From MSc Thesis of Lars Gimmestad Johansen

1

Energy bands in semiconductors

Electrons in an isolated atom can only have discrete energy levels, but when atoms are brought together as in crystalline solids, these degenerate energy levels will split into many separated levels due to the atomic interaction. Because the levels are so closely separated, they may be treated as a continuous band of allowed energy states. The two highest energy bands are the valence band and the conduction band. These bands are separated by a region which designates energies that the electrons in the solid cannot possess. This region is called the forbidden gap, or bandgap Eg . This is the energy difference between the maximum valence band energy EV and the mimimum conduction band energy EC . For insulators the valence electrons form strong bonds between neighbouring atoms. These bonds are difficult to break, and consequently there are no free electrons to participate in current conduction.

Bonds between neighbouring atoms in a semiconductor are only moderately strong. Therefore thermal vibrations may break some bonds. When a bond is broken, an electron is injected from the valence band into the conduction band. This is now a mobile negative charge carrier, and the atom from which the electron emerges is left with a negative charge deficiency, i.e. a positive net charge, also called a hole. The bandgap for Si at 300 K is 1.12 eV. A characteristic property of semiconductors is that the bandgaps have a negative temperature coefficient. The bandgap in Si and GaAs as a function of temperature can be described by: 4.73 × 10−4 T 2 T + 636 5.4 × 10−4 T 2 Eg (T ) = 1.52 − T + 204 Eg (T ) = 1.17 −

for Si

(1)

for GaAs

(2)

as seen in figure 1, where the temperature T is expressed in Kelvin. At room temperature (300 K) and under normal atmosphere, the values of the bangap are 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 of the valence band can move to the next-higher available energy level when they gain kinetic energy (e.g. from an applied electric field). Therefore, current conduction can readily occur in conductors.

2

The Effective Mass Approximation

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

p¯2 2mn

(3)

where p¯2 is the crystal momentum and mn is the electron effective mass. The crystal momentum is analogous to the particle momentum, i.e. the particle momentum 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 occurs at p¯ = 0. The minimum energy of the conduction band electron in GaAs also occurs for p¯ = 0, hence a transition across the bandgap only requires an Eg energy absorbtion or emission. Because of this, GaAs is called a direct semiconductor. Si, on the other hand has its minimum conduction band electron energy along the [100] direction where p¯ 6= 0, and hence an electron transition across 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 indirect semiconductor. As mentioned earlier, Si is not a suitable material for building light-emitting diodes and semiconductor lasers, and the reason is that these devices require direct semiconductors for efficient generation of photons. With a known E − p¯ relationship, one can obtain the effective mass from the second derivative of E with respect to p¯ from equation (3):

mn =

d2 E d¯ p2

!−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, since the electrons and holes then may be treated essentially as classical charged particles. Because of the mutual attraction between the negative electron and the positive atom nucleus, the potential energy of an electron increases as the distance increases. Therefore, if a hole is treated as a classical charged particle, it has a potential energy oppositely directed to that of an electron due to its opposite electric charge. 2.0.1

Intrinsic carrier concentration and Fermi energies

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

n=

Z Etop

n(E) dE =

EC

Z Etop

N (E)F (E) dE

(5)

EC

It can be shown by using equation (3) that the density of allowed energy states per unit energy in the phase space is: 2mn N (E) = 4π h2 

3/2

E 1/2

(6)

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

1 1 + e(E−EF )/kT

(7)

where k is the Boltzmann constant, T is the absolute temperature, and EF is the Fermi energy level. The Fermi level is the energy at which the probability of 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 conduction band in equation (5) can be shown to be:



n = NC e

−

EC −EF kT



where

2πmn kT NC ≡ 2 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 the valence band may be obtained: 

p = NV e

−

EF −EV kT



where

2πmp kT NV ≡ 2 h2

!3/2

(9)

where NV is the effective density of states in the valence band and mp is the hole effective mass. For an undoped semiconductor, the number of electrons per unit volume in the conduction band equals the number of holes per unit volume in the valence band, i.e. n = p = ni , where ni is the intrinsic carrier density (intrinsic and extrinsic semiconductors are defined below). The Fermi level for an intrinsic semiconductor is obtained by equating equations 8 and 9: EC + EV 3kT mp + ln 2 4 mn 

