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PHENITEC 2017/3/3 SEMICONDUCTOR

Physics of Semiconductor Devices Vol.3

Transport of carriers 2 (hard perturbations)

Generation of excess carriers and recombination

In Vol.3, we consider the case of “hard” perturbations, which modify the number of free carriers in the crystal. The

phenomenon of generation-recombination can be regarded as the result of the superposition of several processes.

The main mechanisms are the following: thermal generation-recombination, optical generation-recombination

(transitions via photons) and impact-ionization, non-radiative recombination (transition via phonons), three parti-

cles recombination (Auger recombination), and surface recombination. Generation-recombination is characterized

by a net generation rate G, given in cm-3

s-1

, which specifies the carrier concentration created per second. It is ex-

pressed as the sum of various contributions, which will be described below:

𝐺 = 𝐺𝑡ℎ + 𝐺𝑜𝑝𝑡 + 𝐺𝑖𝑚𝑝 + 𝐺𝑆𝑅𝐻 + 𝐺𝐴𝑢𝑔𝑒𝑟 + 𝐺𝑠𝑢𝑟𝑓

However, in thermal equilibrium the thermal recombination rate (RT) must be balanced by the thermal generation

rate (GT). Therefore, net thermal generation rate (Gth) becomes

𝐺𝑡ℎ = 𝐺𝑇 −𝑅𝑇 = 0.

In the case of thermal equilibrium, a net generation rate G will be change to

𝐺 = 𝐺𝑜𝑝𝑡 + 𝐺𝑖𝑚𝑝 + 𝐺𝑆𝑅𝐻 + 𝐺𝐴𝑢𝑔𝑒𝑟 + 𝐺𝑠𝑢𝑟𝑓 . 3 − 1

Generation of carriers by optical irradiation for a direct-bandgap semiconductor

This process is a direct transition of the carriers between the conduction band and the valence band. The generation

(GL) corresponds to an excitation of an electron of the valence band to the conduction band by absorption of a pho-

ton of appropriate energy. In the same way, the recombination (RL) is the opposite phenomenon, i.e. an electron of

the conduction band relaxes towards the valence band by emitting a photon. This mechanism is very effective in

direct band-gap semiconductors. The rate of direct recombination is expected to be proportional to the number of

electrons available in the conduction band and the number of holes available in the valence band; that is

𝑅𝐿 = 𝐶𝑜𝑝𝑡𝑛𝑝

where Copt is the optical capture and emission rate, characteristics of the material. On the other hand, generation

rate is expected to be proportional to the number of electrons and holes before generation i.e. ni2.

𝐺𝐿 = 𝐶𝑜𝑝𝑡𝑛𝑖2.

Hence the net generation-recombination rate is given by

𝐺𝑜𝑝𝑡 = 𝐶𝑜𝑝𝑡(𝑛𝑖2 − 𝑛𝑝) 3 − 2

Hot carrier generation of carriers

This phenomenon, also called impact ionization, occurs under electric fields higher than 100kV/cm, for example in

the space charge zone of a junction in reverse bias. The kinetic energy of a carrier accelerated by such a field grows

to such a point that the carrier may behave as an ionizing radiation, i.e. it can be lose part of its energy to create an

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electron-hole pair. The resulting carriers are in turn accelerated by the electric field and can create other pairs,

which may give rise to the avalanche phenomenon, and possibly to junction breakdown.

Two parameters characterize the phenomenon of impact ionization:

• The ionization energy threshold, which is the minimum kinetic energy that must be acquired by a carrier to

generate a pair. This energy is appreciably higher than the bandgap energy.

• The ionization coefficient α (cm-1

) which gives the number of pairs generated per centimeter crossed by a

hot carrier. This coefficient depends directly on the carrier energy but is, however, not easily accessible ex-

perimentally. This ionization coefficient is thus usually given as a function of the electric field.

Semi-empirical models have enabled to be evaluated.

The net generation rate given by

𝐺𝑖𝑚𝑝 =1

𝑞[𝛼𝑛|𝐽𝑛| + 𝛼𝑝|𝐽𝑝|] 3 − 3

where αn and αp are the ionization coefficients and Jn and Jp the electron and hole current densities, respectively.

Recombination

When there are excess free electron and holes compared to the thermal equilibrium state, various recombination

processes are likely be activated to restore the equilibrium carrier concentrations. We distinguish direct or

band-to-band recombination and indirect recombination via deep levels (Fig.3.1).

The recombination processes provide energy; the different mechanisms are distinguished by the way of dissipating

this energy. There are essentially three:

• Non-radiative recombination: the energy is dissipated by exciting vibration modes of the crystal lattice, i.e.

by phonon emission (Fig. 3.2a). The crystal is heated. Non-radiative indirect recombination is the dominant

mechanism in silicon. In very pure silicon, the carrier lifetime τn or τp is about 1 ms. It can go down to ap-

proximately 1 ns if indirect recombination centers are introduced, for example by diffusion of gold or plat-

inum. In the case of indirect recombination, the net generation-recombination rate is given by the Shock-

ley-Read-Hall model, i.e.

𝐺𝑆𝑅𝐻 =𝑛𝑖2 − 𝑛𝑝

𝜏𝑝(𝑛 + 𝑛𝑖) + 𝜏𝑛(𝑝 + 𝑛𝑖) 3 − 4

• Auger recombination: the energy is dissipated by transfer to a free carrier (Fig.3.2b); it is opposite mecha-

nism to impact ionization. It is important only at high temperatures or the presence of a strong field. The net

generation-recombination rate is given by

Figure 3.1 Direct and indirect recombination process



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𝐺𝐴𝑢𝑔𝑒𝑟 = (𝑛𝑖2 − 𝑛𝑝)(𝑛𝐶𝑛 + 𝑝𝐶𝑝) 3 − 5

where Cn and Cp are the Auger coefficients, respectively about 3×10-43

m6/s and 10

-43 m

6/s at room temperature in

silicon.

