Definition of specific resistance of semiconductors. Temperature dependence of electrical conductivity of own and impurity semiconductors Temperature conductivity
The study of the electrical properties of materials includes the determination of electrical conductivity and its temperature dependence. For metals, the temperature coefficient of electrical conductivity is negative, that is, the electrical conductivity of metals is reduced by increasing the temperature.
For semiconductors and many dielectrics, the temperature coefficient of own electrical conductivity is positive. The electrical conductivity also grows with the introduction of defects and impurities in its own semiconductor.
The electrical conductivity of ionic crystals is usually increasing with increasing temperature and near T. PL reaches the conductivity of liquid electrolytes (s NaCl at 800 ° C. 10 -3 Ohm -1 × cm -1), while at room temperature is chemically pure NaCl - insulator.
In the crystals of alkali metal halides (for example, NaCl), the cations are more mobile than anions:
Fig. 6 - Migration of cationic vacancies (or Na + ions) in NaCl
therefore, the magnitude of the ionic conductivity NaCl depends on the number of available cationic vacancies.
The amount of cationic vacancies in turn strongly depends on the chemical purity and thermal prehistory of the crystal. Increasing the number of thermodynamically equilibrium own vacancies occurs either when the crystal is heated,
| (22) |
either the introduction of heterovalent impurities can occur vacancies that compensate for the excess charge of impurity cations.
So, when adding small amounts of MnCl 2, NaCl + MnCl 2 ® Na 1-2 X.MN. X.V Na. x. CL (solid solution), where each ion Mn 2+ accounts for one associated cationic vacancy, i.e. An impurity vacancies arise (V Na). Such jobs are called impurity, since in pure NaCl they cannot form.
At low temperatures (~ 25 ° C), the concentration of vacancies of thermal origin is very small. Therefore, despite the high purity of the crystal, the number of own vacancies remains much less impurity. And when the temperature increases, there is a transition from impurity to its own conduction.
The temperature dependence of the ion conductivity is subject to the Arrhenius equation:
s \u003d. \u003d A.exp ( -E -E A./RT), (23)
where E A. - Energy activation of electrical conductivity.
The pre-exponential factor A includes several constants, including the frequency of oscillations of potentially moving ions. The graphic dependence of Ln S from T -1 should be expressed by the straight line with an angle of inclination -e / r. In some cases, a multiplier of 1 / t is introduced in the processing of temperature dependence in the pre-seential factor. In this case, the graphical dependence is taken to be submitted in the coordinates of LN ST - T -1. The slope of the resulting (E / R) can be somewhat different from tilt in Arrhenius coordinates. Arrhenius dependence for NaCl is schematically shown in Fig. 7. In the low-temperature impurity region, the number of vacancies is determined by the impurity concentration and for each concentration level is the magnitude of constant. In fig. 7 This corresponds to a number of parallel straight lines, each of which corresponds to the conductivity of crystals with different content of the alloying additive.
Fig. 7 - The dependence of the ionic conductivity of NaCl on temperature. Parallel lines in the impurity region correspond to different concentrations of alloying impurities
In the impurity region, the dependence S on temperature is determined only by the temperature dependence of the mobility of m cations, which also obeys the Arrhenius equation:
m. = m 0 EXP ( - E. moment / Rt.), (23)
where E. MiG - Energy activation of media migration.
and naCl \u003d 0.564 nm; d na - cl \u003d a / 2 \u003d 0.282 nm; R na + \u003d ~ 0.095 nm; R Cl - \u003d ~ 0.185 nm.
The Na-Cl Communication Length, calculated as the sum of these ion radius, turns out to be ~ 0.28 nm, which is close to the experimentally found value.
Fig. 8 - Na + ion migration path in NaCl
Fig. 9 is a triangular interstice through which the moving Na + ion in NaCl must pass. R / - radius inscribed circle; Circle 1-3 depict CL ions with radius X / 2.
In the impurity region (Fig. 7), the conductivity seems to depend on the concentration of vacancies
s \u003d. nem 0 EXP (- E. moment / Rt.). (24)
At a higher temperature in the field of own conductivity, the concentration of vacancies of thermal origin exceeds the concentration of vacancies due to alloying additives and the number of vacancies n. Depends on the temperature according to the Arrhenius equation:
n \u003d N.× Const × exp ( -E. Obr. / 2RT). (25)
This equation is identical to equation 22, in which E arr / 2light is the energy of activation of the formation of one praying of codament vacancies, that is, half of the energy required for the formation of one praying Schottky defects. The vacancy mobility is still described by equation 23, and thus, in general, electricity in the field of own conductivity is subject to the equation
s. \u003d N.× const × m 0 exp (- E. moment / Rt.) exp (- E. Obr. / 2RT)(26)
.
(27)
Fig. 10 - Temperature dependence of ionic conductivity of "pure" NaCl
Deviations from linear dependence near T. PL are associated with an increase in the mobility of anionic vacancies, as well as with long-range (Debay Hyukkelevs) interactions of cation and anionic vacancies, leading to a decrease in vacancy education energy. Deviations from linearity in the field of low temperatures are determined by the formation of complexes of defects that can be destroyed only at a certain activation energy.
In tab. 7 shows the activation energy of NaCl crystals.
Table 7 - NaCl crystals activation energy
Temperature dependence of electrical conductivity is known for a long time. However, it was not used to predict chemical processes in solids.
In 1987, an unknown regularity of the pyrometallurgical restoration of elements from oxides was experimentally established, which consists in simultaneously changing the type of conductivity of oxides (from impurity to its own) and their reactivity, due to an increase in the concentration of free electrons in the crystal semiconductor oxide lattice. In other words, the reduction of oxides begins at a temperature corresponding to the transition from impurity conductivity to its own.
Dielectrics. Dielectric materials are used in electronics for the manufacture of passive elements (rigid substrates, tanks, masks), as well as active elements (capacitors and electrical insulators).
Dielectrics to which most of the ionic crystals relate are characterized by
High electrical strength, i.e. resistance to degradation (structure change) at high electric field strengths and transition to a conducting state;
Low dielectric losses (TGD), i.e. loss of energy of the variable electric field, which are highlighted in the form of heat.
The dielectric properties of materials are determined when studying flat capacitors representing two flat-parallel conductive plates located apart from each other at a distance of D, which is much less than the size of the plates (Fig. 6).
