1 energy bands in semiconductors - universitetet i...

From MSc Thesis of Lars Gimmestad Johansen

1 Energy bands in semiconductors

Electrons in an isolated atom can only have discrete energy levels, but whenatoms are brought together as in crystalline solids, these degenerate energylevels will split into many separated levels due to the atomic interaction. Be-cause the levels are so closely separated, they may be treated as a continuousband of allowed energy states. The two highest energy bands are the valenceband and the conduction band. These bands are separated by a region whichdesignates energies that the electrons in the solid cannot possess. This regionis called the forbidden gap, or bandgap Eg. This is the energy difference be-tween the maximum valence band energy EV and the mimimum conductionband energy EC .

For insulators the valence electrons form strong bonds between neighbouringatoms. These bonds are difficult to break, and consequently there are no freeelectrons to participate in current conduction.

Bonds between neighbouring atoms in a semiconductor are only moderatelystrong. Therefore thermal vibrations may break some bonds. When a bondis broken, an electron is injected from the valence band into the conductionband. This is now a mobile negative charge carrier, and the atom from whichthe electron emerges is left with a negative charge deficiency, i.e. a positivenet charge, also called a hole. The bandgap for Si at 300 K is 1.12 eV. Acharacteristic property of semiconductors is that the bandgaps have a nega-tive temperature coefficient. The bandgap in Si and GaAs as a function oftemperature can be described by:

Eg(T ) = 1.17− 4.73× 10−4T 2

T + 636for Si (1)

Eg(T ) = 1.52− 5.4× 10−4T 2

T + 204for GaAs (2)

as seen in figure 1, where the temperature T is expressed in Kelvin. At roomtemperature (300 K) and under normal atmosphere, the values of the bangapare 1.12 eV for Si and 1.42 eV for gallium arsenide.

In conductors such as metals, the conduction band is either partially filled

Figure 1: Bandgaps of Si and GaAs as a function of temperature.

(the Fermi energy level EF is located in the middle of the conduction band)or overlaps the valence band so that there is no bandgap. As a consequence,

the uppermost electrons in the partially filled band or electrons at the top ofthe valence band can move to the next-higher available energy level when theygain kinetic energy (e.g. from an applied electric field). Therefore, currentconduction can readily occur in conductors.

2 The Effective Mass Approximation

For conduction electrons in a semiconductor, the electrons are relatively freeto move, but bound by the periodic potential of the lattice nuclei. Thus theeffective mass is different from that of a free electron. The energy-momentumrelationship of a conduction electron can be written as:

E =p̄2

2mn

(3)

where p̄2 is the crystal momentum and mn is the electron effective mass. Thecrystal momentum is analogous to the particle momentum, i.e. the particlemomentum along a given crystal direction as defined previously by the Miller

index.

For both Si and GaAs the maximum valence band energy of an electron occursat p̄ = 0. The minimum energy of the conduction band electron in GaAs alsooccurs for p̄ = 0, hence a transition across the bandgap only requires an Eg

energy absorbtion or emission. Because of this, GaAs is called a direct semi-conductor. Si, on the other hand has its minimum conduction band electronenergy along the [100] direction where p̄ 6= 0, and hence an electron transitionacross the bandgap not only requires the exchange of the energy quantum Eg,but also a change in the crystal momentum. Therefore Si is called an indirectsemiconductor. As mentioned earlier, Si is not a suitable material for buildinglight-emitting diodes and semiconductor lasers, and the reason is that thesedevices require direct semiconductors for efficient generation of photons.

With a known E − p̄ relationship, one can obtain the effective mass from thesecond derivative of E with respect to p̄ from equation (3):

mn =

(d2E

dp̄2

)−1

(4)

Values for the effective mass are 0.19m0 for Si and 0.07m0 for GaAs, where

m0 is the free electron mass. The effective-mass approximation is useful, sincethe electrons and holes then may be treated essentially as classical chargedparticles.

