Statistical physics (2005)

For this intrinsic mechanism, the variation of ni(T ) versus temperature is dominated by the exp(−Eg /2kB T ) term. In fact, here the thermal mechanism creates two charge carriers with a threshold energy Eg .

The very fast variation of the exponential is responsible for the distinction between insulators and semiconductors. Indeed at room temperature T = 300 K, which corresponds to approximately 401 eV, let us compare the diamond situation where Eg = 5.2 eV [exp(−Eg /2kB T ) = exp(−104) = 6.8 × 10−46] and that of silicon, where Eg = 1.1 eV [exp(−22) = 2.8 × 10−10]. In 1 cm3 of diamond, there is no electron in the conduction band, whereas ni(T ) is equal to 1.6 × 1016 m−3 at 300 K in silicon.

A semiconductor is an insulator with a “not too wide” band gap, which allows the excitation of some electrons into the conduction band at room temperature. The most commonly used semiconductors, apart from silicon, are germanium and gallium arsenide GaAs (Eg = 1.52 eV).

The chemical potential of an intrinsic semiconductor is determined by substituting expression (8.47) of ni(T ) into (8.41). One obtains :

The Fermi level εF , which is the chemical potential at T = 0 K, lies in the middle of the band gap and µ does not move much apart from this location when T increases (kB T Eg ). Consequently, the hypotheses that µ lies very far from the band edges as compared to kB T , made when calculating n(T ) and p(T ), are valid.

Under the e ect of an external electric field, both electrons and holes participate in the electrical current : indeed the current includes contributions from both the conduction and the valence bands. As will be studied in Solid State Physics courses, the holes behave like positive charges and the total electrical conductivity in the semiconductor is the sum of the valence band and conduction band currents :

Here τn (respectively τp) is the electron (respectively hole) collision time.

The temperature variation of σ in an intrinsic semiconductor is dominated by the variation of exp(−Eg/2kT ), the other factors varying like powers of T . Consequently, σ increases very fast with T , thus its reciprocal, the resistivity and the measured resistance of a semiconductor decrease with temperature because the number of carriers increases.

c) Doped Semiconductor

The density of intrinsic carriers at room temperature is very weak and practically impossible to attain : for instance in silicon about 1 electron out of 1012 is in the conduction band due to thermal excitation. To achieve the intrinsic concentration the studied sample should contain neither impurities nor defects! It is more realistic to control the conductivity by selected impurities, the dopants.

Thus phosphorus atoms can be introduced into a silicon crystal : phosphorus is an element of column V of the periodic table, which substitutes for silicon atoms in some lattice sites. Each phosphorus atom brings one more electron than required by the tetrahedral bonds of Si. At low temperature this electron remains localized near the phosphorus nucleus, on a level located 44 meV below the bottom of the conduction band. When the temperature increases, the electron is ionized into the conduction band. Phosphorus is a donor.

The concentration of conduction electrons will be determined by the concentration ND in P atoms, of typical value ND 1022 m−3, chosen much larger than the intrinsic and the residual impurity concentrations, but much smaller than the Si atoms concentration.

In such a material, because of (8.46), the hole concentration is very small with respect to ni, the current is carried by the electrons, the material is called of n-type.

In a symmetrical way, the substitution in the silicon lattice of a Si atom by a boron atom creates an acceptor state : boron, an element of column III of the periodic table, only has three peripheral electrons and thus must capture one extra electron in the valence band to satisfy the four tetrahedral bonds of the crystal. This is equivalent to creating a hole, which, at low temperature, is localized 46 meV above the silicon valence band and at higher temperature is delocalized inside the valence band.

In boron-doped silicon, the conductivity arises from the holes, the semiconductor is of p-type.

In Solid State Physics courses, the statistics of doped semiconductors is studied in more detail : one finds in particular that the chemical potential µ moves versus temperature inside the band gap (in a n-type material, at low temperature it is close εc, in the vicinity of the donors levels and returns to the middle of the band gap, like in an intrinsic material, at very high temperature). The semiconductors constituting the electronic compounds are generally doped.

8.5Examples of Semiconductor Devices

Here we only give a flavor of the variety of configurations and uses. In specialized Solid State Physics courses, one will describe in more details the semiconductors properties. The aim here is to show that, just using the basic elements of the present chapter, one can understand the principle of operation of three widespread devices.

8.5.1The Photocopier : Photoconductivity Properties

Illuminating a semiconductor by light of energy larger than Eg promotes an electron into the conduction band and leaves a hole in the valence band. The conductivity increase is ∆σ, proportional to the light flux. In the xerography process, the photocopier drum is made of selenium which, under the illumination e ect, shifts from a very insulator state to a conducting state. This property is utilized to fix the pigments (the toner black powder) which are next transferred to the paper and then cooked in an oven to be stabilized on the paper sheet.

8.5.2The Solar Cell : an Illuminated p − n Junction

When, in the same solid, a p region is placed next to a n region, the electrons of the n region di use toward the holes of the p region : some of these mobile charges neutralize and a so-called space charge region becomes emptied of mobile carriers. The dopants still remain in the region, their charged nuclei create an electric field which prevents any additional di usion. This e ect can also be interpreted from the requirement of equal chemical potentials in the n and p region at equilibrium : in the neighborhood (a few µm) of the

region of doping change, there prevails an intense internal electric field Ei. If the sun is illuminating this region, the excited electron and hole, of opposite

charges, are separated by Ei. An external use circuit will take advantage of the current produced by this light excitation. We will reconsider these devices, called “solar cells,” at the end of next chapter when we will deal with thermal radiation.