Fig. 6 - capacitor with parallel plates and dielectric between them
Capacity capacitor in vacuum
C. 0 = e 0 S / D., (28)
The dielectric permeability of the vacuum in the international system of physical quantities (C) is the size of
e 0 = 10 7 / 4Ps 2 \u003d 8,854 × 10 -12 F / m. (29)
When applied on the potential difference plate V, the capacitor poins the charge Q of equal
Q. 0 = C. 0 V.. (30)
If there is a dielectric between the plates, when the potential is imposed, the charge is increasing to Q 1, and its capacity is up to 1.
For dielectric with charge size Q 1 and capacity C. 1 Dielectric constant is associated with the capacity of the following ratio
e "\u003d. C. 1 / C 0. (31)
For air E "" 1;
for most ionic compounds E "~ 5 ¸ 10;
for ferroelectrics (Btio 3) E "\u003d 10 3 ¸ 10 4.
e "depends on the degree of polarization or shift of charges occurring in the material.
Polarizability of dielectric A - coefficient connecting the dipole moment ( r) and local electric field ( E.).
p \u003d.a. E., (32)
and a \u003d a E. + A. I. + A. D. + A. S., (33)
where A. E. - Displacement of the electronic cloud,
a. I. - ions,
a. D. - dipoles,
a. S. - Volume charge.
Electronic polarizability A. E. It occurs as a result of the displacement of electronic orbitals of atoms relative to the nuclei and inherent in all solid bodies. In some solids, such as diamond, a E. - the only component of polarizability;
Ion polarizability A. I. - associated with the relative displacement or separation of cations and anions in the solid (determines polarization in ionic crystals);
Dipole Polarizability A. D. - It occurs in substances having permanent electric dipoles (H 2 O, HCl), which can be lengthened or change the orientation under the action of the field. At low temperatures a D. frozen.
Volume and Charger A S. It occurs in "bad" dielectrics and is determined by the migration of carriers over long distances. In NaCl, the migration of cations occurs in the cationic vacancies to the negative electrode. As a result, a double electric layer occurs, which leads to an increase of E "(applying e" order 10 6 ... 10 7 appears, which corresponds to the capacity of the double electric layer (18 ... 36 μF / cm 2).
By deposit in the size of polarization and dielectric constant
a. S. \u003e A. D. \u003e A. I. \u003e A. E..
These components of polarizability are found from capacitive, microwave and optical measurements in a wide frequency interval ( f.) (Fig. 7).
|
For f. < 10 3 Гц все aдают вклад в величину p..
For f. \u003e 10 6 In most ion crystals, the bulk charge does not have time to form.
For f.\u003e 10 9 (microwave) there is no polarization of dipoles.
In area f. \u003e 10 12, corresponding to the oscillations of the optical range, the only component of the polarization remains E.Coaching is still observed in the UV region, but disappears at frequencies corresponding to the X-ray range. In good dielectrics that do not have a D. and A. S.The permeability at the low frequency e "0 is determined mainly by ion and electron polarization. The value of E" 0 can be obtained from the measurements of the container using the AC bridge. For this, the container is measured twice - without a substance under study between the plates of the condenser and with the substance (equation 31). The value of E "¥, connected only with electron polarizability, can be found from the measurements of the refractive index in the visible region of the spectrum based on the simple ratio of E". For example, for NaCl E "0 \u003d 5.62; E" ¥ \u003d 2.32.
where w \u003d 2p f. (angular frequency),
t - Relaxation time (currently to describe complex polarization processes in dielectrics. distribution of relaxation times).
Dielectric loss tangent is determined by the ratio
e // / E " \u003d TGD (36)
Fig. 9 - frequency dependence E / and E //
In the range of inter / 0 and e / ¥ The dielectric constant is represented in the form of a complex value of E * \u003d E / - JE // where E // is the real component that is located from the following relationship:
where W-angle frequency equal to 2PF, W p is the frequency of the rear of the current carriers, and P 1 and N 2 -Constructants. The basis of this equation is the idea that individual polarization phenomena, whether the jumps of ions in the conductors or reorientation of dipoles in dielectrics, occur independently from each other, but as a result of cooperative interaction. This means that if a separate dipole in the crystal is reoriented, it thus influences the surrounding dipoles. At the current level of understanding, however, it is unclear how on the basis of the Law of the Jonecher to come to the quantitative description of cooperative phenomena. In more detail, the diagrams in the complex plane are discussed in ch. 13 (but at the same time an acceptance is made on the performance of the conductivity, and not dielectric properties).
In tab. 8 shows the values \u200b\u200bof the dielectric constant of some oxides at different frequencies and temperatures.
Table 8 - Dielectric constant of some oxides
| Oxide | frequency Hz | T.,TO | E " | Oxide | frequency Hz | T.,TO | E " |
| H 2 O (Ice) | 10 8 | 3,2 | Veo | 10 5 | 6,3 | ||
| N 2 About liquid | 10 8 | 88,0 | Al 2 O 3 | ~10 6 | 10–12 | ||
| TiO 2. | 10 4 | ||||||
| H 2 O (Couples) | 10 6 | 1,013 | WO 3. | ~10 8 | |||
| SiO 2. | 3.10 7 | 4,3 | Zno. | 10 6 | |||
| SiO. | >10 8 | 2,6...4,0 | PBO. | 4,5.10 3 | |||
| NB 2 O 5 | ~10 12 | 35…50 | PBO 2. | ~10 8 | |||
| SNO 2. | ~10 12 | 9–24 | TB 4 O 7 | 10 6 | |||
| MNO. | 4.4 × 10 8 | 13,8 |
The relationship between ion and electron polarization is a measure of the ordering of electrons relative to the ions of the crystal lattice
. (39)
From those shown in table. 9 data follows that even a small change h leads to a significant change in the properties of passive elements of microelectronics ( U. Pr - breakdown voltage, D G. 0 - free energy of education). The higher H, the greater the electronic polarization is relatively complete and the greater the possibility of controlling polarization using the electric field.
| |
| Dielectric | FROM, μF / cm | E " | TGD. | U. Pr, B. | H. | -D. G. 0, kj / mole |
| at 10 3 Hz | ||||||
| TA 2 O 5 | 0,15 | 1,5 | 0,48 | |||
| Al 2 O 3 | 0,085 | 1,0 | 0,49 | |||
| Al 2 (SiO 3) 3 | 0,01 | 6,5 | 0,3 | 0,50 | ||
| SiO. | 0,014 | 0,1 | 0,52 | |||
| SiO 2. | 0,0046 | 0,1 | 0,55 | |||
| Aln. | 0,045 | 7,2 | 0,01 | – | 0,75 | |
| Si 3 N 4 | 0,04 | 6,5 | 0,001 | – | 0,94 | |
| LA 2 O 3 | 0,05...1,0 | 0,02 | – | 0,60 | ||
| Natao 3. | 0,6 | 0,01 | – | 0,50 |
The greatest value for assessing the quality of dielectrics at high frequencies has a relationship between ionic and electronic components of polarization, i.e. between and as well as the value of the tangent of dielectric losses (TGD). When the AC passes through the capacitor at low frequencies, the current vector is ahead of phase by 90 ° voltage vector. Then the product of vectorsi × v \u003d 0 and energy is transmitted without loss. When increasing the frequency, ion polarization appears and the current and voltage phases are offset. In this case, the current component of the current I × sind occurs, which is in one phase with a voltage.