Because of the mutual attraction between the negative electron and the positiveatom nucleus, the potential energy of an electron increases as the distanceincreases. Therefore, if a hole is treated as a classical charged particle, it has apotential energy oppositely directed to that of an electron due to its oppositeelectric charge.

2.0.1 Intrinsic carrier concentration and Fermi energies

The electron density of a semiconductor can be obtained by first finding thedensity in an incremental energy range dE. This density n(E) is given by theproduct of the density of allowed energy states per unit volume N(E) and theprobability of occupying that energy range F (E). Thus, the conduction bandelectron density is given by integrating the product from the bottom of theconduction band EC to the top of the conduction band Etop:

n =∫ Etop

EC

n(E) dE =∫ Etop

EC

N(E)F (E) dE (5)

It can be shown by using equation (3) that the density of allowed energy statesper unit energy in the phase space is:

N(E) = 4π(

2mn

h2

)3/2

E1/2 (6)

where h is Planck’s constant. Furthermore F (E) is given by the Fermi-Diracdistribution function:

F (E) =1

1 + e(E−EF )/kT(7)

where k is the Boltzmann constant, T is the absolute temperature, and EF isthe Fermi energy level. The Fermi level is the energy at which the probabilityof occupation by an electron is exactly one half. By assuming that F (E) 'exp[−(E−EF )/kT ] for (E−EF ) > 3kT , the electron density of the conductionband in equation (5) can be shown to be:

n = NCe

(−EC−EF

kT

)where NC ≡ 2

(2πmnkT

h2

)3/2

(8)

where NC is the effective density of states in the conduction band and mn

is the electron effective mass. In a similar manner, the hole density p in thevalence band may be obtained:

p = NV e

(−EF−EV

kT

)where NV ≡ 2

(2πmpkT

h2

)3/2

(9)

where NV is the effective density of states in the valence band and mp is thehole effective mass. For an undoped semiconductor, the number of electronsper unit volume in the conduction band equals the number of holes per unitvolume in the valence band, i.e. n = p = ni, where ni is the intrinsic carrierdensity (intrinsic and extrinsic semiconductors are defined below). The Fermilevel for an intrinsic semiconductor is obtained by equating equations 8 and 9:

EF = Ei =EC + EV

2+

3kT

4ln(mp

mn

)(10)

At room temperature, the second term is much smaller than the bandgap.Hence the intrinsic Fermilevel Ei of a semiconductor generally lies very closeto the middle of the bandgap. The intrinsic carrier density ni is also obtaineddirectly from equations 8 and 9:

np = n2i (11)

n2i = NCNV e

(−EV −EC

kT

)ni =

√NCNV e

(− Eg

2kT

)(12)

The relation in equation (11) is called the mass action law, and is independentof the Fermi energy. The relation is valid for both intrinsic and extrinsic (to beexplained in section 3) semiconductors under a thermal equilibrium condition,and therefore central for studying the generation and recombination currentsin section 4.5 where the increase of one type of carrier tends to reduce thenumber of the other.

3 Doping of semiconductors

Pure semiconductors, such as a Si crystal without any impurities, are calledintrinsic semiconductors. When a semiconductor is doped with impurities,it becomes extrinsic and impurity energy levels are introduced. The dopingoccurs when some atoms in the lattice are replaced with foreign atoms, alteringthe lattice structure.

Figure 2a shows a Si lattice where one of the atoms have been replaced bya type-V atom, e.g. phosphor. Four of the phosphor valence electrons formcovalent bonds to the nearest neighbouring Si-atoms, while the fifth becomesa conduction electron that is ’donated’ to the conduction band. Thus, thematerial is called an n-type donator because of the additional negative chargecarrier, or simply an n-type material. A complementary situation is seen infigure 2b, where a Si atom is replaced by a type-III atom, e.g. boron. Here thelocal lattice deficiency of one electron can receive an electron from one of theneighbouring atoms, or a free conduction band electron. Thus, the material iscalled a p-type acceptor because of the positive charge released by acceptingan electron. The convention of viewing this occurrence as a movement of apositive charge, rather than a negative electron, will become clear in section 4.

