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.

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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.

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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(2πm kT ) 2/3 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.

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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.

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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.

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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.

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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.

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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.

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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. In the high temperature or intrinsic range the Fermi level approaches the mid- gap position.

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Charge carriers in the semiconductors doped with donor impurities

... In the impurity ionisation (low temperature ) range the density of electrons increases with temperature. In the extrinsic range (at middle temperatures ) the density of electrons is almost constant. At last in the intrinsic range (at high temperatures ) the intrinsic carriers predominate and their density increases with temperature. Plotting the variable 1/ T along the x-axis and ln n and ln p along the y-axis, we obtain curves that may be approximated by the segments of a straight line. Wc −Wd fF (Wd )= 2/1 n ≅ Nd 2/ Ts ≅ kln()2Nc / Nd ∆W T = ni = Nd i 2 k ln(Nc Nv / Nd )

VGTU EF ESK [email protected] ELEKTRONIKOS PAGRINDAI 2008 16

Charge carriers in the semiconductors doped with donor impurities

... In the vicinity of room temperature the conduction electron density in the n- type semiconductor is approximately constant and equals to the density of donor impurity atoms. The hole density is much lower than the intrinsic density and is strongly dependent upon temperature.

At normal temperature carrier densities and Fermi level depend on impurity density. n p = i p << p nn ≅ Nd n i i nn

If the donor density is higher, the Fermi level is higher, closer to the bottom of the conduction band; the electron density is higher, the hole density is lower. If the donor concentration is lower, the Fermi level is lower, closer to the middle of the forbidden band.

... In the n-type semiconductor the Fermi level is always over the middle of the forbidden band.

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Charge carriers in the semiconductors doped with donor impurities. Problems

1. How many charges and how many charge carriers appear as a result of ionization of semiconductor atom? 2. How many charges and how many charge carriers appear as a result of donor atom ionization? 3. A germanium specimen is doped with phosphorus. Its density is 10 16 cm –3. Find densities of electrons and holes and position of the Fermi level at temperature 300 K. 4. Silicon is doped with phosphorus. The Fermi level is in the forbidden band at the distance of 0,044 eV from the bottom of the conduction band. Impurity density is 10 16 cm –3. Find the position of the Fermi level at temperature 300 K.

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Charge carriers in the semiconductors doped with acceptor impurities Semiconductors with majority charge carrier density of holes are called p-type. They can be produced adding impurities from column III of the periodic table, e. g. boron (B), aluminium (Al), gallium (Ga) or indium (In) to intrinsic silicon or germanium. These p-type impurities are characterized by three valence electrons in their outer shell.

Let us consider silicon doped with boron. The vacancy thus created by the impurity is not a hole, since it is bound to the atom.

At some temperature above 0 K the electron from a bond of a neighbouring parent atom can fill the vacant site leaving a hole. So dopants from column III accept electrons to create holes for conduction. Therefore they are called acceptors .

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Charge carriers in the semiconductors doped with acceptor impurities The energy required by a valence electron to fill the vacancy created by an impurity atom and thus to create a hole is of the similar magnitude to the ionisation energy of a donor atom. So acceptors create discrete acceptor levels just above the top of the valence band.

At 0 K acceptor levels are free. Electrons occupy the valence band of the semiconductor. There are no charge carriers. Semiconductor has properties of insulator. If the temperature increases, electrons jump from the valence band to the acceptor levels leaving holes in the valence band. So in the impurity ionisation range the density of holes increases with temperature.

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Charge carriers in the semiconductors doped with acceptor impurities

In the extrinsic range : pp ≅ Na 2 2 ni ni Density of electrons is small and increases with temperature: np = ≅ pp Na

... Positive holes are the majority carriers and electrons are the minority carriers in the p-type semiconductor.

As the temperature in the extrinsic range is raised, the Fermi level increases.

Nv WF −Wv = kT ln Na

At high temperatures the material becomes intrinsic and the Fermi level approaches midway between the conduction and valence bands.

VGTU EF ESK [email protected] ELEKTRONIKOS PAGRINDAI 2008 21

Charge carriers in the semiconductors doped with acceptor impurities

1 – low temperature, impurity ionization range; 2 – middle temperature, extrinsic conductivity range; 3 – high temperature, intrinsic conductivity range In the extrinsic range:

n2 n2 N n = i ≅ i v pp ≅ Na p WF −Wv = kT ln pp Na Na

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Charge carriers in the semiconductors doped with acceptor impurities

At normal temperature, Fermi level and carrier densities depend on the impurity density.