The value of TGD for high-quality dielectrics is about 0.001.
For capacitors rated FROM \u003e 50 PF TGD does not exceed 0.0015,
and with a capacity of about 0.01 μF TGD ~ 0.035.
The properties of dielectrics have a significant impact on the quality of the MOS structures used in microelectronics. These properties are determined by voltechotic or voltfarad characteristics ( C-V. or VFC methods).
Segneto, piezo and pyroelectrics. Polarization of crystals belonging to centrosymmetric point groups is removed after removal of the field. However, from 32 point groups 21 does not contain a symmetry center. In this regard, there are phenomena of residual polarization in electrical, mechanical and thermal fields. In accordance with these phenomena, the classes of ferroneto, piezo and pyroelectrics are distinguished.
Segnetoelectricsdiffer from conventional high E dielectrics " and residual polarization, that is, they have the ability to preserve some residual electric polarization after removing the external electric field. Therefore, with equal volumes, condensers from ferroelectrics have 1000 times a large container. In addition, in contrast to conventional dielectrics, which observes a proportional increase in the induced polarization of P or induced charge Q (equation 30), in segnetryelectrics, dependence between polarization value ( R, CL / cm 2) and the electric field strength is characterized by hysteresis. (Fig. 11) The shape of the hysteresis determines the residual polarization value ( P R.) and the coercive field ( N S.), which removes polarization. Segenelectrics are characterized by the presence of polarization of saturation P S at high electrical stresses, for example, for Batio 3 P S. \u003d 0.26 Kl / cm 2 at 23 ° C and residual polarization P R, i.e. polarization persisting after the elimination of an external electric field. In order to reduce polarization to zero, it is necessary to apply the electrical field of the E E reverse sign, called the coercive field.
Fig. 11 - hysteresis loop for typical segnetodielectric. The dashed line passing through the origin shows the behavior of an ordinary dielectric.
Some of the ferroelectrics are shown in Table. 10. All of them possess structures in which one cation, for example, Ti 4+ in Batio 3 can be significantly shifted (~ 0.01 nm) relative to its anionic environment. This shift of charges leads to dipoles and a large value of the dielectric constant, which is characteristic of ferroelectrics.
Table 10 - Curie temperature of some ferroelectrics
In fig. 12 shows the elementary titanate cell of the SRTIO 3 strontium, which has both Batio 3, the structure of the Batio 3 perovskite type. Ti 4+ ions occupy the vertices of this cubic primitive cell, O 2- - in the middle of the ribs, ion strontium in the center of Cuba. However, it is possible to submit the structure of WATIO 3 and otherwise: ions of VA 2+ are placed in the vertices of the cube, Ti 4+ - in the center, and ions O 2- in the center of the faces. However, in no dependence on the choice of elementary cell, the structure is picked from octahedra TiO 6, forming a three-dimensional framework by means of joint vershas of strontium ions in this frame structure occupy emptiness with Kch \u003d 12.
Fig. 12 - PEROVOKITES STRUCTURE SRTIO 3
From a chemical point of view (the possibility of quantum-chemical calculation and experimental control of the properties of dielectrics), the perovskite structure consists of octahedra TiO 6, and Ba 2+ ions are placed in the resulting voids. In such an ideal structure that exists at temperatures above 120 ° C, all charges are arranged symmetrically, its own dipole moment and Batio 3 is a conventional dielectric with high E " . When the temperature decreases, Ti 4+ ions decreases to the octahedron vertex at 0.1 Å (with an average Ti-O \u003d 1.95 m)), which is confirmed by the data of x-ray diffraction analysis, i.e. There are distortions that manifest themselves that the TiO 6 octahedra cease to be symmetrical. There is a dipole moment, and as a result of the interaction of dipoles - spontaneous polarization (Fig. 13).
If such displacements occur simultaneously in all octahedra TiO 6, then the material occurs its own spontaneous polarization. In the segroelectric Watio 3 each of the octahedra TiO 6 is polarized; The effect of the external electric field is reduced to the "forced" orientation of individual dipoles. After building all the dipoles along the direction direction, the saturation polarization state is reached. The distance that titanium ions from octahedra centers are shifted to one of oxygen, according to estimates based on the experimentally observed value of P A, is 0.01 nm, which is also confirmed by x-ray analysis data. As can be seen, this distance is sufficiently small in comparison with the average ti-o communication length in octahedra TiO 6, equal to 0.195 nm. The ordered orientation of dipoles is schematically shown in Fig. 13, and where each arrow corresponds to one distorted octahedron TiO 6.
Fig. 13 - scheme orientation of the polarization vector of structural units in ferroelectrics (A), anti-segmentoelectric (b) segnetelectric (B)
In ferroelectrics like Batio 3, domain structures are formed due to the fact that the adjacent TiO 6 dipoles are spontaneously line up parallel to each other (Fig. 14). The size of the formed domains is varied, but, as a rule, can reach tens in cross section - hundreds of angstrom. Within one domain dipoles are polarized in one crystallographic direction. The own polarization of some sample of the ferroelectric is equal to the vector sum of polarizations of individual domains.
Fig. 14 - ferroelectric domains separated by a blast wall (boundary)
The imposition of an external electric field leads to a change in its own polarization of a segmentoelectric sample; The reason for such changes may be the following processes:
1) Changing the direction of polarization of domains. This will happen if all TiO 6 dipoles within the domain under consideration replace their orientation; For example, all dipoles in the domain (2) (Fig. 14) change the orientation to the parallel dipoles of the domain (1);
2) an increase in polarization within each domain, which is especially likely if there was some disorder in the orientation of the dipoles;
the movement of domain walls, as a result of which the size of domains oriented along the field increases by reducing the domains with adverse orientation. For example, Domain 1 (Fig. 14) can grow when the domain wall shift is one step to the right. To carry out such a shift, dipoles on the domain 2 boundary must take the orientation shown by the stroke arrows.