Si+4Si

+4Si

+4Si

+4Si

+4Si

+4

+4

Si

Si

+4 +5P

CONDUCTION

(a)

ELECTRON

-q

Si

+3B

+4Si

+4Si

+4Si

+4Si

HOLE

+4Si

+4Si

+4Si

+4

+q

(b)

Figure 2: Schematic picture of bonds between atoms in an (a)n-type Si lattice with phosphor doping, and a (b) p-type Si latticewith Boron doping.

The energies associated with shallow dopants in a semiconductor are shownschematically in figure 3.

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IONS

EA

Ep

En

E

E

E

E

D

C

V

iE

g

DONORIONS

CONDUCTIONELECTRONS

CONDUCTIONHOLES

ACCEPTOR

Figure 3: Schematic picture of energies in doped lattices. Adonator is ionized by accepting a valence electron, thus ’injecting’a conduction hole to the valence band.

Conduction electrons in an n-doped semiconductor are called majority carriersand conduction holes are called minority carriers. For a p-doped semiconduc-tor, the conduction electrons become minority carriers and conduction holesbecome majority carriers.

4 Current conduction in semiconductors

4.1 Intrinsic charge carriers

Electric currents may occur in intrinsic silicon. For intrinsic silicon at roomtemperature, ni (the intrinsic carrier density) is about 1.45 × 1010cm−3. Thefree electric charges are created as the thermal ionization of atoms where co-valent bonds are broken. The free electrons may either move around in thecrystal, or recombine with positive Si ions, and under thermal equilibrium, theionization rate equals the recombination rate.

The current is seen as the net movement of electrons through the periodicpotential of the silicon lattice. As the positive charges are all fixed within theatomic nuclei of the lattice, there is no positive electric current in the samesense as for the electrons. As described in the previous section an electron ofa neighbouring atom may be captured by this positive charge, resulting in anionized atom with a positive net charge. If this step is successively repeatedin one direction, only electrons have moved, but it is still regarded a positivecurrent of electron deficiencies, i.e. a positive current of holes.

4.2 Extrinsic charge carriers

The intrinsic conductivity of Si is very low. By doping the material, theconductivity can be increased by many orders of magnitude, even by smallimpurity concentrations. Elements utilized in semiconductor doping shouldpreferrably introduce energy levels into the forbidden bandgap. A shallowdonor will have an energy level ED in the band gap close to the lowest con-duction band energy EC , and a shallow acceptors state EA is similarly locatedin the bandgap close to the highest valence band energy EV . Phosphor is ashallow dopant in silicon, and its dopant level is 0.045eV below EC (where theband gap Eg in Si is 1.12 eV as mentioned earlier). Similarly, boron introducesa shallow acceptor state in silicon, and the acceptor level of boron is 0.045 eVabove EV in silicon.

For shallow dopants in Si and GaAs, there usually is enough thermal energyto supply the energy ED or EA to ionize all donor or acceptor impurities atroom temperature, and thus provide an equal number of electrons or holesin the conduction band or valence band respectively. This condition is calledcomplete ionization, and leads to:

n = ND (13)

p = NA

where n is the electron density, ND is the donor concentration, p is the holedensity and NA is the acceptor concentration. The charge carrier densityfor Si is in a large temperature range (typically 100 - 500 K) approximatelyconstant and dominated by complete ionization of the impurity states (calledthe extrinsic region). Below this range, the density approaches 0, and abovethis range, the charge carrier density is dominated by intrinsic electrons, henceit is called the intrinsic region. Therefore, for given applications, the numberof free charge carriers may be tailored to the specific requirements by theimpurity atom doping concentration for a wide temperature range.

Assuming complete ionization of an n-doped material at room temperature,the Fermi level may be calculated by combining equations 8 and 13:

EC − EF = kT ln(NC

ND

)(14)

Similarly for a p-doped material it is easily shown by combining equations 9and 13 that:

EF − EV = kT ln(NV

NA

)(15)

It is seen that for high donor concentrations, the difference (EC − EF ) isreduced, hence the Fermi level will move closer to the bottom of the conductionband. Similarly, a high acceptor concentration will move the Fermi level closerto the valence band. This means that for extrinsic semiconductors, the Fermilevel is different from the intrinsic Fermi level, i.e. EF 6= Ei.