• Surface recombination: at a surface, or an interface with a dielectric, the crystalline periodicity rupture in-

troduces localized states whose levels can be in the forbidden bandgap (Fig.3.3). These states favor surface

recombination, by a process similar to the indirect recombination previously described.

The net surface generation-recombination rate is given by

𝐺𝑠𝑢𝑟𝑓 =𝑁𝑅𝑠𝑐𝑛𝑐𝑝(𝑛𝑠𝑝𝑠 − 𝑛𝑖

2)

𝑐𝑛(𝑛𝑠 + 𝑛𝑖) + 𝑐𝑝(𝑝𝑠 + 𝑛𝑖) 3 − 6

where NRs is the surface density of the interface centers (in cm-2

), cn and cp are the capture coefficients (in cm3/s),

and ns and ps are the carrier concentrations (in cm-3

) at the surface.

The detailed derivation of indirect recombination rate for indirect-band gap semiconductors

For indirect-band gap semiconductors, such as silicon, a direct recombination process is very unlikely, because the

electrons at the bottom of the conduction band have nonzero momentum with respect to the holes at the top of the

valence band. A direct transition that conserves both energy and momentum is not possible without a simultaneous

lattice interaction. Therefore the dominant recombination process in such semiconductors is indirect transition via

localized energy states in the forbidden energy gap. These states act as stepping stones between the conduction

band and the valence band.

If the concentration of centers in the interface is Nt, the concentration of unoccupied centers is given by Nt (1-F),

where F is the Fermi distribution function for the probability that a center is occupied by an electron. In equilibri-

um,

(a)

ħω EC

EV

(b)

Figure 3.2 Mechanisms of recombination: (a) non-radiative, (b) Auger

Interface centers

EC

EV

Air or dielectric

Figure 3.3 Surface recombination

Ra Rb

Et

Electron capture

Hole capture Hole emission

Electron emission

Rc Rd



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𝐹 =1

1 + 𝑒(𝐸𝑡−𝐸𝐹) 𝑘𝑇⁄or

1

𝐹= 1 + 𝑒(𝐸𝑡−𝐸𝐹) 𝑘𝑇⁄ 3 − 7

where Et is energy level of the center and EF is the Fermi level.

Therefore the capture rate of an electron by a recombination center (Fig.3.3 Ra and Rb) is given by

𝑅𝑎 = 𝑐𝑛𝑛𝑠𝑁𝑅𝑠(1 − 𝐹) 3 − 8

The rate of emission of electrons from the center (Fig.3.3 Rb) is the inverse of the electron capture process. The rate

is proportional to the concentration of centers occupied by electrons, that is, NtF. We have

𝑅𝑏 = 𝑒𝑛𝑁𝑡𝐹. 3 − 9

The proportionality constant en is called the emission probability. At thermal equilibrium the rates of capture and

emission of electrons must be equal (Ra = Rb). Thus, the emission probability can be expressed in terms of the

quantities already defined in equation 3-7:

𝑒𝑛𝑁𝑡𝐹 = 𝑐𝑛𝑛𝑁𝑡(1 − 𝐹) → 𝑒𝑛 =𝑐𝑛𝑛(1 − 𝐹)

𝐹 3 − 10

Since the electron concentration in thermal equilibrium is given by

𝑛 = 𝑛𝑖𝑒(𝐸𝐹−𝐸𝑖) 𝑘𝑇⁄

𝑒𝑛 =𝑐𝑛𝑛(1 − 𝐹)

𝐹= 𝑐𝑛𝑛𝑠 (

1

𝐹− 1) = 𝑐𝑛𝑛(𝑒

(𝐸𝑡−𝐸𝐹) 𝑘𝑇⁄ ) = 𝑐𝑛𝑛𝑖𝑒(𝐸𝐹−𝐸𝑖) 𝑘𝑇⁄ (𝑒(𝐸𝑡−𝐸𝐹) 𝑘𝑇⁄ )

= 𝑐𝑛𝑛𝑖𝑒(𝐸𝑡−𝐸𝑖) 𝑘𝑇⁄ 3 − 11

The transition between the recombination center and valence band are analogous to those described above. The

capture rate of a hole by an occupied recombination center (Fig.3.3 Rc) is given by

𝑅𝑐 = 𝑐𝑝𝑝𝑁𝑡𝐹. 3 − 12

By argument similar to those for electron emission, the rate of hole emission (Fig.3.3 Rd) is

𝑅𝑑 = 𝑒𝑝𝑁𝑡(1 − 𝐹). 3 − 13

The emission probability ep of a hole may be expressed in terms of cp by considering the thermal equilibrium con-

dition for which Rc = Rd.

𝑝 = 𝑛𝑖𝑒(𝐸𝑖−𝐸𝐹) 𝑘𝑇⁄

𝑒𝑝𝑁𝑡(1 − 𝐹) = 𝑐𝑝𝑝𝑁𝑡𝐹 → 𝑒𝑝 = 𝑐𝑝𝑝𝐹

1 − 𝐹= 𝑐𝑝𝑛𝑖𝑒

(𝐸𝑖−𝐸𝐹) 𝑘𝑇⁄1

1𝐹 − 1

= 𝑐𝑝𝑛𝑖𝑒(𝐸𝑖−𝐸𝑡) 𝑘𝑇⁄ 3 − 14

Let us now consider the non-equilibrium case in which an n-type semiconductor is illuminated uniformly to give a

generation rate GL. Thus in addition to the process shown in Fig.3.3, electron-hole pairs are generated as a result of

light. In steady state the electrons entering and leaving the conduction band must be equal. This is called the princi-

ple of detailed balance, and yields

𝑑𝑛𝑛𝑑𝑡

= 𝐺𝐿 − (𝑅𝑎 − 𝑅𝑏) = 0. 3 − 15

Similarly, in steady state the detailed balance of holes in valence band leads to

𝑑𝑝𝑛𝑑𝑡

= 𝐺𝐿 − (𝑅𝑐 − 𝑅𝑑) = 0. 3 − 16



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Under, state-state non-equilibrium conditions Ra ≠ Rb and Rc ≠ Rd. From equation 3-15 and 3-16 we obtain