The ferroelectric state is usually observed at low temperatures, since the heat movement, increasing with increasing temperature, disrupts the consistent nature of the displacement in neighboring octahedra, and therefore violates the domain structure. The temperature at which this destruction occurs is called a ferroelectric point of Curie T A (Table 10). The above T with materials become paleeclates (i.e., "nonsenselelectrics"); Their dielectric permeability is still high values \u200b\u200b(Fig. 15), but the residual polarization in the absence of an external field is no longer observed.
Above with the value of E "is usually described by the law of Curie - Weiss:
e / \u003d C / (T-Q) (37)
where C is a permanent Curie and Q - the temperature of Curie -weiversa. As a rule, T C and Q coincide or differ in just a few degrees. The transition from the ferroelectric to a paraleclectric state at T C is an example of a phase transition order - a mess. However, in contrast to transitions, the order - the disorder observed, say, in bronze, does not occur the diffusion displacement of ions over long distances. Below, the ordering is carried out by preferential distortion or agreed inclination of polyhedra and is thus referred to phase transitions with displacement ( gL 12). In the high-temperature paleelectric phase of distortion and the slope of polyhedra, if present, then at any case are random character.
The required condition for spontaneous polarization and ferroelectric properties in the crystal is that the latter should relate to a spatial group that does not have a symmetry center ( gL 6.). Paraelectric phases, stable above T C, are often centrosymmetric, and the ordering occurring during cooling is reduced to a decrease in symmetry to a non-centrosymmetric spatial group.
Currently, several hundred segainelectric materials are known, among which a large group of oxide compounds are distinguished with a distorted (non-commic) perovskite structure. These compounds contain such cations that "feel" conveniently in a distorted octahedral environment - Ti, Ni, Ta; The non-equality of connections within such distorted octahedra MO 6 is the cause of polarization and dipole moment. Not all perovskite-ferroelectrics, for example, as opposed to Batio 3 and RBTio 3 Satio 3 does not exhibit ferroelectric properties, which, apparently, is associated with the difference in the size of two-chain cations. A large radius of iona of Va 2+ causes an extension of the elementary cell compared to Satio 3, which in turn leads to large lengths of Ti-O in Watio 3 and greater displacement of Ti 4+ ions inside the TiO 6 octahedra. The composition of other oxides with ferroelectric properties includes cations whose bonds with oxygen ions are unequal due to the presence of a free e-pair of a path of the outer shell; These may be heavy p-element cations that meet the degrees of oxidation, two units smaller than the limit for this group, such as Sn 2+, Pb 2+, Bi 3+, etc.
Segroesoelectric oxides are used to make capacitors due to high dielectric constant, which is especially large near T C (Fig. 15). Therefore, pursuing a practical goal to increase should create materials with points of Curie close to room temperature. In particular, the Curie temperature for Batio 3 120 ° C (Fig. 15) can be significantly reduced, and the temperature interval of the transition is expanded by partial replacement of BA 2+ or Ti 4+ by other cations: Replacement of BA 2+ on SR 2 + Causes the compression of the elementary cell of the structure and decrease T C; The substitution of "active" Ti 4+ -IONs with other "inactive" quadrol charges by ions, in particular Zr 4+ and Sn 4 +, leads to a sharp drop of T with.
Fig. 15 - Temperature dependence of the dielectric constant of ceramic Batio 3
In antsygroelectrics, spontaneous polarization is also observed, similar in nature with the polarization of ferroelectrics. Individual defenses of anticegietoelectrics are ordered relative to each other in such a way that each dipole is aityia-parallel adjacent dipoles (Fig. 14, b). As a result, its own spontaneous polarization of the material turns out to be zero. Above the anti-eelectric point of Curie, the material becomes a normal paraelectric. PBZRO 3 (233 ° C), niobat sodium nanbo 3 (638 ° C) and ammonium dihydrophosphate NH 4 H 2 PO 4 (-125 ° C) are examples of substances with anti-selection properties (numbers in brackets indicate the corresponding points of Curie) .
| | ¯¯¯¯¯ | |
| | ¯¯¯¯¯ | |
| | ¯¯¯¯¯ | |
| Segnetoelectric Batio 3. | Antsegnetoelectric PBZRO 3. | Segnetelectrics (BI 4 Ti 3 O 12, Tartrates) |
Fig. 16 - scheme orientation of the polarization vector of structural units in specific representatives of ferroelectrics (A), anti-segnetoelectrics (b) segnetelectrics (B)
In anti-seepoelectrics, spontaneous polarization occurs ( P S. \u003d 0), hysteresis is missing, but near T. kr also observed the maximum E " .
The magnitude of the electric field intensity can affect the phase
transitions of the second kind in ferroelectrics (Fig. 14).
Fig. 1 - Effect of temperature on orientational phase transitions
type order of confusion in PBZRO 3
Fig. 16 - dependence of the transition temperature of the antsegnetoelectric -zegnetoelectric in PBZRO 3 from the applied voltage (A) and the behavior of polarization with this transition (b)
|  |
|
| but | B. |
Fig. 17 - the structures of the ferroelectric KH 2 PO 4 (A) and the anti-ethylenectress NH 4 H 2 PO 4 (b) (projection on the plane)
In pyroelectrics In contrast to segroelectrics, the direction of the polarization vector cannot be changed by an external electric field, and polarization depends on temperature change:
D. P s \u003d.pd. T., (38)
where P is a pyroelectric coefficient.
Pyroelectric properties are detected when heated as a result of the expansion of the crystal lattice and the change in the length of the dipoles. An example of a pyroelectric compound is a ZnO crystal, which includes layers of oxygen ions (hexagonal tight packaging) and Zn 2+ ions in tetrahedral voids. All Zno tetrahedra are oriented in one direction and have a dipole moment, as a result of which the crystal is in a polarized state. The pyroelectric effect is masked by the adsorption of water and is detected when heated.
Fig.18 - ordered tetrahedral structures of Wurzit. The oxygen ions layer are shown and the placement of Ti + cations by interstitial.
Piezoelectrics Also belong to non-centrosymmetric point groups of crystals. Polarization and electrical charge on the opposite edges of the crystal occur under the action of mechanical fields and depend on the direction of the field. In quartz, polarization occurs when compressed along the direction (100) and is absent when compressed along the axis (001).
Piezoelectrics Many crystals with a tetrahedral structure, the distortion of which leads to polarization (quartz, ZNS, ZnO). A similar piezoelectric effect (PEE) is observed in LA 2 S 3. An important group of piezoelectrics is solid solutions PBTIO 3 and PBZRO 3. All ferroelectrics are pyro- and piezoelectrics, but not all pyro and piezoelectrics are ferroelectrics.