4.3 Drift current

4.3.1 Mobility

Under thermal equilibrium, the average thermal energy of a conduction elec-tron may be obtained from the theorem of equipartition of energy. The elec-trons in a semiconductor have three degrees of freedom, since they can move

about in three dimensional space. Therefore, the kinetic energy of the electronsis given by:

1

2mnv

2th =

3

2kT (16)

where mn is the effective electron mass, vth is the average thermal velocity,k is the Boltzmann’s constant and T is the absolute temperature. At roomtemperature the thermal velocity in equaiton 16 is about 107cm/s for Si andGaAs. Without the presence of either an electric field, or a charge gradientin the material, the random motion leads to a zero net displacement of anelectron over a sufficiently long period of time. The Drude model, introducedin 1900 by P. Drude, assumes the conduction electrons to move around freelyin a static lattice of positive ions, forming a ’gas’ of conduction electrons whichis then treated using the method of kinetic theory.

When an electric field is applied, each electron is accelerated in the field direc-tion until it undergoes a collision (considered by Drude to be a collision withthe lattice), after which its velocity is randomized again. But during the meanfree path, the momentum applied to an electron is given by −qEτc, and themomentum gained is mnvn, where vn is called the drift velocity. We have:

− qEτc = mnvn ⇒ vn = −(qτcmn

)E (17)

which states the proportionality of the drift velocity to the applied electricfield. The proportionality factor is called the electron mobility µn in units ofcm2/Vs, or:

µn ≡qτcmn

⇒ vn = −µnE (18)

A similar expression can be written for holes in the valence band (with signinversion due to the positive charge):

vp = µpE (19)

The mobility in equation (18) is related directly to the mean free time betweencollisions, which in turn is determined by the various scattering mechanisms,the two most important mechanisms being lattice scattering and impurityscattering.

Lattice scattering results from thermal vibrations of the lattice atoms at anytemperature above zero. These vibrations disturb the lattice periodic poten-tial and allow energy to be transferred between the carriers and the lattice.Since lattice vibration increases with increasing temperature, lattice scatteringbecomes dominating at high temperatures, hence the mobility decreases withincreasing temperature. Theoretical analysis shows that the mobility due tolattice scattering µL will decrease in proportion to T−3/2.

Impurity scattering results when a charge carrier travels past an ionized dopantimpurity (donor or acceptor). The charge carrier path will be deflected due toCoulomb force interaction. The probability of impurity scattering depends onthe total concentration of ionized impurities, i.e. the sum of the concentrationof the negatively and positively charged ions. However, impurity scattertingbecomes less significant at higher temperatures, since the carriers then remainnear the impurity atom for a shorter time due to higher carrier velocity. Hence,they are less effectively scattered. The mobility due to impurity scattering µI

can be shown to vary as T 3/2/NT , whereNT is the total impurity concentration.

Impurity scattering not only predicts the mobility reduction for high temper-atures, it also explains the fact that semiconductor mobility has a maximumvalue for low doping concentrations. Increasing the doping concentration above

this limit will increase the concentration of ionized impurities and thus decreasethe mobility.

4.3.2 Resistivity

The transport of carriers under the influence of an applied electric field pro-duces a current called drift current. Considering free electrons in a semicon-ductor, they will experience a force −qE which is equal to the negative gradientof the potential energy:

− qE = −dEC

dx⇒ E =

1

q

dEC

dx=

1

q

dEi

dx(20)

and if the material is homogeneous, E is constant throughout the sample. Aconduction electron is accelerated in this field, gaining kinetic energy at thesame rate as the potential energy loss. When lattice scattering occurs, theelectron loses all, or parts of its kinetic energy, and the total energy dropstowards its thermal equilibrium position. This process is repeated as long asthe field exists. Conduction by holes may be described in a similar manner,

but in the opposite direction.