𝐺𝐿 = 𝑅𝑎 − 𝑅𝑏 = 𝑅𝑐 −𝑅𝑑 . 3 − 17

Using this relation and equation 3-8, 3-9, 3-12, and 3-13 we can eliminate F as follows

𝑅𝑎 − 𝑅𝑏 = 𝑅𝑐 − 𝑅𝑑 → 𝑐𝑛𝑛𝑛𝑁𝑡(1 − 𝐹) − 𝑐𝑛𝑛𝑖𝑒(𝐸𝑡−𝐸𝑖) 𝑘𝑇⁄ 𝑁𝑡𝐹 = 𝑐𝑝𝑝𝑛𝑁𝑅𝑠𝐹 − 𝑐𝑝𝑛𝑖𝑒

(𝐸𝑖−𝐸𝑡) 𝑘𝑇⁄ 𝑁𝑡(1 − 𝐹)

𝑐𝑛𝑛𝑛𝑁𝑡 + 𝑐𝑝𝑛𝑖𝑒(𝐸𝑖−𝐸𝑡) 𝑘𝑇⁄ 𝑁𝑡 = (𝑐𝑛𝑛𝑛𝑁𝑡 + 𝑐𝑛𝑛𝑖𝑒

(𝐸𝑡−𝐸𝑖) 𝑘𝑇⁄ 𝑁𝑡 + 𝑐𝑝𝑝𝑛𝑁𝑡 + 𝑐𝑝𝑛𝑖𝑒(𝐸𝑖−𝐸𝑡) 𝑘𝑇⁄ 𝑁𝑡)𝐹

So that F and 1-F are denoted by

𝐹 =𝑐𝑛𝑛𝑛𝑁𝑡 + 𝑐𝑝𝑛𝑖𝑒

(𝐸𝑖−𝐸𝑡) 𝑘𝑇⁄ 𝑁𝑡

(𝑐𝑛𝑛𝑛𝑁𝑡 + 𝑐𝑛𝑛𝑖𝑒(𝐸𝑡−𝐸𝑖) 𝑘𝑇⁄ 𝑁𝑡 + 𝑐𝑝𝑝𝑠𝑁𝑡 + 𝑐𝑝𝑛𝑖𝑒

(𝐸𝑖−𝐸𝑡) 𝑘𝑇⁄ 𝑁𝑡)

1 − 𝐹 =𝑐𝑛𝑛𝑛𝑁𝑡 + 𝑐𝑛𝑛𝑖𝑒

(𝐸𝑡−𝐸𝑖) 𝑘𝑇⁄ 𝑁𝑡 + 𝑐𝑝𝑝𝑛𝑁𝑡 + 𝑐𝑝𝑛𝑖𝑒(𝐸𝑖−𝐸𝑡) 𝑘𝑇⁄ 𝑁𝑡 − 𝑐𝑛𝑛𝑛𝑁𝑡 − 𝑐𝑝𝑛𝑖𝑒


𝑐𝑛𝑛𝑛𝑁𝑡 + 𝑐𝑛𝑛𝑖𝑒(𝐸𝑡−𝐸𝑖) 𝑘𝑇⁄ 𝑁𝑡 + 𝑐𝑝𝑝𝑛𝑁𝑡 + 𝑐𝑝𝑛𝑖𝑒


=𝑐𝑛𝑛𝑖𝑒

(𝐸𝑡−𝐸𝑖) 𝑘𝑇⁄ 𝑁𝑡 + 𝑐𝑝𝑝𝑛𝑁𝑡



Hence,

𝐺𝐿 = 𝑐𝑝𝑝𝑛𝑁𝑡 (𝑐𝑛𝑛𝑛𝑁𝑡 + 𝑐𝑝𝑛𝑖𝑒


(𝑐𝑛𝑛𝑛𝑁𝑡 + 𝑐𝑛𝑛𝑖𝑒(𝐸𝑡−𝐸𝑖) 𝑘𝑇⁄ 𝑁𝑡 + 𝑐𝑝𝑝𝑛𝑁𝑡 + 𝑐𝑝𝑛𝑖𝑒

(𝐸𝑖−𝐸𝑡) 𝑘𝑇⁄ 𝑁𝑡))

− 𝑐𝑝𝑛𝑖𝑒(𝐸𝑖−𝐸𝑡) 𝑘𝑇⁄ 𝑁𝑡 (

𝑐𝑛𝑛𝑖𝑒(𝐸𝑡−𝐸𝑖) 𝑘𝑇⁄ 𝑁𝑡 + 𝑐𝑝𝑝𝑛𝑁𝑡


(𝐸𝑖−𝐸𝑡) 𝑘𝑇⁄ 𝑁𝑡)

=(𝑐𝑝𝑝𝑛𝑁𝑡)(𝑐𝑛𝑛𝑛𝑁𝑡) − (𝑐𝑝𝑛𝑖𝑒

(𝐸𝑖−𝐸𝑡) 𝑘𝑇⁄ 𝑁𝑡)(𝑐𝑛𝑛𝑖𝑒(𝐸𝑡−𝐸𝑖) 𝑘𝑇⁄ 𝑁𝑡)



=𝑐𝑛𝑐𝑝𝑝𝑛𝑛𝑛𝑁𝑡 − 𝑐𝑛𝑐𝑝𝑛𝑖

2𝑁𝑡

𝑐𝑛𝑛𝑛 + 𝑐𝑛𝑛𝑖𝑒(𝐸𝑡−𝐸𝑖) 𝑘𝑇⁄ + 𝑐𝑝𝑝𝑛 + 𝑐𝑝𝑛𝑖𝑒

(𝐸𝑖−𝐸𝑡) 𝑘𝑇⁄=

𝑁𝑡𝑐𝑛𝑐𝑝(𝑝𝑛𝑛𝑛 − 𝑛𝑖2)

𝑐𝑛(𝑛𝑛 + 𝑛𝑖𝑒(𝐸𝑡−𝐸𝑖) 𝑘𝑇⁄ ) + 𝑐𝑝(𝑝𝑛 + 𝑛𝑖𝑒

(𝐸𝑡−𝐸𝑖) 𝑘𝑇⁄ )