Fig. 19- phase diagram of the CTS system
For semiconductors with one charge carrier, the electrical conductivity γ is determined by the expression
where n is the concentration of free charge carriers, M -3; Q is the charge value of each of them; μ is the mobility of charge carriers equal to the average rate of charge carrier (υ) to field strength (E): υ / E, m 2 / (b ∙ c).
Figure 5.3 shows the temperature dependence of the concentration of carriers.
In the field of low temperatures, the dependence of the relationship between points A and B characterizes only the concentration of carriers due to impurities. With increasing temperature, the number of carriers supplied by impurities increases until electronic resources of impurity atoms (point b) are expelled. On the section B-in impurities have already been exhausted, and the transition of the electrons of the main semiconductor through the forbidden zone is not yet detected. The portion of the curve with a constant concentration of charge carriers is called the impurity exhaustion area. In the future, the temperature increases so much that the rapid increase in the concentration of carriers begins due to the transition of electrons through the prohibited zone (section in G). The slope of this area characterizes the width of the forbidden semiconductor zone (tanglex angle α angle gives ΔW value). The slope of the A-B section depends on the ionization energy of the impurities ΔW n.
Fig. 5.3. Typical dependence of charge carrier concentration
in semiconductor on temperature
Figure 5.4 presents the temperature dependence of the mobility of the charge carrier for the semiconductor.
Fig. 5.4. Temperature dependence of carrier mobility
charge in semiconductor
An increase in the mobility of free charge carriers with an increase in temperature is due to the fact that the higher the temperature, the greater the thermal speed of the free carrier υ. However, with further increasing temperature, thermal oscillations of the lattice and charge carriers begin to face it more and more often, the mobility falls.
Figure 5.5 presents the temperature dependence of the electrical conductivity for the semiconductor. This dependence is more complicated, since the electrical conductivity depends on the mobility and number of media:
In the area of \u200b\u200bAB, the height of the specific electrical conductivity with an increase in temperature is caused by an admixture (tilting the line on this section determine the activation energy of the impurities W P). At the BW section, saturation occurs, the number of carriers does not grow, and the conductivity falls due to a decrease in the mobility of charge carriers. In the Vg section, the conductivity growth is due to an increase in the number of electrons of the main semiconductor overcoming the prohibited zone. Tilt the straight on this area determine the width of the forbidden zone of the main semiconductor. For approximate calculations, you can use the formula
where the width of the forbidden zone W is calculated in eV.
Fig. 5.5. Temperature dependence of the electrical conductivity
for semiconductor
In laboratory work, a silicon semiconductor is investigated.
Silicon, like Germany, refers to the IV group of Table D.I. Mendeleeva. It is one of the most common elements in the earth's crust, its content in it is about equal to 29%. However, in a free state in nature, it is not found.
Technical silicon (about one percent of impurities), obtained by recovery from dioxide (SiO 2) in an electrical arc between graphite electrodes, is widely used in ferrous metallurgy as a doping element (for example, in electrical steel). Technical silicon as a semiconductor is used can not be. It is the initial raw material for the production of silicon semiconductor purity, the content of impurities in which should be less than 10 -6%.
The technology of obtaining silicon semiconductor purity is very complex, it includes several stages. The final silicon cleaning can be performed by the method of zone melting, while there are a number of difficulties, since the melting point of silicon is very high (1414 ° C).
Currently, silicon is the main material for the manufacture of semiconductor devices: diodes, transistors, stabilids, thyristors, etc. In silicon, the upper limit of the operating temperature of the devices can be dependent on the degree of purification of materials 120-200 o C, which is significantly higher than in Germany.
As we have already seen, the specific conductivity is expressed by the formula
where n is the concentration of charge carriers that determine the conducting properties of this body, and U is the mobility of these carriers. Charge carriers can be both electrons and holes. It is interesting to note that, although, as you know, the majority of metals are free chargers of charges are electrons, in some metals the role of free chargers of charge performs holes. Typical representatives of metal with hole conductivity are zinc, beryllium and some others.
To determine the dependence of the conductivity on temperature, it is necessary to know the temperature dependence of the concentration of free carriers and their mobility. In metals, the concentration of free charge carriers does not depend on temperature. Therefore, the change in the conductivity of metals depending on the temperature is fully determined by the temperature dependence of carriers mobility. In semiconductors, on the contrary, the concentration of carriers dramatically depends on temperature, and the temperature changes of mobility are practically imperceptible. However, in those areas of temperature, where the concentration of carriers is constant (the area of \u200b\u200bdepletion and the area of \u200b\u200bsaturation of impurities), the course of temperature dependence of the conductivity is fully determined by the temperature change of carrier mobility.
The meaning of the mobility itself is determined by the processes of carrier scattering on various defects of the crystal lattice, that is, by changing the velocity of the directional movement of carriers when they interact with various defects. The interaction of carriers with ionized atoms of various impurities and thermal fluctuations of the crystal lattice are most significant. In various areas of temperature, the scattering processes caused by these interactions affect differently.
In the field of low temperatures, when thermal fluctuations of atoms are so small that they can be neglected, scattering on ionized impurity atoms is basic. In the region of high temperatures, when in the process of thermal oscillations, the lattice atoms are significantly shifted from the position of a stable equilibrium in the crystal, thermal scattering is performed on the fore.
Scattering on ionized impurity atoms. In impurity semiconductors, the concentration of impurity atoms is many times the concentration of impurities in metals. Even with a sufficiently low temperature, most of the impurity atoms are in an ionized state, which seems quite natural, since the origin of the conductivity of semiconductors is primarily associated with the ionization of impurities. The scattering of carriers on the ions of the impurity is much stronger than scattering on neutral atoms. This is explained by the fact that if the scattering of the carrier on a neutral atom occurs with a direct collision, then for scattering on an ionized atom, a sufficient carrier to get into the area of \u200b\u200bthe electrical field created by ion (Fig. 28). When the electron flies through the area of \u200b\u200bthe electric field created by a positive ion, its flight trajectory undergoes a change, as shown in the figure; In this case, the rate of its directional movement υ E, acquired by exposing the external field, decreases before if the electron passes closely near the ion, then after scattering the direction of the electron motion may be generally the opposite direction of the external electric field.