Considering a sample with a homogeneous electric field and a cross-sectionalarea A, the electron current density Jn may be found by summing the productof the charge on each electron multiplied by the electron’s velocity over allelectrons per unit volume n:

Jn =InA

=n∑

i=1

(−qvi) = −qnvn = qnµnE (21)

where In is the electron current, and equation (18) is used for the relationshipbetween vn and E . A similar argument applies to holes yielding Jp = qpµpE ,thus the total current density due to the applied field may be written as:

J = Jn + Jp = (qnµn + qpµp)E = σE (22)

The quantity σ is known as the conductivity. The corresponding resistivity isdefined as the reciprocal of σ:

ρ ≡ 1

σ=

1

q(nµn + pµp)(23)

Generally, in extrinsic semiconductors, only one of the current componentsis significant due to the large difference between the two carrier densities.Therefore, equation (23) reduces to:

ρ ' 1

qnµn

for an n-type semiconductor (24)

ρ ' 1

qpµp

for a p-type semiconductor (25)

4.4 Diffusion current

Diffusion currents occur in a semiconductor at locations with a nonuniformcharge distribution, i.e. where the spatial charge distribution gradient isnonzero. In effect, the charge carriers tend to move away from a high con-centration of charge carriers of the same sign and towards a direction with

lower concentration. Diffusion currents of carriers of the opposite sign movein the opposite direction.

In regions with a nonzero spatial charge distribution gradient, drift currentswill also exist due to the electric field produced by the nonuniformity. But sincethe field originates from the electrically interacting charge carriers themselves,the field will change continuously, unlike the case for an externally appliedstatic field.

Drift and diffusion currents are superimposed components, and the diffusionmay be studied in more detail by a one dimensional first order approximation.Considering a nonuniform electron distribution along the x-axis at the pointsx = −l, x = 0 and x = l, where l is the mean free path of a particle withthermal velocity vth and mean free time τc, i.e. l = vthτc. Electrons at x = −lhave equal probability of moving a distance l in either direction within a timeperiod of τc, hence the average rate of electron flow per unit area F1 crossingx = 0 from the x = −l is:

F1 = n(−l)12

l

τc= n(−l)1

2vth (26)

where n(x) is the electron density at x. Similarly, the average rate of electronflow per unit area F2 from x = l is:

F1 = n(l)1

2vth (27)

The net rate of carrier flow F from left to right (across x = 0) is then

F = F1 − F2 =1

2vth[n(−l)− n(l)] (28)

An approximation of n(x) at x = ±l by a first order Taylor series expansiongives:

F =1

2vth

{[n(0)− l

dn

dx

]−[n(0) + l

dn

dx

]}

= −vthldn

dx≡ −Dn

dn

dx(29)

where Dn ≡ vthl is called the diffusivity. The flow of electrons results in acurrent:

Jn = −qF = qDndn

dx(30)

and it is seen that for an increasing electron concentration in positive x-direction, there is a (positive) current in the same direction, i.e. the electronsflow in the opposite direction. The same relations hold for diffusion of holes,and the total drift and diffusion current for holes and electrons are simplyadditive, giving the total conduction current density Jcond as:

Jn = qµnnE + qDndn

dx

Jp = qµppE − qDpdp

dx⇓

Jcond = Jn + Jp (31)

4.5 Generation and recombination currents

In thermal equilibrium the relationship (equation (11)) pn = n2i is valid. If

exess carriers are introduced to a semiconductor so that pn > n2i , we have a

nonequilibrium situation, and the process of introducing exess carriers is calledcarrier injection. Excess carriers may be injected by various means, includ-ing optical excitation and ionizing particles. Carrier injection by ionizationrequires energies larger than the band gap Eg in order to inject electrons fromthe valence to the conduction band.

