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CHARGE CARRIERS in SEMICONDUCTORS Objectives ELEKTRONIKOS PAGRINDAI 2008 1 CHARGE CARRIERS IN SEMICONDUCTORS Objectives: • Discovery of the nature of charge carriers in intrinsic and extrinsic semiconductors • Finding on what, how and why densities of charge carriers in semiconductors depend • Calculation methods of of charge carrier densities in semiconductors Content: Charge carriers in intrinsic semiconductors Nature of charge carriers Fermi level in an intrinsic semiconductor Densities of carriers Charge carriers in extrinsic semiconductors n-type semiconductors p-type semiconductors Compensation doping Excess carriers and lifetime VGTU EF ESK [email protected] ELEKTRONIKOS PAGRINDAI 2008 2 Nature of charge carriers in intrinsic semiconductors Carefully refined semiconductors are called intrinsic semiconductors. In a silicon crystal each atom is surrounded by four neighbour atoms. At 0 K all valence electrons take part in covalent bonding and none are free to move through the crystal. At very low temperatures the valence band is full (filled to capacity) and the conduction band is completely empty. Thus, at a very low temperature (close to 0 K) the crystal behaves as an insulator. As the temperature increases, the lattice vibrations arise. Some of the energy of the lattice vibrations is transferred to the valence electrons. If sufficient energy is given to an electron, it leaves a bond and becomes free. VGTU EF ESK [email protected] ELEKTRONIKOS PAGRINDAI 2008 3 Nature of charge carriers in intrinsic semiconductors The jump of an electron from the valence band to the conduction band corresponds to the release of an electron from covalent bonding. The minimum energy required for that is equal to the width of the forbidden band. If the width of the forbidden band is less, electrons are released at lower temperature. When an electron is released, a positively charged vacancy appears. This vacancy may be considered as a positive hole . According to the energy band diagram an uncompleted allowed energy level in the valence band corresponds to a hole. VGTU EF ESK [email protected] ELEKTRONIKOS PAGRINDAI 2008 4 Nature of charge carriers in intrinsic semiconductors Above occupied levels there are unoccupied energy levels in the conduction and valence bands. The situation is similar to that in conductors: electrons are able to accept energy, change their velocity and participate in conductivity. … a hole behaves as a positive charge carrier having a positive charge equal in magnitude to the electronic charge. ... In the intrinsic semiconductor the number of electrons in the conduction band equals to the number of holes in the valence band. Charge carriers appear as a result of charge carrier generation. Positive holes attract negative electrons. If an electron is drawn into the bond, it recombines with a hole. The jump of an electron from the conduction band to the valence band corresponds to the recombination process. VGTU EF ESK [email protected] ELEKTRONIKOS PAGRINDAI 2008 5 Fermi level in an intrinsic semiconductor Wv +Wc 3 mp WF = + kT ln 2 4 mn The Fermi level in an intrinsic semiconductor lays at the middle of the forbidden band. If the Fermi level is below the bottom of the conduction band, it is possible to use the simplified formula −(W −WF k/) T fF (W ) ≅ e ... The Maxwell-Boltzmann distribution function can be used for calculation of the probability of occupation of energy levels in the conduction band. The same conclusion can be made for holes in the valence band. VGTU EF ESK [email protected] ELEKTRONIKOS PAGRINDAI 2008 6 Densities of charge carriers in intrinsic semiconductors We have derived the expression for density of electrons in a non-generate system. Now it is necessary to take into account that conduction electrons exist in a crystal. We must consider the effective mass: 2/3 2(2πmn kT ) −(Wc −WF k/) T −(Wc −WF k/) T n = e = Nc e h3 Acting in a similar way we can find density or holes in the valence band. 2/3 2(2πm kT ) p −(WF −Wv k/) T −(WF −Wv k/) T p = e = Nv e h3 −(Wc −Wv k/) T −∆W k/ T np = Nc Nv e = Nc Nv e 2 2 np = ni pi = ni = pi ... The densities of electrons and holes in an intrinsic semiconductor −∆W 2/ kT ni = pi = np = Nc N v e are strongly dependent on temperature and gap energy. VGTU EF ESK [email protected] ELEKTRONIKOS PAGRINDAI 2008 7 Densities of charge carriers in intrinsic semiconductors −∆W 2/ kT ni = pi = np = Nc N v e The number of conduction electrons and holes increases rapidly with the increase of temperature and decreases with the increase of the gap energy . 1 ∆W 1 1 ln n = ln p = ln()N N − ≅ a − b i i 2 c v 2k T T ∆W tanα ≅ 2k ... Using the expressions for the densities of electrons and holes and taking into account the condition n = p , it is possible to derive the formula for the Fermi level in an intrinsic semiconductor. VGTU EF ESK [email protected] ELEKTRONIKOS PAGRINDAI 2008 8 Densities of charge carriers in intrinsic semiconductors. Problems 1. Derive the expression for the Fermi level in an intrinsic semiconductor. 