Considering the task of scattering of charged particles on charged centers, an outstanding English physicist E. Rutherford concluded that the length of the free mileage of the particles is proportional to the fourth degree of their speed:
The use of this dependence on the scattering of carriers in semiconductors led to very interesting and, at first glance, an unexpected result: the mobility of carriers in the field of low temperatures should grow with increasing temperature. In fact, the carrier mobility turns out to be proportional to the speed of their movement:
At the same time, the average kinetic energy of charge carriers in semiconductors is proportional to temperature A, it means that the average thermal speed is proportional to the root square 
Consequently, media mobility is located in the following temperature:
In the field of low temperatures, when scattering on ionized impurities plays the main role and when the thermal fluctuations of the lattice atoms can be neglected, the carrier mobility increases as the temperature increases in proportion to the left branch of the U (T) curve in Figure 29). Qualitatively, such a dependence is quite explained: the greater the thermal speed of the carriers, the less time they are in the field of an ionized atom and the less distortion of their trajectory. Due to this, the length of the free path of carriers increases and their mobility increases.
Scattering on thermal oscillations. With increasing temperature, the average speed of thermal carrier movement increases so much that the probability of their scattering on ionized impurities becomes very small. At the same time, the amplitude of thermal oscillations of the lattice atoms increases, so that the scattering of carriers on thermal fluctuations is performing. Thanks to the growth of scattering on heat oscillations, the length of the carrier free path is heated as the semiconductor is heated and, therefore, their mobility.
The specific course of dependence in the field of high temperatures for various semiconductors of non-refinery. It is determined by the nature of the semiconductor, the width of the prohibited zone, the concentration of impurities and some other factors. However, for typical covalent semiconductors, in particular for Germany and silicon, with not too large concentrations of impurity, the dependence U (T) has the form:
(See the right branch of the curve in Figure 29).
So, the mobility of media in semiconductors in the field of low temperatures is growing directly proportionally and in the region of high temperatures, it falls back proportionally
Semiconductor conductivity dependence on temperature. Knowing the temperature dependence of the movement and concentration of carriers in semiconductors, the nature of the temperature dependence of the conductivity of semiconductors can be established. Schematically addiction 
Showing in Figure 30. The course of this curve is very close to the course of the curve 
shown in Figure 25. Since the dependence of the carrier concentration on the temperature is much stronger than the temperature dependence of their mobility, then in the regions of the impurity conductivity (section AB) and its own conductivity (section CD), the dependence of the specific conductivity σ (t) is almost entirely determined by the dependence of the dependence of the carrier concentration on temperature . The angles of inclination of these sections of the graph are dependent on the energy of the ionization of the donor impurity atoms and on the width of the forbidden semiconductor zone. Tangent angle of inclination Γ N proportional to the energies of the separation of the fifth valence electron atom of the donor impurity. Therefore, having received an experimentally graph of the change in the conductivity of the semiconductor when heated on the impurity section AB, it is possible to determine the value of the activation energy of the donor level, that is, the energy distance of the donor level W D from the bottom of the conduction zone (see Fig. 20). Tangent The angle of inclination γ I is proportional to the electron transition energy from the valence zone to the conduction zone, that is, the energies of creating their own carriers in the semiconductor. Thus, having obtained experimentally prior to the dependence of the conductivity on the temperature on its own segment CD, it is possible to determine the width of the forbidden zone W G (see Fig. 17). The values \u200b\u200bof W d and W g are the most important characteristics of the semiconductor.
The main difference between the dependences σ (T) and N (T) is observed on the BC section located between the depletion temperature of the impurities T s and the temperature of the transition to its own conductivity t i. This area corresponds to the ionized state of all impurity atoms, and to create their own conductivity, the energy of thermal fluctuations is still insufficient. Therefore, the concentration of carriers, being almost equal to the concentration of impurity atoms, does not change with increasing temperature. The movement of the temperature dependence of the conductivity in this area is determined by the course of dependence on the temperature of the carrier mobility. In most cases, at a reasonable concentration of impurities, the main mechanism of media scattering in this temperature range is scattering on the heat oscillations of the lattice. This mechanism determines the reduction of carrier mobility and, consequently, the conductivity of semiconductors with increasing temperature on the BC site.
In degenerate semiconductors, due to the large concentration of impurities due to the overlap of electrical ions, the scattering of carriers on ionized impurity atoms retains the main value up to high temperatures. And for this mechanism, the scattering mechanism is characterized by an increase in carrier mobility with an increase in temperature.
Semiconductor is called materials, the main feature of which is the dependence of the specific electrical conductivity from external energy impacts, as well as on the concentration and type of impurities.
Qualitative differences in the properties of semiconductors and
Vodnov are determined by the type of their chemical bonds. In metals, the valence electrons of the atoms of the crystal lattice are part of the collective of equivalent charge carriers, called electron gas (metal communication). The number of these is
charges of charge, which corresponds to the number of atoms in
Nice volume of the crystal lattice. It is impossible to change this concentration of charge carriers to the influence of the external factor (temperature, irradiation, administration of impurities, deformation, etc.). Hence all the features of conductivity conductors: a positive temperature coefficient of specific resistance, independence of the concentration of charge carriers from impurities in the lattice, superconductivity, etc.
In semiconductors, all valence electrons of atoms participate in the formation of a covalent (or ion-covalent) of a saturated chemical bond. With in semiconductor crystals, there is not a single quasi-free charge carrier capable of participating in a directional movement when exposed to an external factor, that is, at the absolute zero temperature, the semiconductor does not have electrical conductivity. The strength of the covalent (ion-covalent) communication (communication energy) corresponds to the width of the prohibited semiconductor zone. At temperatures other than 0 K, part of the charge carriers, having thermal energy can break the chemical bond, which leads to the formation of an equal amount of electrons in the conduction zone and holes in the valence zone. The process of thermoteration of charge carriers is probabilistic, and in the case of generating own carriers The charge of their concentration is determined by the relation
where I. - effective densities of states, given, respectively, to the bottom of the conduction zone to the ceiling of the free zone.