EF = Ei =



(10)

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

np = n2i

(11) 

n2i = NC NV e ni =

q

E −E − VkT C



NC NV e

E

g − 2kT





(12)

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

3

Doping of semiconductors

Pure semiconductors, such as a Si crystal without any impurities, are called intrinsic semiconductors. When a semiconductor is doped with impurities, it becomes extrinsic and impurity energy levels are introduced. The doping occurs when some atoms in the lattice are replaced with foreign atoms, altering the lattice structure. Figure 2a shows a Si lattice where one of the atoms have been replaced by a type-V atom, e.g. phosphor. Four of the phosphor valence electrons form covalent bonds to the nearest neighbouring Si-atoms, while the fifth becomes a conduction electron that is ’donated’ to the conduction band. Thus, the material is called an n-type donator because of the additional negative charge carrier, or simply an n-type material. A complementary situation is seen in figure 2b, where a Si atom is replaced by a type-III atom, e.g. boron. Here the local lattice deficiency of one electron can receive an electron from one of the neighbouring atoms, or a free conduction band electron. Thus, the material is called a p-type acceptor because of the positive charge released by accepting an electron. The convention of viewing this occurrence as a movement of a positive charge, rather than a negative electron, will become clear in section 4.

+4 Si

+4 Si

+4 Si

+4 Si

+4 Si

CONDUCTION ELECTRON

+4 Si HOLE

-q +q +4 Si

+5 P

+4 Si

+4 Si

+3 B

+4 Si

+4 Si

+4 Si

+4 Si

+4 Si

+4 Si

+4 Si

(a)

(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 lattice with Boron doping.

The energies associated with shallow dopants in a semiconductor are shown schematically in figure 3. EC ED

11111111111111111111111 00000000000000000000000 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111

CONDUCTION ELECTRONS

11111111111111111111111 00000000000000000000000 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111

DONOR IONS Eg

Ei

Ep

En ACCEPTOR IONS EA

11111111111111111111111 00000000000000000000000 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 Figure 3: Schematic picture of energies in doped lattices. A donator is ionized by accepting a valence electron, thus ’injecting’ a conduction hole to the valence band. EV

CONDUCTION HOLES

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

4 4.1

Current conduction in semiconductors Intrinsic charge carriers

Electric currents may occur in intrinsic silicon. For intrinsic silicon at room temperature, ni (the intrinsic carrier density) is about 1.45 × 1010 cm−3 . The free electric charges are created as the thermal ionization of atoms where covalent bonds are broken. The free electrons may either move around in the crystal, or recombine with positive Si ions, and under thermal equilibrium, the ionization rate equals the recombination rate. The current is seen as the net movement of electrons through the periodic potential of the silicon lattice. As the positive charges are all fixed within the atomic nuclei of the lattice, there is no positive electric current in the same sense as for the electrons. As described in the previous section an electron of a neighbouring atom may be captured by this positive charge, resulting in an ionized atom with a positive net charge. If this step is successively repeated in one direction, only electrons have moved, but it is still regarded a positive current 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, the conductivity can be increased by many orders of magnitude, even by small impurity concentrations. Elements utilized in semiconductor doping should preferrably introduce energy levels into the forbidden bandgap. A shallow donor will have an energy level ED in the band gap close to the lowest conduction band energy EC , and a shallow acceptors state EA is similarly located in the bandgap close to the highest valence band energy EV . Phosphor is a shallow dopant in silicon, and its dopant level is 0.045eV below EC (where the band gap Eg in Si is 1.12 eV as mentioned earlier). Similarly, boron introduces a shallow acceptor state in silicon, and the acceptor level of boron is 0.045 eV above EV in silicon. For shallow dopants in Si and GaAs, there usually is enough thermal energy to supply the energy ED or EA to ionize all donor or acceptor impurities at room temperature, and thus provide an equal number of electrons or holes in the conduction band or valence band respectively. This condition is called complete ionization, and leads to:

n = ND p = NA

(13)

where n is the electron density, ND is the donor concentration, p is the hole density and NA is the acceptor concentration. The charge carrier density for Si is in a large temperature range (typically 100 - 500 K) approximately constant and dominated by complete ionization of the impurity states (called the extrinsic region). Below this range, the density approaches 0, and above this range, the charge carrier density is dominated by intrinsic electrons, hence it is called the intrinsic region. Therefore, for given applications, the number of free charge carriers may be tailored to the specific requirements by the impurity 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 9 and 13 that: EF − EV = kT ln