Carrier injection by ionization produces an equal amount of excess electronsand holes ∆n = ∆p. For extrinsic semiconductors, e.g. n-doped, the injectedelectron density may be many orders of magnitude smaller than the ionizeddonor concentration ∆n � ND, but the injected hole concentration may becomparable or larger than the minority carrier concentration. This condition isreferred to as low-level injection. High-level injection is the situation where theinjected carrier concentration is comparable, or much larger than the majoritycarrier concentration.

4.5.1 Direct semiconductors

In thermal equilibrium, the electron and hole concentration in an n-dopedmaterial is nn0 and pn0 respectively. In a nonequilibrium carrier injectionsituation, the rate at which conduction electrons are generated by injection tothe conduction band is called the generation rate G. This is the sum of thethermally generated electrons from the valence band Gth and any unspecifiedmechanism that generates conduction electrons GM . The recombination rateR is proportional to the density of conduction electrons e and recombinationsites p (holes):

G = Gth +GM (32)

R = βnnpn = β(nn0 + ∆n)(pn0 + ∆p) (33)

where β is the proportionality constant. Recombination is the process where aconduction electron is captured by an ionized atom, releasing energy either asa photon or as thermal lattice energy. The net recombination rate is definedas U ≡ R − Gth = GM , and it is easily verified that for low-level injection insteady state that:

U ' β(nn0 + pn0 + ∆p)∆p ' βnn0∆p (34)

This shows that the net recombination rate is proportional to the excess minor-ity carrier concentration. Also, because of this proportionality, an exponentialdecay is expected. Considering a steady state nonequilibrium situation whereGM 6= 0 and then suddenly set to G = 0 at t = 0. The solution to thedifferential equation dpn/dt = −U = −βnn0∆p is:

pn(t) = pn0 + τpGMe−t/τp where τp ≡

1

βnn0

(35)

4.5.2 Indirect semiconductors

Since the Eg transition of an electron in an indirect semiconductor requires anonzero crystal momentum exchange, a direct generation or recombination isvery unlikely to occur. Rather an indirect process is more likely where elec-trons step via localized intermediate energy states in the forbidden bandgap.These intermediate states are called generation-recombination centers, or sim-

ply recombination centers.

The generation and recombination currents have a far more complex descrip-tion than for direct semiconductors. Electrons captured by recombinationcenters may either go to the conduction or to the valence band. This leadsto four different currents, and the probability of these transitions depend onthe electron and hole capture crossections σn and σp as well as the location ofthe recombination center Ei in the forbidden bandgap and the density of thecenters Nt. The net recombination rate U in this case may be written as:

U ≡ Ra −Rb =vthσnσpNt(pnnn − n2

i )

σp[pn + nie(Ei−Et)/kT ] + σn[nn + nie(Et−Ei)/kT ](36)

where Ra and Rb are the electron capture and electron emission rates respec-tively, and Et is the recombination energy level. However, by applying similarassumtions as for the direct semiconductor case above, and also assuming thatthe recombination centers are located near the middle of the bandgap, equation36 may be reduced to:

U ' vthσpNt(pn − pn0) (37)

which has the same form as for the direct semiconductor in equation 34.

4.6 High-Field operation

One assumption so far has been that the mean free time τc is constant, whichholds for low values of E . But for high electric fields, as vn becomes comparableto vth, this additional velocity component will result in a shorter average timebetween lattice scatterings, thus the mobility is not longer constant. Themobility will finally saturate for high electric fields, and may be approximatedby the empirical expression:

vn, vp =vs

[1 + (E0/E)]1/γ(38)

where vs is the saturation velocity (107cm/s for Si at 300 K), and E0 is aconstant equal to 7× 103V/cm for electrons and 2× 104V/cm for holes, and γis 2 for electrons and 1 for holes.

Another current component that may occur for high voltage operation of semi-conductors results from an avalanche process. This is governed by conduction

electrons with sufficient energy to ionize atoms in the lattice. These electronsgain kinetic energy from the strong electric field between each interaction withthe lattice, and the ionized electron-hole pair may gain sufficient kinetic energyin the field to ionize other atoms. This chain reaction gives rise to a very largecurrent increase for a small increase of the electric field around a critical value.