To control the type of electrical conductivity and the conductivity value of the semiconductor into the nodes of its crystal lattice are introduced in a low concentration of impurities with valence,
differing in a large or smaller side of the valence of the main semiconductor atoms. Such impurities in the prohibited semiconductor zone correspond to additional energy levels: donor - near the bottom of the conduction zone and acceptor - near the ceiling of the valence zone. The energy required for the thermogeneration of charge carriers caused by the presence of impurities (impurities ionization energy) is 50-100 times smaller than the width of the prohibited zone:
The process of thermogeneration of impurity carriers is also probabilistic and described by formulas
where is the concentration of donor impurities, and - acceptor impurities. While the temperature is low, not all impurities ionisovans and the concentration of carriers are calculated by formulas (4). However, in typical cases, already at a temperature significantly below room (about -60 0 c), all impurities are ionized and, with further heating, the concentration does not change and equal to the concentration of the impurities introduced (each atom of the impurity "gave" by one charge carrier. Therefore, in some Temperature range The concentration of carriers, practically, does not depend on temperature (region II. Figure 4). However, with a significant increase in temperature (for silicon, for example, one hundred about 120 ° C), the breakdown of its own bonds on the mechanism represented by formula (3) and the concentration of charge carriers begins to grow sharply. The considered illustrates Fig. 4, which presents the temperature dependences of the charge carrier concentrations in a semi-luggage scale from the inverse temperature (the convenience of such a scale becomes apparent after logarithming expressions (3) and (4)).
Here - the temperature of impurity exhaustion is the transition temperature to its own conductivity. Formulas for RAS
Fig. 4. Temperature dependence of the concentration of key charge carriers in the impurity semiconductor n. - type. I. - region of weak ionization of impurities (impurity conduit) (); II. - an area of \u200b\u200bimpurity exhaustion (); III - area of \u200b\u200bown conductivity ().
chet of these temperatures are shown below. In area .III. The generation of charge carriers occurs in accordance with formula (3). At lower temperatures, this process is negligible, and therefore in the region .I. The generation of carriers is determined only by formula (4). As follows from expressions (3) and (4), the angle is the greater, the greater the width of the forbidden semiconductor zone, and the angle is the greater, the greater the energy of the ionization of donors (acceptors). Considering that, we conclude that\u003e.
Quasi-free charge carriers (and electrons and holes), possessing the average heat energy make a chaotic movement with a thermal speed. External impact (electric field, electromagnetic field, temperature gradient, etc.) only "order" by this chaos, slightly directing charge carriers , mainly in accordance with the attached effect. If this external influence is the electric field, the directional motion of charge carriers - drift occurs. At the same time density of drift current
where - the electrical conductivity - the concentration of charge carriers - the rate of directional movement under the influence of an external electric field intensity E..
As a rule, when the Ohm law is performed, appointment E - Only send charge carriers without changing their energy (the fields are weak). Thus, the rate of motion of charge carriers remains equal, and the drift speed characterizing the efficiency of the directional movement of the charge carrier team depends on the different defects in the crystal lattice strongly interfere with this movement. The parameter characterizing the effectiveness of the directional movement of charge carriers is called mobility:
Obviously, the larger in the crystal lattice of defects involved in the scattering of charge carriers, the less. Under scattering, changes in the quasi-pulse directional movement of charge carriers caused by the influence of defects. In addition, since in the crystal there are always various types of defects (thermal fluctuations of atoms, impurities, etc.), the mobility of the charge carrier "is controlled" by the most efficient scattering mechanism:
where M σ is the resulting mobility of charge carriers in the semiconductor; M i - mobility due to i.scattering mechanism. For example, in the high temperatures, M σ is controlled by the contribution to the scattering of heat oscillations of the lattice, and with increasing temperature decreases. In the field of low temperatures, when the contribution of lattice scattering in M \u200b\u200bσ is small, charge carriers with a small, long time turn out to be in the field of Coulomb forces (attraction or repulsion) ionized impurities. It is this mechanism of scattering "controls" M Σ in semiconductors at low temperatures. Therefore, the mobility of charge carriers depending on the temperature is determined by the semi-empirical ratio of the form:
where a.and b. - Permanent values.
The qualitative dependence of LNM Σ (T) in the crystals of the form (7) is presented in Fig. 5. In this figure, the curves 1 and 2 illustrate the fact that increasing the concentration of impurities ( N. PR1.<N. PR2) decreases M σ in the field of low temperatures, leaving the continuous mechanism of lattice scattering in the crystal.
Lattice scattering on acoustic phonons prevails with T\u003e100 K. In this case, in the region of impurities, when you can put
Fig. 5. Temperature dependence of the mobility of charge carriers the electrical conductivity can decrease with an increase in temperature by reducing the mobility of carriers M σ ( T.) Due to the scattering of charge carriers on acoustic phonons. Task for settlement schedule Work Purpose of work: semiconductor Radiation Electrical conductivity Metal E \u003d 2K (Ln (R) / (1 / T)) where k \u003d 1.38 * 10-23 J / K, T - temperature in Kelvin, R (OM) - resistance. 3. Determine the slope of the linear part of the graph and calculate the linear expansion coefficient for the metal and compare it with a table value. For metals and semiconductors, the effect of changes in conductivity is known when temperatures change. The mechanism of the phenomenon in these substances is varied. As is known, metals with increasing temperature resistance increases as a result of an increase in the scattering of the energy of the current carriers on the lattice oscillations by law RT \u003d RO (1 + A (T - TOT)), where RO is the resistance at 0 ° C (273 K); RT - resistance at temperatures T1, A - temperature coefficient. For various metals, its value is different. So for platinum a \u003d 3.9 · 10-3 K-1, for nickel a \u003d 5.39 · 10-3 K-1. The resistance thermometers are created on the resistance resistance property, allowing to measure the resistance temperature in the range from -200 ° C to +850 ° C. The most common are the thermometers based on nickel and platinum resistance: PT-100 or Ni-100. Their resistance at 0 ° C is chosen equal to 100 ohms. Standard are also resistance in 500 ohms and 1 com. To transfer the measured value, resistance to temperature values \u200b\u200bthere are special tables. 1. Temperature dependence of resistivity The movement of free electrons in the metal can be considered as the propagation of flat waves, the length of which is determined by the de Broglyl ratio: where V is the average speed of heat movement, E is the particle energy. Such a flat wave in the strict periodic potential of the perfect crystal lattice applies without scattering of energy, i.e. without attenuation. Thus, the free path of the electron in the perfect crystal is equal to?, And the electrical resistance is zero. The scattering of energy leading to the resistance is associated with the defects of the structure. Effective scattering of the waves occurs when the size of the scattering centers exceeds. In metals, electrons energy is 3? 15 eV, i.e. L \u003d 3? 