Thus we can get net generation-recombination rate

𝐺𝐿 =𝑁𝑡𝑐𝑛𝑐𝑝(𝑝𝑛𝑛𝑛 − 𝑛𝑖

2)

𝑐𝑛(𝑛𝑛 + 𝑛𝑖𝑒(𝐸𝑡−𝐸𝑖) 𝑘𝑇⁄ ) + 𝑐𝑝(𝑝𝑛 + 𝑛𝑖𝑒

(𝐸𝑡−𝐸𝑖) 𝑘𝑇⁄ ) 3 − 18

If we use poor approximation that in equilibrium state n = ni and p = pi. The result of recalculation of 3-18 is


2)

𝑐𝑛(𝑛𝑛 + 𝑛𝑖) + 𝑐𝑝(𝑝𝑛 + 𝑛𝑖). 3 − 19

This method is convenient to prove other equations in this volume. For example, if we change nn to ns, pn to ps and

Nt to NRs, then we can get the equation 3-6. And we modify the equation 3-19 like that


2)

𝑐𝑛(𝑛𝑛 + 𝑛𝑖) + 𝑐𝑝(𝑝𝑛 + 𝑛𝑖)×𝑁𝑡𝑁𝑡=

𝑁𝑡𝑐𝑛𝑁𝑡𝑐𝑝(𝑝𝑛𝑛𝑛 − 𝑛𝑖2)

𝑁𝑡𝑐𝑛(𝑛𝑛 + 𝑛𝑖) + 𝑁𝑡𝑐𝑝(𝑝𝑛 + 𝑛𝑖)

and if we change 𝑁𝑡𝑐𝑛 → −1 𝜏𝑛⁄ and 𝑁𝑡𝑐𝑝 → −1 𝜏𝑝⁄ , it change to

𝑁𝑡𝑐𝑛𝑁𝑡𝑐𝑝(𝑝𝑛𝑛𝑛 − 𝑛𝑖2)

𝑁𝑡𝑐𝑛(𝑛𝑛 + 𝑛𝑖) + 𝑁𝑡𝑐𝑝(𝑝𝑛 + 𝑛𝑖)→

1𝜏𝑛

1𝜏𝑝(𝑛𝑖

2 − 𝑝𝑛𝑛𝑛)

1𝜏𝑛(𝑛𝑛 + 𝑛𝑖) +

1𝜏𝑝(𝑝𝑛 + 𝑛𝑖)

=(𝑛𝑖

2 − 𝑝𝑛𝑛𝑛)

𝜏𝑝(𝑛𝑛 + 𝑛𝑖) + 𝜏𝑛(𝑝𝑛 + 𝑛𝑖).

This result is same as the equation 3-4.



6 / 13


Transport in devices

The semiconductor material properties presented until now were essentially focused on the distribution of electrons

and holes in the various energy levels in the crystal. The relative positions of these levels were considered, which

allow us to express the concentrations n and p of electrons and holes according to the difference between the Fermi

level EF and the intrinsic Fermi level Ei.

{

𝑛 = 𝑛𝑖exp(𝐸𝐹 − 𝐸𝑖𝑘𝐵𝑇

)

𝑝 = 𝑛𝑖exp(𝐸𝑖 − 𝐸𝐹𝑘𝐵𝑇

)

3 − 20

As these quantities were defined relative to energy differences, a common energy reference was not necessary. For

device study, the carriers are subjected to external actions (bias voltage, irradiation, etc.). It thus becomes necessary

to define these internal energies relative to the external environment by choosing a common origin.

Consider an electron gas forming a thermodynamic system in interaction with the external environment. The varia-

tion of internal energy U, when increasing the number of particles by dn and dp (n and p are extensive variables), is

given for an isothermal and adiabatic reaction (dT = 0 and dQ = 0) by the following expression (for dp = 0):

𝑑𝑈 = 𝜇𝑐𝑛𝑑𝑛 − 𝑞𝛹𝑑𝑛 = �̃�𝑛𝑑𝑛 3 − 21

with the intensive variables:

𝜇𝑐𝑛: electron chemical potential energy; 𝛹 ∶ electrostatic potential intoduced externally;�̃�𝑛: electron electrochemical potential energy.

We similarly define for holes the energies μcp and �̃�𝑝, leading to the following expressions

{�̃�𝑛 = 𝜇𝑐𝑛 − 𝑞𝛹�̃�𝑝 = 𝜇𝑐𝑝 + 𝑞𝛹

3 − 22

System at the thermodynamic equilibrium

The system is at thermal equilibrium when it does not exchange energy with the external environment (or when the

net balance of exchange is zero). It can be shown that the condition of thermal equilibrium is position independent

of the intensive variables T, �̃�𝑛, �̃�𝑝, i.e.

{

𝑇(𝒓) = constant�̃�𝑛(𝒓) = constant

�̃�𝑝(𝒓) = constant 3 − 23

Moreover, it is shown in statistical mechanics that [1]

{

𝜇𝑐𝑛 = 𝑘𝐵𝑇 ln (

𝑛

𝑛𝑖) = 𝑘𝐵𝑇 ln(exp(

𝐸𝐹 − 𝐸𝑖𝑘𝐵𝑇

)) = 𝐸𝐹 − 𝐸𝑖

𝜇𝑐𝑝 = 𝑘𝐵𝑇 ln (𝑝

𝑛𝑖) = 𝑘𝐵𝑇 ln(exp(

𝐸𝑖 − 𝐸𝐹𝑘𝐵𝑇

)) = 𝐸𝑖 − 𝐸𝐹

3 − 24

where Ei is associated with Ψ through

𝐸𝑖 = −𝑞𝛹 3 − 25

The negative sign in 3-25 just means that the energy is defined for electrons. Finally,



7 / 13


{�̃�𝑛 = 𝜇𝑐𝑛 − 𝑞𝛹 = 𝐸𝐹 �̃�𝑝 = 𝜇𝑐𝑝 + 𝑞𝛹 = −𝐸𝐹

3 − 26

Thus, expressions 3-23 allow us to state that the Fermi level EF is flat for a semiconductor crystal at thermal equi-

librium.

