PN junction - doping and carrier concentration - concept and formulas
An intrinsic or pure semiconductor (eg., pure Silicon) is an insulator at T=0K. This is because all electrons are bound to their atoms and no free electron is available for conduction.
At higher temperatures (T>0K), for example, at room temperature(T=300K), some electrons gain enough thermal energy to break their bonds, cross the band gap energy and enter the conduction band. These electrons are free to conduct current.
Free electrons and holes are always generated in pairs. So, for every free electron in the conduction band, there is a hole in the valance band. Let us denote the free electron concentration byn and free hole concentration by p . Since free electrons and holes are generated in pairs, we can say, n=p=n_i where n_i is called intrinsic carrier concentration.Also, np=n_i^2
Typically, number of free electrons in an intrinsic Si at room temperature is of the order of 1010/cm3. But the number of Si atoms per cm3 is of the order of 1022. Comparing these numbers, we find that there is not a large number of free electrons available for conduction in a pure semiconductor. As a result, intrinsic semiconductors do not exhibit much conductivity. Hence, they may appear useless.
Fortunately, the conductivity of an intrinsic semiconductor can be altered by increasing the number of free electrons/holes by doping. Doping is the process of controlled addition of impurities to an intrinsic semiconductor. Doping increases the number of carriers (free electrons/holes) and hence increases the conductivity of the material. The doped semiconductor is called an "Extrinsic Semiconductor". There are two types of extrinsic semiconductors namely, n-type and p-type. n-type semiconductor is obtained by doping silicon with a group 5 element from the periodic table (eg., P, As). Silicon has 4 electrons in its outermost shell. Each of these electrons take part in covalent bonding with neighboring atoms to give stable octet configuration to Si. Group 5 elements have 5 electrons in their valence band. When doped with a group 5 element, for example, Arsenic, the As atom replaces a Si atom in the lattice. 4 valence electrons of As take part in covalent bonding and the fifth electron is free to conduct. Thus As has contributed an electron to the Si lattice, thus enhancing its conductivity. Since As donated an electron, As, and all group 5 impurities are also called as donor atoms. Similarly, a p-type semiconductor is a piece of silicon doped with a group 3 element like Boron(B). Boron has 3 electrons in its valence shell. When it bonds with silicon, one of the bond lacks a free electron and acts as a hole. Since this hole contributed by boron atom can accept an electron, Boron and all third group impurities are called acceptors.
np=n_i^2 . This equation is known as mass action law. For silicon,
n_i will be given in the question. It is useful to remember that for Si, n_i = 1.5\times10^{10}/cm^3 . The mass action law applies to extrinsic semiconductors also. For an n-type semiconductor, the mass action law can be written as, p_p=N_A and n_p=\dfrac{n_i^2}{N_A}
Similarly, for an n-type semiconductor,n_n=N_D and p_n=\dfrac{n_i^2}{N_D} With all these formulas, we are all set to solve some problems in the next post.
At higher temperatures (T>0K), for example, at room temperature(T=300K), some electrons gain enough thermal energy to break their bonds, cross the band gap energy and enter the conduction band. These electrons are free to conduct current.
Free electrons and holes are always generated in pairs. So, for every free electron in the conduction band, there is a hole in the valance band. Let us denote the free electron concentration by
Typically, number of free electrons in an intrinsic Si at room temperature is of the order of 1010/cm3. But the number of Si atoms per cm3 is of the order of 1022. Comparing these numbers, we find that there is not a large number of free electrons available for conduction in a pure semiconductor. As a result, intrinsic semiconductors do not exhibit much conductivity. Hence, they may appear useless.
Fortunately, the conductivity of an intrinsic semiconductor can be altered by increasing the number of free electrons/holes by doping. Doping is the process of controlled addition of impurities to an intrinsic semiconductor. Doping increases the number of carriers (free electrons/holes) and hence increases the conductivity of the material. The doped semiconductor is called an "Extrinsic Semiconductor". There are two types of extrinsic semiconductors namely, n-type and p-type. n-type semiconductor is obtained by doping silicon with a group 5 element from the periodic table (eg., P, As). Silicon has 4 electrons in its outermost shell. Each of these electrons take part in covalent bonding with neighboring atoms to give stable octet configuration to Si. Group 5 elements have 5 electrons in their valence band. When doped with a group 5 element, for example, Arsenic, the As atom replaces a Si atom in the lattice. 4 valence electrons of As take part in covalent bonding and the fifth electron is free to conduct. Thus As has contributed an electron to the Si lattice, thus enhancing its conductivity. Since As donated an electron, As, and all group 5 impurities are also called as donor atoms. Similarly, a p-type semiconductor is a piece of silicon doped with a group 3 element like Boron(B). Boron has 3 electrons in its valence shell. When it bonds with silicon, one of the bond lacks a free electron and acts as a hole. Since this hole contributed by boron atom can accept an electron, Boron and all third group impurities are called acceptors.
| Type of semiconductor | Majority carrier | Minority carrier |
|---|---|---|
| p-type | Holes | Free electrons |
| n-type | Free electrons | Holes |
Notation
Common questions that appear in GATE from the concept of doping ask you to solve for concentration of carriers. Before we start solving some questions, let us get familiar with the notations. The subscripts indicate the type of extrinsic semiconductor.| n | Free electron concentration in an intrinsic semiconductor/cm3 |
| p | Hole concentration in an intrinsic semiconductor/cm3 |
| n_n | Free electron concentration in an n-type semiconductor/cm3 |
| n_p | Free electron concentration in a p-type semiconductor/cm3 |
| p_p | Hole concentration in a p-type semiconductor/cm3 |
| p_n | Hole concentration in an n-type semiconductor/cm3 |
| N_D | Donor concentration in an n-type semiconductor/cm3 |
| N_A | Acceptor concentration in a p-type semiconductor/cm3 |
Formulas
Now that we are familiar with the notation, let us look at the formulas we need in order to tackle problems under this topic. For intrinsic semiconductor,n_np_n = n_i^2
Similarly, for a p-type semiconductor, the mass action law can be written as,n_pp_p = n_i^2
In general, for an extrinsic semiconductor,- majority carrier concentration x minority carrier concentration =
majority carrier concentration = dopant concentration
minority carrier concentration =
Similarly, for an n-type semiconductor,
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