_i in silicon at room temperature are of the order of 1.5·10¹⁰cm⁻³ [5].

The balance between the two charge carriers in intrinsic semiconductors can be modified adding dopants to the pure crystal. Introducing atoms with one more or one less valence electron an excess of electrons or holes is created. From the band structures point of view the presence of atoms with three valence electrons (acceptors), such as boron or gallium, introduces an energy level slightly above the valence band that can be populated by free electrons creating an excess of holes. The presence of atoms with five valence electrons (donors), such as phosphorus or arsenic, introduces extra electrons that cannot populate the valence band. These electrons will then fill an energy level slightly below the conduction band with en effective energy gap of about 0.05 eV in silicon. The extra electrons are easily promoted to the conduction band creating an excess of free electrons in the semiconductor. Doped semiconductors with an excess of holes are calledp-type while those with excess of electrons are called n-type.

Doped semiconductors remain electrically neutral because the electric charge of the extra electrons or holes is balanced by the nuclei having one more or less proton than silicon. The total sum of positive and negative charge density have therefore to be equal, if N_A and N_D are the acceptors and donors concentrations, the electrical balance can be written as:

N_D+p=N_A+n. (2.7)

In a p-type semiconductorpnand N_D = 0. Combining the last equation with equation 2.6 the charge carrier density can be written in terms of dopant concentration as

p'NAandn' n²_i

N_A. (2.8)

Conversely in an n-type semiconductor n ' ND and p ' n²_i/ND. Meaning that in a doped semiconductor the concentration of the majority charge carriers is approxi-mately equal to the dopant concentration.

Under the effect of an electric field E, free electrons and holes move with a drift velocity of

v_e/h=µ_e/hE, (2.9)

where µ_e/h is the mobility of electrons or holes, whose typical values for high purity silicon at room temperature are 1350 cm²/Vs and 450 cm²/Vs, respectively [6, 7].

The drift velocity cannot grow indefinitely and will reach a saturation value for large electric field. In an intrinsic semiconductor electrons and holes have the same density n_i and both contributes to the current density as J =e n_i(µ_e+µ_h)E. The current density can be expressed in terms of resistivity as J = E/ρ. The resistivity of a

2.1. SOLID STATE DETECTOR WORKING PRINCIPLE 9

semiconductor is therefore given by

ρ= 1

eni(µe+µ_h). (2.10)

In doped semiconductors the current contribution of minority charge carriers can be neglected and the charge carrier density can be replaced by the dopant density as shown in equation 2.8. As a result equation 2.10 can be written as

ρn−type= 1 eNDµe

, (2.11a)

ρp−type= 1 eNAµh

. (2.11b)

Both intrinsic and doped semiconductors have a too large density of free carriers to be used as particle detectors. But, combining semiconductors with opposed doping, a region without free charges can be created and exploited for particle detection.