When the crystal is not uniformly doped, the quantities n and p vary, therefore μcn and μcp also vary and bands are

vent around the Fermi level. Indeed, n varies along the crystal, μcn also varies (see expression 3-24), and Ψ (expres-

sion 3-22) adjusts itself so that �̃�𝑛 (or EF) remains constant in the crystal.

The variations of Ei along the crystal show that there is a potential gradient and thus of the electric field:

𝑬 =1

𝑞(𝐢𝜕𝐸𝑖𝜕𝑥

+ 𝐣𝜕𝐸𝑖𝜕𝑦

+ 𝐤𝜕𝐸𝑖𝜕𝑧) ≡

1

𝑞∇𝐸𝑖 3 − 27

At equilibrium, this field is opposed to carrier diffusion; both condition and diffusion current exist and exactly

compensate each other making the total current zero.

Out-of-equilibrium systems

As seen in the preceding sections, a state of non-equilibrium, i.e. a gradient of electrochemical potential, can have

various causes, in particular:

† the application of a potential difference, inducing a gradient Ψ;

† local generation of excess carriers (e.g. an ionizing radiation), inducing a gradient of μcn.

Assuming sufficiently week perturbation, the resulting carrier flows are proportional to the gradient of intensive

variables of the system. In the case of an isothermal semiconductor, the following current densities result:

{

𝑱𝑛 = 𝜎𝑛∇ (�̃�𝑛𝑞)

𝑱𝑝 = −𝜎𝑝∇(�̃�𝑝

𝑞)

3 − 28

with σn and σp the electron and hole conductivities, respectively.

It should be noted that for a system with uniform carrier density ∇(�̃�𝑛 𝑞⁄ ) = −∇𝛹, therefore Ohm’s law, 𝑱 = 𝜎𝑬,

obtained.

The Fermi pseudo-potentials, φn and φp for electrons and holes, respectively, are defined by

{

𝜑𝑛 = −�̃�𝑛𝑞

𝜑𝑝 =�̃�𝑝

𝑞

3 − 29

There is no reason for these pseudo-potentials to be equal at all points of a device in the presence of a perturbation,

in contrast to the equilibrium situation. We have thus to define locally a quasi-Fermi level for each type of carrier

by

{𝐸𝐹𝑛 = �̃�𝑛𝐸𝐹𝑝 = −�̃�𝑝

3 − 30

Then the current density may be rewritten as



8 / 13


{

𝑱𝑛 = −𝜎𝑛∇𝜑𝑛 =𝜎𝑛𝑞∇𝐸𝐹𝑛

𝑱𝑝 = −𝜎𝑝∇𝜑𝑝 =𝜎𝑝

𝑞∇𝐸𝐹𝑝

3 − 31

We see that −∇𝐸𝐹𝑛 and−∇𝐸𝐹𝑝 are the only true electromotive fields for free carriers. The difference in electromo-

tive potential applied between two points A and B of a crystal using an external generator is thus equal to the line

integral[2] of −∇𝐸𝐹𝑛 or −∇𝐸𝐹𝑝, (see Fig.3.4).

𝑉𝐵 − 𝑉𝐴 = −∫ ∇𝜑𝑛 ∙ 𝑑𝒓

𝐴𝐵

= −∫ (𝜕𝜑𝑛𝜕𝑥

𝑑𝑥 +𝜕𝜑𝑛𝜕𝑦

𝑑𝑦 +𝜕𝜑𝑛𝜕𝑧

𝑑𝑧) =

𝐴𝐵

−∫ 𝑑𝜑𝑛

𝐵

𝐴

= −(𝜑𝑛(𝐵) − 𝜑𝑛(𝐴))

= 𝜑𝑛(𝐴) − 𝜑𝑛(𝐵) = (−�̃�𝑛𝑞(𝐴)) − (−

�̃�𝑛𝑞(𝐵)) =

1

𝑞(𝐸𝐹𝑛(𝐵) − 𝐸𝐹𝑛(𝐴)) 3 − 32

The carrier densities remain related to the chemical potentials and are given by

{

𝑛 = 𝑛𝑖exp(𝜇𝑐𝑛𝑘𝐵𝑇

) = 𝑛𝑖exp(𝐸𝐹𝑛 − 𝐸𝑖𝑘𝐵𝑇

)

𝑝 = 𝑛𝑖exp (𝜇𝑐𝑝

𝑘𝐵𝑇) = 𝑛𝑖exp(

𝐸𝑖 − 𝐸𝐹𝑝

𝑘𝐵𝑇)

3 − 33

which finally in the general case to

𝑛𝑝 = 𝑛𝑖2exp(

𝐸𝐹𝑛 − 𝐸𝐹𝑝

𝑘𝐵𝑇) 3 − 34

Equation to be solved for the study of device

The parts of expression 3-22 and 3-24 related to electrons can be substituted into equation 3-28, which gives

𝑱𝑛 =𝑘𝐵𝑇

𝑞𝜎𝑛∇n

𝑛− 𝜎𝑛∇(𝛹) 3 − 35

∵ 𝑱𝑛 = 𝜎𝑛∇(�̃�𝑛𝑞) = 𝜎𝑛∇(

𝜇𝑐𝑛 − 𝑞𝛹

𝑞) = 𝜎𝑛∇ (

𝜇𝑐𝑛𝑞) − 𝜎𝑛∇(𝛹) = 𝜎𝑛∇(

𝑘𝐵𝑇 ln (𝑛𝑛𝑖)

𝑞) − 𝜎𝑛∇(𝛹)

=𝑘𝐵𝑇

𝑞𝜎𝑛∇(ln (

𝑛

𝑛𝑖)) − 𝜎𝑛∇(𝛹) =

𝑘𝐵𝑇

𝑞𝜎𝑛∇(ln 𝑛 − ln𝑛𝑖) − 𝜎𝑛∇(𝛹) =

𝑘𝐵𝑇

𝑞𝜎𝑛∇(ln 𝑛) − 𝜎𝑛∇(𝛹)

