9.7: Semiconductors and Doping (2024)

In the preceding section, we considered only the contribution tothe electric current due to electrons occupying states in theconduction band. However, moving an electron from the valence bandto the conduction band leaves an unoccupied state orhole in the energy structure of the valence band,which a nearby electron can move into. As these holes are filled byother electrons, new holes are created. The electric currentassociated with this filling can be viewed as the collective motionof many negatively charged electrons or the motion of thepositively charged electron holes.

To illustrate, consider the one-dimensional lattice in Figure\(\PageIndex{1}\). Assume that each lattice atom contributes onevalence electron to the current. As the hole on the right isfilled, this hole moves to the left. The current can be interpretedas the flow of positive charge to the left. The density of holes,or the number of holes per unit volume, is represented byp. Each electron that transitions into the conduction bandleaves behind a hole. If the conduction band is originally empty,the conduction electron density p is equal to the holedensity, that is, \(n = p\).

As mentioned, a semiconductor is a material with a filledvalence band, an unfilled conduction band, and a relatively smallenergy gap between the bands. Excess electrons or holes can beintroduced into the material by the substitution into the crystallattice of an impurity atom, which is an atom of a slightlydifferent valence number. This process is known as doping. Forexample, suppose we add an arsenic atom to a crystal of silicon(Figure \(\PageIndex{2a}\)).

Arsenic has five valence electrons, whereas silicon has onlyfour. This extra electron must therefore go into the conductionband, since there is no room in the valence band. The arsenic ionleft behind has a net positive charge that weakly binds thedelocalized electron. The binding is weak because the surroundingatomic lattice shields the ion’s electric field. As a result, thebinding energy of the extra electron is only about 0.02 eV. Inother words, the energy level of the impurity electron is in theband gap below the conduction band by 0.02 eV, a much smaller valuethan the energy of the gap, 1.14 eV. At room temperature, thisimpurity electron is easily excited into the conduction band andtherefore contributes to the conductivity (Figure\(\PageIndex{3a}\)). An impurity with an extra electron is known asa donor impurity, and the doped semiconductor iscalled an n-type semiconductor becausethe primary carriers of charge (electrons) are negative.

By adding more donor impurities, we can create animpurity band, a new energy band created bysemiconductor doping, as shown in Figure \(\PageIndex{3b}\). TheFermi level is now between this band and the conduction band. Atroom temperature, many impurity electrons are thermally excitedinto the conduction band and contribute to the conductivity.Conduction can then also occur in the impurity band as vacanciesare created there. Note that changes in the energy of an electroncorrespond to a change in the motion (velocities or kinetic energy)of these charge carriers with the semiconductor, but not the bulkmotion of the semiconductor itself.

Doping can also be accomplished using impurity atoms thattypically have one fewer valence electron than thesemiconductor atoms. For example, Al, which has three valenceelectrons, can be substituted for Si, as shown in Figure\(\PageIndex{2b}\). Such an impurity is known as anacceptor impurity, and the doped semiconductor iscalled a p-type semiconductor, becausethe primary carriers of charge (holes) are positive. If a hole istreated as a positive particle weakly bound to the impurity site,then an empty electron state is created in the band gap just abovethe valence band. When this state is filled by an electronthermally excited from the valence band (Figure\(\PageIndex{1a}\)), a mobile hole is created in the valence band.By adding more acceptor impurities, we can create an impurity band,as shown in Figure \(\PageIndex{1b}\).

The electric current of a doped semiconductor can be due to themotion of a majority carrier, in which holes arecontributed by an impurity atom, or due to a minoritycarrier, in which holes are contributed purely by thermalexcitations of electrons across the energy gap. In ann-type semiconductor, majority carriers are free electronscontributed by impurity atoms, and minority carriers are freeelectrons produced by thermal excitations from the valence to theconduction band. In a p-type semiconductor, the majoritycarriers are free holes contributed by impurity atoms, and minoritycarriers are free holes left by the filling of states due tothermal excitation of electrons across the gap. In general, thenumber of majority carriers far exceeds the minority carriers. Theconcept of a majority and minority carriers will be used in thenext section to explain the operation of diodes andtransistors.

Hall Effect

In studying p- and n-type doping, it isnatural to ask: Do “electron holes” really act like particles? Theexistence of holes in a doped p-type semiconductor isdemonstrated by the Hall effect. The Hall effect is the production of a potentialdifference due to the motion of a conductor through an externalmagnetic field. A schematic of the Hall effect is shown in Figure\(\PageIndex{5a}\).

A semiconductor strip is bathed in a uniform magnetic field(which points into the paper). As the electron holes move from leftto right through the semiconductor, a Lorentz force drivesthese charges toward the upper end of the strip. (Recall that themotion of the positively charged carriers is determined by theright-hand rule.) Positive charge continues to collect on the upperedge of the strip until the force associated with the downwardelectric field between the upper and lower edges of the strip(\(F_E = E_q\)) just balances the upward magnetic force (\(F_B =qvB\)). Setting these forces equal to each other, we have \(E =vB\). The voltage that develops across the strip is therefore

\[V_H = vBw, \nonumber \]

where \(V_H\) is the Hall voltage; \(v\) is the hole’sdrift velocity, or average velocity of a particlethat moves in a partially random fashion; B is themagnetic field strength; and w is the width of the strip.Note that the Hall voltage is transverse to the voltage thatinitially produces current through the material. A measurement ofthe sign of this voltage (or potential difference) confirms thecollection of holes on the top side of the strip. The magnitude ofthe Hall voltage yields the drift velocity (v) of themajority carriers.

Additional information can also be extracted from the Hallvoltage. Note that the electron current density (the amount ofcurrent per unit cross-sectional area of the semiconductor strip)is

\[j = nqv, \label{eq3} \]

where q is the magnitude of the charge, n isthe number of charge carriers per unit volume, and v isthe drift velocity. The current density is easily determined bydividing the total current by the cross-sectional area of thestrip, q is charge of the hole (the magnitude of thecharge of a single electron), and u is determined byEquation \ref{eq3}. Hence, the above expression for the electroncurrent density gives the number of charge carriers per unitvolume, n. A similar analysis can be conducted fornegatively charged carriers in an n-type material (seeFigure \(\PageIndex{5}\)).