5 The p-n junction

The basic constituent of the ATLAS semiconductor tracker is the detectorelement which utilizes some important properties of the p-n junction. The p-njunction, usually called a diode, is a nonlinear element that is most commonlyused in electronics as a rectifying, current limiting or voltage stabilizing device.

5.1 Steady state thermal equilibrium operation

The diode is a two-terminal device with one n-doped and one p-doped region.During fabrication the diode is produced on a single piece of crystal, either by

epitaxial growth, diffusion or ion implantation. However, it may be useful tovisualize the junction as the face of two elements brought in close contact witheach other.

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Ip n

E

EC

V

EF

EF

EC

EV

EC

EV

EF

EE

E

F

C

V

Drift

Diffusion

Diffusion

Drift

(a) (b)

p

ε

ndrift

Idiff

Figure 4: The p-n junction in thermal equilibrium. (a) Uni-formly doped samples and their energy levels when separated. (b)Energy levels and currents in a p-n junction.

Considering the n-material initially, before the two samples are brought to-gether (fig. 4a), the n-side is electrically neutral, since an equal amount ofpositive ionized atoms and negative conduction electrons exists. After the twosamples are joined, there will be a large charge concentration gradient at thejunction due to all the free electrons that exist at the n-side of the junction,which results in a diffusion current across the junction of free electrons intothe p-region. The positively charged ionized dopant atoms at the n-side, how-ever, are bound by the crystal lattice and may not diffuse. As more and moreelectrons diffuse across the junction, a positive net charge is created at then-side, and the positive ions are said to be uncovered.

The same process occurs in the p-material, where positive holes diffuse into then-region, leaving behind uncovered bound negatively charged acceptor ions inthe p-lattice. The uncovered charges on both sides govern an electric field Eacross the junction region directed from n to p as seen in figure 4b. This fieldwill produce a drift current of free electrons and holes, which is oppositelydirected to the initial diffusion currents; electrons will drift from the p-sideacross the junction and holes will drift from the n-side towards the p-side.

In thermal equilibrium and with no bias applied, the current density of holesand electrons across the p-n junction must be zero. For the hole current density

this yields:

Jp = Jp(drift) + Jp(diffusion)

= qµppE − qDpdp

dx

= qµpp

(1

q

dEi

dx

)− kTµp

dp

dx= 0 (39)

where equation (20) and the Einstein relation Dp = kTµp/q have been used.It can be verified from equation (9) that:

p = nie(Ei−EF )/kT (40)

⇓dp

dx=

p

kT

(dEi

dx− dEF

dx

)(41)

Inserted into equation (39) yields the net hole current density:

Jp = µppdEF

dx= 0 or

dEF

dx= 0 (42)

Thus, the Fermi level must be constant throughout the sample as illustratedin figure 4b. Close to the junction on each side, there are now two regionswhere all bound excited states are left uncompensated. Going outwards onereaches the transition region where the ions are partially compensated by freecharge carriers. The region from the junction and outwards on both sides,including the transition region, is called the space charge region, or depletionregion, since it is depleted for free charge carriers (see figure 5). Outside thetransition region charge neutrality is maintained.

The transition region is usually small compared to the total space charge re-gion, thus the space charge distribution may be approximated by a rectangularwhere the n-side depletion region extends uniformly up to xn, and at the p-sideup to xp.

The electrostatic potential ψ is defined in order that its negative gradient equalsthe electric field:

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Neutral p

x

region

Neutral nregion

TransitionRegion

N -ND A

Depletion Region

xn

xp

0

Figure 5: Space charge distribution of a p-n junction withabrupt doping changes at the metallurgical junction.

E ≡ −dψdx

⇒ ψ = −Ei

q(43)

according to equation (20). In the neutral p-region the electrostatic potential

calculated with respect to the Fermi level ψp will be:

ψp ≡ −1

q(Ei − EF )

∣∣∣x≤−xp

= −kTq

lnNA

ni

(44)

where equation (40) has been used by first setting p = NA. This is valid sincePoisson’s equation yields (for the neutral region):

dEdx

=ρs

εS

=q

εS

(ND −NA + p− n) (45)

assuming all donors and acceptors to be ionized. And due to the p-type ma-terial; ND = 0 and p� n. εS is the permittivity of Si.