2. Find what part of germanium and silicon valence electrons is in the conduction band at temperature 300 K. 3. Find the ratio of carrier densities in germanium and silicon at room temperature ( T = 300 K). 4. Find how many times carrier density in the intrinsic germanium increases, if the temperature increases from 20 to 100 0C. Repeat the calculations for the intrinsic silicon. Comment on the results. VGTU EF ESK [email protected] ELEKTRONIKOS PAGRINDAI 2008 9 Extrinsic (doped) semiconductors The intrinsic carrier densities are very small and depend strongly on temperature. In order to fabricate devices such as diodes or transistors, it is necessary to increase the free electron or hole population. This is done intentionally doping the semiconductor, i. e. adding specific impurities in controlled amounts. Doped semiconductor are called extrinsic semiconductors . ... The percentage of impurity in non-degenerate semiconductors must be small (for example, about 10 -5 % in the substrates for integrated circuits). Then impurity atoms are isolated from each other by semiconductor atoms. In order to have necessary conductivity type, donor and acceptor impurities are used. VGTU EF ESK [email protected] ELEKTRONIKOS PAGRINDAI 2008 10 Charge carriers in the semiconductors doped with donor impurities The elements from the column V of the periodic table, e. g. phosphorus (P), arsenic (As) and antimony (Sb) are added to an intrinsic elemental semiconductor to modify the semiconductor into an n-type semiconductor. Let us consider silicon doped with phosphorus. Each pentavalent atom occupies a site usually occupied by a silicon atom. Four valence electrons are in covalent bonds, the fifth electron rotates round the positively charged impurity atom. The binding energy of this extra electron is small. In the case of phosphorus in silicon it is only about 0.044 eV. If the fifth electron gets such energy, it leaves the impurity atom and becomes free. The impurity atom becomes ionized. ... An electron and a positive ion appear as a result of ionization of a donor impurity atom. VGTU EF ESK [email protected] ELEKTRONIKOS PAGRINDAI 2008 11 Charge carriers in the semiconductors doped with donor impurities The free electron is in the conduction band. So donor energy levels are in the forbidden band near the bottom of the conduction band At room temperature each donor impurity atom donates an additional charge carrier – a negative conduction electron. According to this semiconductors that contain donor impurities are called donor-type , electronic or n-type semiconductors. Because usually impurity density is small, donor levels are isolated and are shown by the dashed line. VGTU EF ESK [email protected] ELEKTRONIKOS PAGRINDAI 2008 12 Charge carriers in the semiconductors doped with donor impurities Charge carrier densities and Fermi level in extrinsic semiconductors strongly depend on temperature and impurity density. At 0 K all allowed energy levels in the valence band are filled by electrons. All donor levels are filled by unbound electrons. The conduction band is free. So charge carriers do not exist, and the semiconductor behaves as an insulator. At 0 K the Fermi level is between the donor levels and the bottom of the conduction band. If the temperature increases, unbound electrons obtain energy and jump to the conduction band. So, in the low temperature or impurity ionisation range, density of conduction electrons increases. VGTU EF ESK [email protected] ELEKTRONIKOS PAGRINDAI 2008 13 Charge carriers in the semiconductors doped with donor impurities At some temperature, that is sufficiently lower than 300 K, all impurity atoms become ionised and the density of conduction electrons becomes equal to the donor density Nd. If the temperature further increases, in the wide range the density of conduction electrons remains constant. This temperature range is called the extrinsic range . n p = i p << p nn ≅ Nd n i i nn ... In the n-type semiconductor electrons are the majority carriers and holes are the minority carriers. VGTU EF ESK [email protected] ELEKTRONIKOS PAGRINDAI 2008 14 Charge carriers in the semiconductors doped with donor impurities In the extrinsic range nn = Nc exp[− (Wc −WF ) k/ T ] = Nd Nc WF = Wc − kT ln Nd ... The Fermi level in the extrinsic range falls as the temperature increases. As the density of the intrinsic charge carriers increases with temperature, at some temperature that is sufficiently higher than 300 K it becomes equal to Nd. At higher temperatures thermally exited intrinsic carriers predominate. Then the semiconductor obtains the properties of an intrinsic semiconductor.
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