NV NA



(15)

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

4.3 4.3.1

Drift current Mobility

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

about in three dimensional space. Therefore, the kinetic energy of the electrons is given by: 1 3 2 mn vth = kT 2 2

(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 room temperature the thermal velocity in equaiton 16 is about 107 cm/s for Si and GaAs. Without the presence of either an electric field, or a charge gradient in the material, the random motion leads to a zero net displacement of an electron over a sufficiently long period of time. The Drude model, introduced in 1900 by P. Drude, assumes the conduction electrons to move around freely in a static lattice of positive ions, forming a ’gas’ of conduction electrons which is then treated using the method of kinetic theory. When an electric field is applied, each electron is accelerated in the field direction until it undergoes a collision (considered by Drude to be a collision with the lattice), after which its velocity is randomized again. But during the mean free path, the momentum applied to an electron is given by −qEτc , and the momentum gained is mn vn , where vn is called the drift velocity. We have:

− qEτc = mn vn

⇒

vn = −



qτc E mn 

(17)

which states the proportionality of the drift velocity to the applied electric field. The proportionality factor is called the electron mobility µn in units of cm2 /Vs, or: µn ≡

qτc mn

⇒

vn = −µn E

(18)

A similar expression can be written for holes in the valence band (with sign inversion due to the positive charge): v p = µp E

(19)

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

Lattice scattering results from thermal vibrations of the lattice atoms at any temperature above zero. These vibrations disturb the lattice periodic potential and allow energy to be transferred between the carriers and the lattice. Since lattice vibration increases with increasing temperature, lattice scattering becomes dominating at high temperatures, hence the mobility decreases with increasing temperature. Theoretical analysis shows that the mobility due to lattice scattering µL will decrease in proportion to T −3/2 . Impurity scattering results when a charge carrier travels past an ionized dopant impurity (donor or acceptor). The charge carrier path will be deflected due to Coulomb force interaction. The probability of impurity scattering depends on the total concentration of ionized impurities, i.e. the sum of the concentration of the negatively and positively charged ions. However, impurity scatterting becomes less significant at higher temperatures, since the carriers then remain near 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 , where NT is the total impurity concentration. Impurity scattering not only predicts the mobility reduction for high temperatures, it also explains the fact that semiconductor mobility has a maximum value for low doping concentrations. Increasing the doping concentration above

this limit will increase the concentration of ionized impurities and thus decrease the mobility. 4.3.2

Resistivity

The transport of carriers under the influence of an applied electric field produces a current called drift current. Considering free electrons in a semiconductor, they will experience a force −qE which is equal to the negative gradient of the potential energy: − qE = −

dEC dx

⇒

E=

1 dEC 1 dEi = q dx q dx

(20)

and if the material is homogeneous, E is constant throughout the sample. A conduction electron is accelerated in this field, gaining kinetic energy at the same rate as the potential energy loss. When lattice scattering occurs, the electron loses all, or parts of its kinetic energy, and the total energy drops towards its thermal equilibrium position. This process is repeated as long as the 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-sectional area A, the electron current density Jn may be found by summing the product of the charge on each electron multiplied by the electron’s velocity over all electrons per unit volume n: Jn =

n In X = (−qvi ) = −qnvn = qnµn E A i=1

(21)

where In is the electron current, and equation (18) is used for the relationship between vn and E. A similar argument applies to holes yielding Jp = qpµp E, 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 is defined as the reciprocal of σ:

ρ≡

1 1 = σ q(nµn + pµp )

(23)

Generally, in extrinsic semiconductors, only one of the current components is significant due to the large difference between the two carrier densities. Therefore, equation (23) reduces to: 1 for an n-type semiconductor qnµn 1 for a p-type semiconductor ρ ' qpµp ρ '

4.4

(24) (25)