A similar expression may be obtained for the electrostatic potential at x > xn

in an n-type material with respect to the Fermi level:

ψn ≡ −1

q(Ei − EF )

∣∣∣x≥xn

= kT lnND

ni

(46)

Hence the total electrostatic potential difference between the p-side and the

n-side neutral regions at thermal equilibrium is called the built-in potentialVbi:

Vbi = ψn − ψp =kT

qlnNAND

n2i

(47)

5.2 Highly doped abrupt junctions

As will be described in another chapter, the p−n junction of the ATLAS SCTtracker is of a special design with a large difference in doping concentrationbetween the p and the n region. The p-material is heavily doped, in the orderof 1019cm−3, whereas the n-material is lightly doped, in the order of 1016cm−3.To indicate the heavy doping concentration, the p region is often designatedas p+-type, and the lightly doped n-region simply as n-type.

In unbiassed thermal equilibrium, the total space charge must be neutral. Andsince there are no free charge carriers in the depleted regions, the total p andn space charge on each side must be equal:

��

��

��

��

Depletion Region

Neutral nregion

N -ND A

xn0 x

px

Neutral pregion

Lightly dopedn region

Heavily dopedp region

Figure 6: Space charge distribution of a highly doped abrupt p-njunction with abrupt doping changes at the metallurgical junction.

NAxp = NDxn (48)

where the space charge regions are approximated by rectangular regions, alsocalled an abrupt junction, with −xp and xn as the lower and upper limitsrespectively. The total depletion region width d is then given by d = xn + xp.d is easily found by first calculating the electric field of the space charge regionfrom Poisson’s equation:

E = − qNA

εS(xp + x) ,−xp ≤ x ≤ 0 (49)

E = qND

εS(x− xn) , 0 ≤ x ≤ xn (50)

and then the built-in potential Vbi:

Vbi = −∫ xn

−xp

E(x) dx = −∫ 0

−xp

E(x) dx−∫ xn

0E(x) dx

=qNAx

2p

2 + εS

+qNDx

2n

2εS

(51)

and from this, the depletion width d is given by:

d =

√2εS

q

(NA +ND

NAND

)Vbi (52)

For approximate abrupt space charge regions where one impurity type is dom-inant (as for the p+–n junction mentioned above), the depletion width willalmost entirely depend on the light dopant in accordance with the above equa-tions. These junctions are called one-sided abrupt junctions, and equation (52)may be approximated by:

d ≡√

2εSVbi

qND

(53)

for a one-sided abrupt p+-n junction where NA � ND.

5.3 Biassed p-n junction in thermal equilibrium

If an additional electric field is introduced to the juntion by an external voltage,this will shift the depletion width depending on the sign of the bias voltage. Ifthe applied potential has the opposite sign as that of the built-in electrostaticpotential, the total electrostatic potential will be reduced by |VF | to Vbi + VF ,where VF is negative. The junction is now forward biassed. The reduction ofthe electrostatic potential across the junction reduces the drift current, andthere is no longer a balance between the drift and diffusion current; there willbe a nonzero net diffusion current across the junction. Since p-n junctions ofa Si microstrip detector are reverse biassed, the forward bias properties willnot be discussed any further here.

If the junction is biassed with opposite polarity, the electrostatic potential willbe additional to the built-in potential, and thus increasing the total electro-static potential. The junction is said to be reverse biassed. As a result thedepletion width will increase accordingly for a one-sided abrupt junction as:

d ≡

√√√√2εS(Vbi + V )

qNeff

(54)

where Neff is the effective doping concentration of the lightly doped bulk, andV is positive for reverse bias voltages. A method for measuring this value isoutlined in a later section. The space charge region that results from a reversebias voltage is important when using the p-n junction as a particle detector,as will be explained in a later chapter.

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