2 pp ≅ Na np ≅ ni / Na

Nv WF −Wv = kT ln Na

If acceptor density is higher, the Fermi level is lower, closer to the top of the valence band; the hole density is higher, the electron density is lower. If the acceptor density is lower, the Fermi level is higher, closer to the middle of the forbidden band.

... The Fermi level below the middle of the forbidden band is the characteristic feature of the band model of the p-type semiconductor.

VGTU EF ESK [email protected] ELEKTRONIKOS PAGRINDAI 2008 23

Charge carriers in the semiconductors doped with acceptor impurities. Problems

1. How many charges and how many charge carriers appear as a result of acceptor ionization? 2. Silicon plate is doped by boron. Its density is 10 16 cm –3. Find carrier densities and Fermi level at temperature 300 K.

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Compensation doping When both acceptor and donor impurities are added simultaneously to an intrinsic semiconductor, the compensation takes place. At higher donor density the crystal is n-type semiconductor since n-type impurity predominates. The free carriers supplied by the less concentrated dopant recombine with an equal number of carriers of the opposite type. So some of the donor states are cancelled by acceptor states.. If The process is called compensation .

At higher donor impurity density when ND '>> N A ,' 2 nn = Nd ≅ Nd '−Na ' pn = ni / Nd

If N A'>> ND ,' 2 pp = Na ≅ Na '−Nd ' np = ni / Na

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Compensation doping

p-type material can be converted to the n-type and vice-versa, by the addition of excess dopant atoms of the appropriate type.

If ni > ND − N A , the intrinsic charge carriers predominate. The semiconductor has properties of the intrinsic compensated or near-fully compensated material. The Fermi level lies near the middle of the forbidden gap.

Compensation doping is widely used in manufacturing of semiconductor devices and integrated circuits.

The compensation is possible, if impurity density is not very high. Then the distance between impurity atoms is relatively great and impurity atoms cannot interact. If impurity density increases, the distance between impurity atoms decreases. When impurity density in silicon becomes approximately 10 19 cm -3, degeneration of the semiconductor arises. Then impurity levels split and allowed bands appear.

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Compensation doping

W In the degenerate n-type semiconductor, conduction and donor bands overlap and form the hybrid WF conduction band. Energy levels at the bottom of the Wc hybrid conduction band are occupied by electrons. The Fermi level is above the bottom of the conduction band. Wv In the degenerate p-type semiconductor, we have overlapping of the valence and acceptor bands. Then energy levels at the top of the hybrid valence W band are not occupied by electrons. The Fermi level is below the top of the valence band Wc

In both cases there are unoccupied allowed energy levels over the levels that are filled by electrons. The Wv situation is very similar to that in conductors. WF Therefore conduction is possible even at 0 K in degenerate semiconductors and they are sometimes called semimetals .

VGTU EF ESK [email protected] ELEKTRONIKOS PAGRINDAI 2008 27

Compensation doping. Problem

Germanium is doped with phosphorus and boron. Their densities are 10 17 cm –3 and 10 16 cm -3, respectively. Find carrier densities and position of the Fermi level at 300 K.

VGTU EF ESK [email protected] ELEKTRONIKOS PAGRINDAI 2008 28

Excess carriers and lifetime

Under thermal equilibrium the generation rate and the recombination rate are equal and the carrier densities and remain constant. The equilibrium may be disturbed by light or carrier injection.

Excess electrons and holes are always equal in number: 2 n = n0 + ∆n p = p0 + ∆p ∆n = ∆p. pn > ni . If the external excitation (activation) stops, the density of the excess carriers reduces exponentially:

∆n(t) = ∆p(t) = ∆n0 exp(−t / τ) = ∆p0 exp(−t / τ)

The recombination of the excess carriers may be radiative or nonradiative, direct or indirect, band-to-band or through recombination centres and traps.

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Excess carriers and lifetime

n0 = nn + ∆n0, p0 = pn + ∆p0.

∆n(t) = ∆p(t) = ∆n0 exp(−t / τ) = ∆p0 exp(−t / τ)

The lifetime τ represents the average time a carrier remains free before it recombines. During the lifetime the number of excess carriers reduces e times.

The recombination rate of the excess carriers is dependent on the lifetime: d n d(∆n) ∆n d p d(∆p) ∆p = ... = − = ... = − dt dt τ dt dt τ

In practical device design small lifetime may be desirable for high-speed switching applications. Nanosecond switching speed is realised using gold doping.

VGTU EF ESK [email protected]