8.5.3Compact Disk Readers : Semiconductor Quantum Wells

It was seen in §8.3.3 that starting from the energy levels of the constituting atoms one gets the energy bands of a solid. Their location depends of the initial atomic levels, thus of the chemical nature of the material. For almost thirty years it has been possible, in particular using Molecular Beam Epitaxy (MBE), to sequentially crystallize di erent semiconductors, which have the same crystal lattice but di erent chemical composition : a common example is GaAs and Ga1−xAlxAs.

In such semiconductor sandwiches the bottom of the conduction band (or the top of the valence band ) does not lie at the same energy in GaAs and in Ga1−xAlxAs. One thus tailors quantum wells, to fashion. Their width usually ranges from 5 to 20 nm, their energy depth is of a few hundreds of meV, and the energies of their localized levels is determined by their “geometry,” as taught in Quantum Mechanics courses.

These quantum wells are much used in optoelectronics, for example as diodes or lasers emitters, in the telecommunication networks by optical fibers, as readers of compact disks, of code-bars in supermarkets, etc.

9.1.1Thermodynamics of the Boson Gas

Let us recall the average occupation number nk [Eq.(6.26)] by bosons of

As nk is positive or zero, the denominator exponent should be positive for any εk , which implies that the chemical potential µ must be smaller than any εk , including ε1, the energy of the fundamental state. At fixed T and µ, nk decreases when εk increases. The µ value is fixed by the condition

k

where N is the total particles number of the considered physical system.

From nk one can calculate all the thermodynamical parameters and in particular the average energy :

In the large volume limit, (9.1) is replaced by the Bose-Einstein distribution, a continuous function given by

The physical parameters are calculated in this case using the density of states D(ε), then expressing the constraint replacing (9.2) on the particles number and verifying that µ < ε1. Here (§9.1) one is concerned by free bosons, with no

mutual interaction, therefore their energy is reduced to ε = p2

2m

the “box” potential producing the quantization, leading to the expression of D(ε)), so that ε1 = 0 and µ < 0.

Then for particles with a spin s, in the three-dimensional space, one gets,

Integrating (9.9) by parts, one finds (9.10) which is equal to (9.8) to the factor

which takes the same form as for the fermions (7.13), although the physical properties are totally di erent.

9.1.2Bose-Einstein Condensation

The above expressions are only valid when the condition µ < 0 is fulfilled (see above). Now when T decreases at constant density N/Ω, since β increases the di erences (ε −µ) must decrease in order that N/Ω keeps its value (9.7). This means that µ gets closer to zero.

Relation (9.15) defines a specific temperature TB = 1/kB βB which is only related to the density n = N/Ω :

spin of the 42He particles is zero.

The temperature TB , or Bose temperature, would correspond in the above calculation to µ = 0, whereas the situation of §9.1.1 is related to µ < 0. Since the integrals in (9.7) and (9.13) are both equal to N/CΩ, to maintain this value constant it is necessary that β (corresponding to µ < 0) be smaller than

βB (associated to µ = 0). Consequently, (9.7) is only valid for T > TB . When lowering the temperature to TB , µ tends to zero, thus getting closer to the fundamental state of energy ε1 = 0.

For T < TB the analysis has to be reconsidered : the population of level ε1 has to be studied separately, as it would diverge if one were using expression (9.1) without caution! Now the population N1 of ε1 is at most equal to N . Expression (9.1) remains valid but µ cannot be strictly zero.

µ is very close to zero because it is the ratio of a microscopic energy to a macroscopic number of particles. The first excited level ε2 has the energy

2 Ω−2/3, whereas the chemical potential is in Ω−1, so that µ is very

2mL2

small as compared to ε2.

The N − N1 particles which are not located in ε1 are distributed among the excited states, for which one can take µ = 0 without introducing a large error,

Indeed shifting the limit of the integral from ε2 to ε1 is adding a relative contribution to (9.20) of the order of (ε2/kB T )1/2. Since the volume Ω is macroscopic, this quantity is very small, ε2 being much less than kB T .

One recognizes in (9.21) the integral (9.13) under the condition of replacing βB by β. Whence :

Thus, when lowering the temperature, di erent regimes take place :

–as long as T > TB , the chemical potential is negative and the particles are distributed among all the microscopic states of the system according to the Bose-Einstein distribution (9.6);

–when T = TB , µ almost vanishes (it is very small but negative);

–when T < TB , the particles number N1 on the fundamental level becomes macroscopic, this is the “Bose condensation.” One thus realizes a macroscopic quantum state. The distribution of particles on the excited levels corresponds to µ 0. When T continues to decrease, N1 increases and tends to N when T tends to zero; µ is practically zero.

Going through TB corresponds to a phase transition : for T < TB the Bose gas is degenerate, which means that a macroscopic particles number is located in the same quantum state. This corresponds for 42He to the superfluid state. The phenomenon of helium superfluidity is very complex and the interactions between atoms, neglected up to now, play a major part in this phase. The above calculation predicts a transition to the superfluid state at 3.2 K in standard conditions (4He specific mass =140 kg/m3 under the atmospheric pressure) whereas the transition occurs at Tλ = 2.17 K. This is a second-order phase transition (the free energy is then finite, continuous and derivable; the specific heat is discontinuous).

The superconductivity of some solids also corresponds to a Bose condensation (2003 Nobel prize of A. Abrikosov and V. Ginzburg). It was possible to observe the atoms’ Bose condensation by trapping and cooling atomic beams at temperatures of the order of a mK in “optical molasses” (1997 Nobel prices of S. Chu, C. Cohen-Tannoudji and W.D. Phillips).