=𝑘𝐵𝑇

𝑞𝜎𝑛∇n

𝑛− 𝜎𝑛∇(𝛹)

Since 𝜎𝑛 = 𝑞𝑛𝜇𝑛 and 𝐷𝑛 =𝑘𝐵𝑇

𝑞𝜇𝑛 (see equation 2-19 and 2-25 of Vol.2), and using relation 3-25, 3-27 and 3-31

A

B ∇𝜑𝑛

𝑑𝒓

𝒓 = 𝑥𝐢+ 𝑦𝐣+ 𝑧𝐤 → 𝑑𝒓 = 𝑑𝑥𝐢+ 𝑑𝑦𝐣+ 𝑑𝑧𝐤

Figure 3.4 Line integral

∫ ∇𝜑𝑛 ∙ 𝑑𝒓

𝐴𝐵



9 / 13


equation 3-35 becomes

𝑱𝑛 = 𝑞𝐷𝑛∇𝑛 + 𝑞𝑛𝜇𝑛𝑬 = 𝑛𝜇𝑛∇𝐸𝐹𝑛 3 − 36

and in the same way for the hole current

𝑱𝑝 = −𝑞𝐷𝑝∇𝑝 + 𝑞𝑝𝜇𝑝𝑬 = 𝑝𝜇𝑝∇𝐸𝐹𝑝 3 − 37

Expressions 2-27 and 2-28 of Vol.2 are recovered again. Two additional comments can be made:

† for a zero electrostatic field, a current may exist only if there is a density gradient;

† the current density Jn can be zero even in the presence of an electrostatic field if 𝑬 = −𝑘𝐵𝑇

𝑞𝑛∇𝑛.

Continuity equation

The equation known as the continuity equation is very important in device theory. It expresses the local conserva-

tion of charge density, taking account current densities and the

generation-recombination phenomena at a given time t.

Unlike in a circuit, in the semiconductor device, currents can

flow any directions. So we consider a fixed volume in a n-type

silicon (Fig.3.5). Suppose that the domain V is enclosed by the

closed surface S, t is the normal unit vector on the surface S,

and positive direction is same as direction of the normal unit

vector t’s direction which means inward direction of current

density is negative for holes and positive for electrons.

Net charge variation per second in the fixed volume V is described by

𝑑𝑄(𝒓, 𝑡)

𝑑𝑡=𝑑

𝑑𝑡∭𝑞𝑛(𝒓, 𝑡)

𝑉

𝑑𝑉 =∭ 𝑞𝜕𝑛(𝒓, 𝑡)

𝜕𝑡

𝑉

𝑑𝑉

If Gn(r,t) is the rate of generation-recombination of electrons (/cm3s), that charge created by genera-

tion-recombination per second is denoted by

∭𝑞𝐺𝑛(𝒓, 𝑡)𝑑𝑉

𝑉

Total current which flow into the domain V per second is denoted by

∯𝑱𝑛(𝒓, 𝑡) ∙ 𝒕(𝒓)𝑑𝑆

𝑆

The net charge variation is result of generation-recombination and current from outside, this means that

∭𝑞𝜕𝑛(𝒓, 𝑡)

𝜕𝑡

𝑉

𝑑𝑉 =∭𝑞𝐺𝑛(𝒓, 𝑡)𝑑𝑉

𝑉

+∯𝑱𝑛(𝒓, 𝑡) ∙ 𝒕(𝒓)𝑑𝑆

𝑆

Using Gauss’s theorem [2] the second term of right side become

∯𝑱𝑛(𝒓, 𝑡) ∙ 𝒕(𝒓)𝑑𝑆

𝑆

=∭div𝑱𝑛(𝒓, 𝑡)𝑑𝑉

𝑉

Note that ‘div’ is one of the vector operators, called divergence

div𝑱 ≡ ∇ ∙ 𝑱 =𝜕𝐽𝑥𝜕𝑥

+𝜕𝐽𝑦

𝜕𝑦+𝜕𝐽𝑧𝜕𝑧.

J t

S

Figure 3.5 the domain V



10 / 13


And also ‘∇’ or ‘grad’ is a vector operator called gradient.

grad𝜙 ≡ ∇𝜙 = 𝐢𝜕𝜙

𝜕𝑥+ 𝐣

𝜕𝜙

𝜕𝑥+ 𝐤

𝜕𝜙

𝜕𝑥

Hence

∭𝜕𝑛(𝒓, 𝑡)

𝜕𝑡

𝑉

𝑑𝑉 =∭𝐺𝑛(𝒓, 𝑡)𝑑𝑉

𝑉

+∭1

𝑞div𝑱𝑛(𝒓, 𝑡)𝑑𝑉

𝑉

=∭ (𝐺𝑛(𝒓, 𝑡) +1

𝑞div𝑱𝑛(𝒓, 𝑡))𝑑𝑉

𝑉

Since, domain V is arbitrary, we can get following result.

𝜕𝑛(𝒓, 𝑡)

𝜕𝑡= 𝐺𝑛(𝒓, 𝑡) +

1

𝑞div𝑱𝑛(𝒓, 𝑡)

A similar equation can be derived for holes, which finally leads to the two continuity equations

{

𝜕𝑛(𝒓, 𝑡)

𝜕𝑡= 𝐺𝑛(𝒓, 𝑡) +

1

𝑞div𝑱𝑛(𝒓, 𝑡)

𝜕𝑝(𝒓, 𝑡)

𝜕𝑡= 𝐺𝑝(𝒓, 𝑡) −

1

𝑞div𝑱𝑝(𝒓, 𝑡)

3 − 38

To study a device, it is necessary to solve these two continuity equations together with Gauss’s law [3]:

div𝑬 = ∇ ∙ 𝑬 =𝑞(𝑁𝐷 + 𝑝 − 𝑛 − 𝑁𝐴)

𝜀𝑠𝑐 3 − 39

where εsc is the dielectric permittivity of the semiconductor, NA and ND are acceptor and donor densities respective-

ly. In the steady state, E is described as follow

𝑬 = −grad𝜙 = −∇𝜙

where ϕ is a scalar potential.