Diffusion current

Diffusion currents occur in a semiconductor at locations with a nonuniform charge distribution, i.e. where the spatial charge distribution gradient is nonzero. In effect, the charge carriers tend to move away from a high concentration of charge carriers of the same sign and towards a direction with

lower concentration. Diffusion currents of carriers of the opposite sign move in the opposite direction. In regions with a nonzero spatial charge distribution gradient, drift currents will also exist due to the electric field produced by the nonuniformity. But since the field originates from the electrically interacting charge carriers themselves, the field will change continuously, unlike the case for an externally applied static field. Drift and diffusion currents are superimposed components, and the diffusion may be studied in more detail by a one dimensional first order approximation. Considering a nonuniform electron distribution along the x-axis at the points x = −l, x = 0 and x = l, where l is the mean free path of a particle with thermal velocity vth and mean free time τ c, i.e. l = vth τ c. Electrons at x = −l have equal probability of moving a distance l in either direction within a time period of τc , hence the average rate of electron flow per unit area F1 crossing x = 0 from the x = −l is: F1 = n(−l)

1 l 1 = n(−l) vth 2 τc 2

(26)

where n(x) is the electron density at x. Similarly, the average rate of electron flow per unit area F2 from x = l is: 1 F1 = n(l) vth 2

(27)

The net rate of carrier flow F from left to right (across x = 0) is then 1 F = F1 − F2 = vth [n(−l) − n(l)] 2

(28)

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

("

#

"

dn dn n(0) − l − n(0) + l dx dx dn dn = −vth l ≡ −Dn dx dx

1 F = vth 2

#)

(29)

where Dn ≡ vth l is called the diffusivity. The flow of electrons results in a current: Jn = −qF = qDn

dn dx

(30)

and it is seen that for an increasing electron concentration in positive xdirection, there is a (positive) current in the same direction, i.e. the electrons flow in the opposite direction. The same relations hold for diffusion of holes, and the total drift and diffusion current for holes and electrons are simply additive, giving the total conduction current density Jcond as: dn dx dp = qµp pE − qDp dx ⇓ = Jn + Jp

Jn = qµn nE + qDn Jp Jcond

(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 called carrier injection. Excess carriers may be injected by various means, including optical excitation and ionizing particles. Carrier injection by ionization requires energies larger than the band gap Eg in order to inject electrons from the valence to the conduction band. Carrier injection by ionization produces an equal amount of excess electrons and holes ∆n = ∆p. For extrinsic semiconductors, e.g. n-doped, the injected electron density may be many orders of magnitude smaller than the ionized donor concentration ∆n  ND , but the injected hole concentration may be comparable or larger than the minority carrier concentration. This condition is referred to as low-level injection. High-level injection is the situation where the injected carrier concentration is comparable, or much larger than the majority carrier concentration.

4.5.1

Direct semiconductors

In thermal equilibrium, the electron and hole concentration in an n-doped material is nn0 and pn0 respectively. In a nonequilibrium carrier injection situation, the rate at which conduction electrons are generated by injection to the conduction band is called the generation rate G. This is the sum of the thermally generated electrons from the valence band Gth and any unspecified mechanism that generates conduction electrons GM . The recombination rate R is proportional to the density of conduction electrons e and recombination sites p (holes):

G = Gth + GM R = βnn pn = β(nn0 + ∆n)(pn0 + ∆p)

(32) (33)

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

U ' β(nn0 + pn0 + ∆p)∆p ' βnn0 ∆p

(34)

This shows that the net recombination rate is proportional to the excess minority carrier concentration. Also, because of this proportionality, an exponential decay is expected. Considering a steady state nonequilibrium situation where GM 6= 0 and then suddenly set to G = 0 at t = 0. The solution to the differential equation dpn /dt = −U = −βnn0 ∆p is: pn (t) = pn0 + τp GM e−t/τp

4.5.2

where

τp ≡

1 βnn0

(35)

Indirect semiconductors

Since the Eg transition of an electron in an indirect semiconductor requires a nonzero crystal momentum exchange, a direct generation or recombination is very unlikely to occur. Rather an indirect process is more likely where electrons 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 description than for direct semiconductors. Electrons captured by recombination centers may either go to the conduction or to the valence band. This leads to four different currents, and the probability of these transitions depend on the electron and hole capture crossections σn and σp as well as the location of the recombination center Ei in the forbidden bandgap and the density of the centers Nt . The net recombination rate U in this case may be written as: U ≡ Ra − Rb =

vth σn σp Nt (pn nn − n2i ) σp [pn + ni e(Ei −Et )/kT ] + σn [nn + ni e(Et −Ei )/kT ]

(36)

where Ra and Rb are the electron capture and electron emission rates respectively, and Et is the recombination energy level. However, by applying similar assumtions as for the direct semiconductor case above, and also assuming that the recombination centers are located near the middle of the bandgap, equation 36 may be reduced to: U ' vth σp Nt (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, which holds for low values of E. But for high electric fields, as vn becomes comparable to vth , this additional velocity component will result in a shorter average time between lattice scatterings, thus the mobility is not longer constant. The mobility will finally saturate for high electric fields, and may be approximated by the empirical expression: vn , v p =

vs [1 + (E0 /E)]1/γ

(38)

where vs is the saturation velocity (107 cm/s for Si at 300 K), and E0 is a constant equal to 7 × 103 V/cm for electrons and 2 × 104 V/cm for holes, and γ is 2 for electrons and 1 for holes. Another current component that may occur for high voltage operation of semiconductors results from an avalanche process. This is governed by conduction

electrons with sufficient energy to ionize atoms in the lattice. These electrons gain kinetic energy from the strong electric field between each interaction with the lattice, and the ionized electron-hole pair may gain sufficient kinetic energy in the field to ionize other atoms. This chain reaction gives rise to a very large current 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 detector element which utilizes some important properties of the p-n junction. The p-n junction, usually called a diode, is a nonlinear element that is most commonly used 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 to visualize the junction as the face of two elements brought in close contact with each other. ε 1111 0000 00000 11111 0000 1111 00000 11111 I 0000 1111 00000 000011111 1111 00000 11111 0000 1111 00000 11111 I 0000 1111 00000 11111 drift

p

p

n

n

diff

Drift EC

EF 00000000000 11111111111 00000000000 EV 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111

EC EF

000000000000 111111111111 000000000000 E111111111111 V 000000000000 111111111111 000000000000 111111111111

(a)

EC Diffusion EC E F 0000000000000000000000 1111111111111111111111 EF 0000000000000000000000 1111111111111111111111 E 0000000000000000000000 1111111111111111111111 V 0000000000000000000000 1111111111111111111111 0000000000000000000000 1111111111111111111111 Diffusion 0000000000000000000000 1111111111111111111111 0000000000000000000000 1111111111111111111111 0000000000000000000000 1111111111111111111111 0000000000000000000000 1111111111111111111111 0000000000000000000000 1111111111111111111111 EV 0000000000000000000000 1111111111111111111111 0000000000000000000000 1111111111111111111111 Drift 0000000000000000000000 1111111111111111111111 0000000000000000000000 1111111111111111111111 (b)

Figure 4: The p-n junction in thermal equilibrium. (a) Uniformly 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 together (fig. 4a), the n-side is electrically neutral, since an equal amount of positive ionized atoms and negative conduction electrons exists. After the two samples are joined, there will be a large charge concentration gradient at the junction 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 into the p-region. The positively charged ionized dopant atoms at the n-side, however, are bound by the crystal lattice and may not diffuse. As more and more electrons diffuse across the junction, a positive net charge is created at the n-side, and the positive ions are said to be uncovered. The same process occurs in the p-material, where positive holes diffuse into the n-region, leaving behind uncovered bound negatively charged acceptor ions in the p-lattice. The uncovered charges on both sides govern an electric field E across the junction region directed from n to p as seen in figure 4b. This field will produce a drift current of free electrons and holes, which is oppositely directed to the initial diffusion currents; electrons will drift from the p-side across 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 holes and electrons across the p-n junction must be zero. For the hole current density

this yields:

Jp = Jp (drift) + Jp (diffusion) dp = qµp pE − qDp !dx 1 dEi dp =0 = qµp p − kT µp q dx dx

(39)

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

p = ni e(Ei −EF )/kT ⇓ ! dp p dEi dEF = − dx kT dx dx Inserted into equation (39) yields the net hole current density:

(40)

(41)

J p = µp p

dEF =0 dx

or

dEF =0 dx

(42)

Thus, the Fermi level must be constant throughout the sample as illustrated in figure 4b. Close to the junction on each side, there are now two regions where all bound excited states are left uncompensated. Going outwards one reaches the transition region where the ions are partially compensated by free charge carriers. The region from the junction and outwards on both sides, including the transition region, is called the space charge region, or depletion region, since it is depleted for free charge carriers (see figure 5). Outside the transition region charge neutrality is maintained. The transition region is usually small compared to the total space charge region, thus the space charge distribution may be approximated by a rectangular where the n-side depletion region extends uniformly up to xn , and at the p-side up to xp . The electrostatic potential ψ is defined in order that its negative gradient equals the electric field:

Depletion Region ND -N A Neutral n region

Neutral p region

111111111111 000000000000 000000000000 111111111111 000000000000 111111111111 000000000000 111111111111 x 000000000000 111111111111 0000000000 1111111111 0 x 0000000000 1111111111 Transition 0000000000 1111111111 Region 1111111111 0000000000 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 p

n

x

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

E ≡−

dψ dx

⇒

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: 1 kT NA ψp ≡ − (Ei − EF ) =− ln x≤−x p q q ni

(44)

where equation (40) has been used by first setting p = NA . This is valid since Poisson’s equation yields (for the neutral region): dE ρs q = = (ND − NA + p − n) dx εS εS

(45)

assuming all donors and acceptors to be ionized. And due to the p-type material; 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: 1 ND ψn ≡ − (Ei − EF ) = kT ln x≥xn q 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 potential Vbi : Vbi = ψn − ψp =

5.2

kT NA ND ln q n2i

(47)

Highly doped abrupt junctions

As will be described in another chapter, the p − n junction of the ATLAS SCT tracker is of a special design with a large difference in doping concentration between the p and the n region. The p-material is heavily doped, in the order of 1019 cm−3 , whereas the n-material is lightly doped, in the order of 1016 cm−3 . To indicate the heavy doping concentration, the p region is often designated as p+ -type, and the lightly doped n-region simply as n-type. In unbiassed thermal equilibrium, the total space charge must be neutral. And since there are no free charge carriers in the depleted regions, the total p and n space charge on each side must be equal:

Depletion Region ND -N A

Neutral p region

Lightly doped n region

111111111111 000000000000 000000000000 111111111111 000000000000 111111111111 000000000000 x 111111111111 000000000000 111111111111 0 1 0 x 0 1 0 1 doped 0 Heavily 1 p region 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

Neutral n region

p

n

x

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

NA x p = ND x n

(48)

where the space charge regions are approximated by rectangular regions, also called an abrupt junction, with −xp and xn as the lower and upper limits respectively. 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 region from Poisson’s equation:

A E = − qN (xp + x) εS

E=

qND (x εS

− xn )

, −xp ≤ x ≤ 0

(49)

, 0 ≤ x ≤ xn

(50)

and then the built-in potential Vbi :

Vbi = − =

Z xn

−xp qNA x2p

2 + εS

E(x) dx = −

Z 0 −xp

+

qND x2n 2εS

E(x) dx −

Z xn

E(x) dx

0

(51)

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

d=

2εS q



NA + ND Vbi NA ND 

(52)

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

d≡

2εS Vbi qND

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

(53)

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. If the applied potential has the opposite sign as that of the built-in electrostatic potential, the total electrostatic potential will be reduced by |VF | to Vbi + VF , where VF is negative. The junction is now forward biassed. The reduction of the electrostatic potential across the junction reduces the drift current, and there is no longer a balance between the drift and diffusion current; there will be a nonzero net diffusion current across the junction. Since p-n junctions of a Si microstrip detector are reverse biassed, the forward bias properties will not be discussed any further here. If the junction is biassed with opposite polarity, the electrostatic potential will be additional to the built-in potential, and thus increasing the total electrostatic potential. The junction is said to be reverse biassed. As a result the depletion width will increase accordingly for a one-sided abrupt junction as:

d≡

v u u 2εS (Vbi t

+V)

qNef f

(54)

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