Hence 3-39 becomes

∇ ∙ 𝑬 ≡ −∇ ∙ ∇𝜙 = −(𝜕2𝜙

𝜕𝑥2+𝜕2𝜙

𝜕𝑦2+𝜕2𝜙

𝜕𝑧2) =

𝑞(𝑁𝐷 + 𝑝 − 𝑛 − 𝑁𝐴)

𝜀𝑠𝑐 3 − 40

Equation 3-40 is called Poisson’s equation. Note that, this equation is valid in the case of steady state otherwise the

scalar potential cannot be expressed by simple equation such as Poisson’s equation.

Column

In general, an electric field E(r,t) is described as

𝑬(𝒓, 𝑡) = −𝜕𝑨(𝒓, 𝑡)

𝜕𝑡− ∇𝜙(𝒓, 𝑡) (1)

where A(r,t) is called the vector potential or the current potential. And a magnetic flux density B(r,t) is expressed as

𝑩(𝒓, 𝑡) = ∇ × 𝑨(𝒓, 𝑡) ≡ 𝐢 (𝜕𝐴𝑧𝜕𝑦

−𝜕𝐴𝑦

𝜕𝑧) + 𝐣 (

𝜕𝐴𝑥𝜕𝑧

−𝜕𝐴𝑧𝜕𝑥

) + 𝐤(𝜕𝐴𝑦

𝜕𝑥−𝜕𝐴𝑥𝜕𝑦

) ≡ rot𝑨(𝒓, 𝑡) or curl𝑨(𝒓, 𝑡)

If we operate ‘rot’ to both side of equation (1), then we will get an interesting result.

rot𝑬(𝒓, 𝑡) = −𝜕rot𝑨(𝒓, 𝑡)

𝜕𝑡− ro∇𝜙(𝒓, 𝑡) = −

𝜕𝑩(𝒓, 𝑡)

𝜕𝑡− ∇ × ∇𝜙(𝒓, 𝑡)



11 / 13


= −𝜕𝑩(𝒓, 𝑡)

𝜕𝑡− 𝐢 (

𝜕2

𝜕𝑦𝜕𝑧−

𝜕2

𝜕𝑧𝜕𝑦)𝜙(𝒓, 𝑡) + 𝐣 (

𝜕2

𝜕𝑧𝜕𝑥−

𝜕2

𝜕𝑧𝜕𝑥)𝜙(𝒓, 𝑡) + 𝐤(

𝜕2

𝜕𝑥𝜕𝑦−

𝜕2

𝜕𝑥𝜕𝑦)𝜙(𝒓, 𝑡) = −

𝜕𝑩(𝒓, 𝑡)

𝜕𝑡

Finally we get Faraday’s law

rot𝑬(𝒓, 𝑡) +𝜕𝑩(𝒓, 𝑡)

𝜕𝑡= 0.

This three-equation system is generally impossible to solve analytically. It is therefore necessary to make approxi-

mations, or to implement a computation-based numerical solution. However, the analytical solution can be obtained

of one of the two following approximation is made:

† Either the considered region is a space-charge zone (SCZ) where the electron and hole density can be neglected.

We then have to solve Poisson’s equation to determine the distribution potential.

† Or the considered region is a quasi-neutral zone (QNZ) where the density of the majority carriers (electrons in

an N-type zone, or holes in a P-type zone) is equal to the ionized impurities and where the electric field is zero. The

continuity equations are then solved.

We now study for instance the reaction of a system to a small local, space or time, and we make further assump-

tions like that

𝜕𝐽𝑛𝑦𝜕𝑦

=𝜕𝐽𝑛𝑧𝜕𝑧

= 0,𝜕𝐸𝑦𝜕𝑦

=𝜕𝐸𝑧𝜕𝑧

= 0.

Then G = 0 the equation system to be solved is reduced to

{

𝜕𝑛

(𝒓, 𝑡)

𝜕𝑡=1

𝑞

𝜕𝐽𝑛𝑥(𝒓, 𝑡)

𝜕𝑥

𝜕𝐸𝑥(𝒓)

𝜕𝑥=𝑞(−𝑛(𝒓, 𝑡))

𝜀𝑠𝑐

3 − 41

Moreover,

𝐽𝑛𝑥(𝒓, 𝑡) = 𝜎𝑛𝐸𝑥(𝒓) + 𝑞𝐷𝑛𝜕𝑛(𝒓, 𝑡)

𝜕𝑥. 3 − 42

By assuming that the domain considered has a uniform conductivity, system 3-41 is reduced to

𝜕𝑛(𝒓, 𝑡)

𝜕𝑡=1

𝑞

𝜕

𝜕𝑥(𝜎𝑛𝐸𝑥(𝒓) + 𝑞𝐷𝑛

𝜕𝑛(𝒓, 𝑡)

𝜕𝑥) =

1

𝑞(𝜎𝑛

𝜕𝐸𝑥(𝒓)

𝜕𝑥+ 𝑞𝐷𝑛

𝜕2𝑛(𝒓, 𝑡)

𝜕𝑥2)

=−𝜎𝑛𝑛(𝒓, 𝑡)

𝜀𝑠𝑐+ 𝐷𝑛

𝜕2𝑛(𝒓, 𝑡)

𝜕𝑥2

Considering that 𝜎𝑛 = 𝑞𝑛(𝒓, 𝑡)𝜇𝑛 ≈ 𝜎0 = 𝑞𝑛(𝒓, 0)𝜇𝑛, we obtain

𝜕𝑛(𝒓, 𝑡)

𝜕𝑡+𝜎0𝜀𝑠𝑐𝑛(𝒓, 𝑡) − 𝐷𝑛

𝜕2𝑛(𝒓, 𝑡)

𝜕𝑥2= 0. 3 − 43

We will determine the solution of this differential equation in two particular cases, which will result in the defini-

tion of two important physical quantities: the dielectric relaxation time and the Debye length.

Time perturbation: dielectric relaxation time

Assuming that initial condition 𝑛(𝒓, 0) = 𝑛0, and 𝜕2𝑛(𝒓,𝑡)

𝜕𝑥2= 0. Equation 3-43 becomes



12 / 13


𝜕𝑛(𝒓, 𝑡)

𝜕𝑡+𝜎0𝜀𝑠𝑐

𝑛(𝒓, 𝑡) = 0 3 − 45

This equation is first order differential equation and integrable.

∫1

𝑛(𝒓, 𝑡)

𝑑𝑛(𝒓, 𝑡)

𝑑𝑡𝑑𝑡 = −∫

𝜎0𝜀𝑠𝑐

𝑡

0

𝑡

0

𝑑𝑡 → ∫1

𝑛(𝒓, 𝑡)𝑑𝑛(𝒓, 𝑡)

𝑛(𝒓,𝑡)

𝑛0

= −𝜎0𝜀𝑠𝑐𝑡

ln 𝑛(𝒓, 𝑡) − ln𝑛0 = ln(𝑛(𝒓, 𝑡)

𝑛0) = −

𝜎0𝜀𝑠𝑐𝑡 →

𝑛(𝒓, 𝑡)

𝑛0= exp(−

𝜎0𝜀𝑠𝑐

𝑡) → 𝑛(𝒓, 𝑡) = 𝑛0exp(−𝜎0𝜀𝑠𝑐𝑡)

Hence we can define the dielectric relaxation time τD like that

1

𝜏𝐷=𝜎0𝜀𝑠𝑐

=𝑞𝑛0𝜇𝑛𝜀𝑠𝑐

.

If we assume, that there is the equilibrium state at t = 0, the equilibrium state is recovered with the characteristic

time constant τD, i.e.,

(𝒓, 𝑡) = 𝑛0exp(−𝑡

𝜏𝐷) 3 − 46

Space perturbation: Debye length

Assuming that initial condition (𝟎, 𝑡) = 𝑛0, 𝜕𝑛(𝒓,𝑡)

𝜕𝑡= 0, 𝑛(𝒓, 𝑡) = 𝑛(𝑥)𝑛(𝑦)𝑛(𝑧), and 𝑛(𝑦)𝑛(𝑧) ≠ 0. Equation

3-43 becomes

𝜎0𝜀𝑠𝑐𝑛(𝑥)𝑛(𝑦)𝑛(𝑧) − 𝐷𝑛

𝜕2𝑛(𝑥)𝑛(𝑦)𝑛(𝑧)

𝜕𝑥2= 𝑛(𝑦)𝑛(𝑧) (

𝜎0𝜀𝑠𝑐𝑛(𝑥) − 𝐷𝑛

𝜕2𝑛(𝑥)

𝜕𝑥2) = 0.

Since 𝑛(𝑦)𝑛(𝑧) ≠ 0, we get

𝜎0𝜀𝑠𝑐

𝑛(𝑥) − 𝐷𝑛𝜕2𝑛(𝑥)

𝜕𝑥2= 0. 3 − 47

This is a second order differential equation which solutions generally have the form [4]

𝑛(𝑥) = 𝑒𝑚𝑥. 3 − 48

The solutions of 3-47 are linear combination of the roots (i.e. 𝑐1𝑒𝑚1𝑥 + 𝑐2𝑒

𝑚2𝑥, ci are arbitrary constant).

Substitution 3-48 into the equation 3-47 gives

𝜎0𝜀𝑠𝑐

𝑒𝑚𝑥 − 𝐷𝑛𝜕2𝑒𝑚𝑥

𝜕𝑥2=𝜎0𝜀𝑠𝑐𝑒𝑚𝑥 −𝐷𝑛𝑚

2𝑒𝑚𝑥 = 0 →𝜎0𝜀𝑠𝑐

− 𝐷𝑛𝑚2 = 0

𝑚 = ±√𝜎0𝜀𝑠𝑐𝐷𝑛

= ±√

𝑞𝑛0𝜇𝑛

𝜀𝑠𝑐𝑘𝐵𝑇𝑞 𝜇𝑛

= ±√𝑞2𝑛0𝜀𝑠𝑐𝑘𝐵𝑇

.

Hence the solutions become

𝑛(𝑥) = 𝑐1exp(+√𝑞2𝑛0𝜀𝑠𝑐𝑘𝐵𝑇

𝑥) + 𝑐2exp(−√𝑞2𝑛0𝜀𝑠𝑐𝑘𝐵𝑇

𝑥) = 𝑐1exp(+𝑥

𝐿𝐷) + 𝑐2exp (−

𝑥

𝐿𝐷) 3 − 49

where 𝐿𝐷 = √𝜀𝑠𝑐𝑘𝐵𝑇

𝑞2𝑛0 is the Debye length.



13 / 13


There are many (infinite) possible solutions, even though initial condition is 𝑛(0) = 𝑛0. Followings are example of

the solutions.

Example 1: If c1 = c2 = n0/2, then the solution becomes

𝑛(𝑥) =𝑛0exp (+

𝑥𝐿𝐷) + 𝑛0exp(−

𝑥𝐿𝐷)

2= 𝑛0cosh (

𝑥

𝐿𝐷).

Example 2: If c1 = 0, then the solution is

𝑛(𝑥) = 𝑛0exp(−𝑥

𝐿𝐷).

In this case, n(LD) = n0/e.

Reference

[1] Ivo Sachs, Siddhartha Sen, James Sexton, ELEMENTS OF STATISTICAL MECHNICS With an Introduction to

Quantum Field Theory and Numerical Simulation, Cambridge University Press, 2006

[2] Jerrold E. Marsden, Anthony Tromba, Vector Calculus Sixth Edition, W.H. Freeman and Company, 2012

[3] Edward M. Purcell, David J. Morin, Electricity and Magnetism 3rd

edition, Cambridge University Press, 2013

[4] Jon Mathews, R. L. Walker, MATHEMATICAL METHODS OF PHYSICS 2nd

edition, W. A. BENJAMIN, INC.,

1973


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