7 A. Therefore, any micronegeneration prevents the spread of the wave. In pure metals, the only reason for the scattering and the limiting length of the free mileage of electrons is thermal oscillations of the lattice, i.e. Atoms. With the increasing temperature of the amplitude of thermal oscillations grows. If we believe it is simplified that the intensity of the scattering is directly proportional to the cross section of the volume of the sphere, which is occupied by the oscillating atom, and S section DA2, where Da is the amplitude of thermal oscillations, then the free path length: where n is the number of atoms per unit volume. The potential energy of an atom deviating to DA from the node is determined by elasticity. Elastic energy, EUPR, is written as where the KUPR is the coefficient of elasticity. The average energy of a one-dimensional harmonic oscillator is equal to KT KT\u003e (DA) 2 \u003d (4) In the field of low temperatures, not only the amplitude of oscillations is reduced, but also the frequency of oscillations of atoms and the scattering becomes not effective, i.e. The interaction with the lattice only slightly changes the pulse of electrons. The maximum frequency of thermal oscillations Vmax is determined by the Debye temperature, thermal energy In classical theory, specific conductivity where VF is an electron speed near the Fermi level, N is the electron concentration per unit volume. considering that Fig. one. Dependence of the resistivity of metals: a) - in a wide range of temperatures, b) - for various materials. Linear approximation of the temperature dependence Rt (T) is valid to T ~, and ~ 400-450 K for most metals. Therefore, the linear approximation is valid at temperatures from room and higher. With T.< Tкомн. cпад rT обусловлен выключением фононных частот и rT ~ Т5 - закон Блоха - Грюнайзена (участок степенной зависимости очень мал) (Рис. 1). In this way, Rt \u003d r about performed in a certain temperature range (Fig. 1.). Platinum measuring resistor on a ceramic basis type PT-100 works in the range 0? 400С, while the amount of resistance varies from 100 to 247.04 ohms almost linearly. 2. Basics of the zone theory of the crystal. The solid, as is known, consists of atoms, i.e. From the nuclei of atoms and electrons. Atomic cores form a crystal lattice that has spatial frequency. The movement of electrons in a solid is equivalent to the movement of electrons in a spatial periodic field. When describing the movement of the electron in the periodic field of the crystal lattice, the quantum mechanic gives such results that are convenient to compare with the quantum mechanical results for an isolated atom. Electrons in an isolated atom have discrete energy values, and the spectrum of the free atom represents a set of discrete spectral lines (Fig. 2). When combining n identical atoms forming a solid, each energy level is split into n closely lying levels that form zone (Fig. 2-b). Thus, instead of the system of individual energy levels in the solid, the system of energy zones appears, each of which consists of closely located levels. The zones of the pervolored energies are separated from each other by some interval, called the forbidden zone (Fig. 2). Energy "distances" between the allowed zones (i.e., the width of the forbidden zones) is determined by the electron bond energy with the lattice atoms. Fig. 3. If the elements part of the levels are free or on the main zone is imposed on a free, unoccupied zone, then such elements have pronounced metal properties. The distribution of electrons by energy in the metal is determined by the Fermi Dirac statistics. The distribution function is: K - Permanent Boltzmann, T - absolute temperature, E - kinetic electron energy located at a given energy level, EF - Fermi level energy. The graphical dependence on E is shown in Fig. 3. The curve depicts this dependence for T \u003d 0. The schedule shows that all states with energy, less EF, will be occupied by electrons. In states with E\u003e EF electron enerphones. At temperatures above absolute zero (T\u003e 0), the distribution of electrons by energies is given curve 2. In this case there are electrons with an E\u003e EF energy. Fig. four. In semiconductors and dielectrics, the zone of valence electrons is completely filled, and the nearest free zone - the conduction zone is separated from it a prohibited zone. For dielectric width, the forbidden E reaches several electron-volts, for semiconductors it is significantly less, for example, for Germany E \u003d 0.72 eV. The width of the forbidden zone is the most important parameter of the semiconductor or dielectric material and largely determines its properties. Electronic conductivity in semiconductors, as well as in metals, are considered as perfect gas and subordinate to Fermi Dirac statistics. The distribution function is viewed. Atoms of foreign substances in the crystal lattice are on the properties of semiconductors. The impurity disrupts the periodicity of the crystal and forms additional levels in the energy spectrum of the semiconductor located in the forbidden zone. If the energy level of the impurity is near the bottom of the conduction zone (Fig. 5), then the heat transfer of electrons from these levels into the conduction zone will be more likely than the transition from the filled zone, since The concentration of electrons in the conduction zone in this case will be larger than the concentration of holes in the vacant zone. Such impurities are called donor, and the conductivity of electronic or n-type. If the impurity levels are near the boundary of the valence zone, the electrons falling on them under the action of thermal movement will be connected. In this case, the main carriers of the current will be holes in the filled zone. Such impurities are called acceptor, and the semiconductor has hole conductivity or p-type. Fig. five. a) donor; b) acceptor Let us explain the example of the elementary semiconductor Germany, located in a 4-subgroup of the Mendeleev table. Each of its atoms has four valence electrons and four tetrahedral communications-oriented communication. Thanks to the pairwise electronic (covalent) interaction of neighboring atoms, its V-zone turns out to be fully occupied. The substitution of the atoms of the main substance atoms of impurity elements 5 subgroups - antimony, arsenic, phosphorus - means the inclusion in the system of pair - electronic bonds of atoms with "superfluous" electrons. These electrons are associated with the surrounding atoms much weaker than the rest and relatively easily can be free about valence ties. In the energy language, this means the appearance near the lower edge of the conduction zone of donor levels with the ionization energy. A similar result is obtained with the introduction of the impurities of the 3-subgroups - aluminum, India, Gallium: the lack of electrons from the V-zone on acceptor levels. It is essential that the concentration of impurities atoms is much less atoms of the main substance - in this case, the energy levels of atoms can be considered local. Temperature dependence of the electrical conductivity of semiconductors. In its own semiconductor, free media arise only due to the breaking of valence bonds, so the number of holes is equal to the number of free electrons, i.e. n \u003d p \u003d ni, where Ni is its own ending, and electrical conductivity at this temperature is equal to: where Mn and MP - electron mobility and holes, e - Electron charge. In the donor semiconductor, the electrical conductivity is determined In case of prevalence of acceptor impurities The temperature dependence of the electrical conductivity is determined by the dependence of the concentration of N on the mobility of the charge carriers from temperature. Own semiconductor. For its own semiconductor, the concentration of charge carriers (n \u003d p \u003d ni) can be expressed by the ratio: where relatively weakly depends on temperature. From (3) it can be seen that the ending of the free Ni carriers depends on the temperature T, the width of the prohibited zone E, the values \u200b\u200bof the effective mass of charge carriers M * n and M * p. The temperature dependence of the Ni concentration at E \u003e\u003e KT is determined mainly by an exponential member of the equation. Since weakly depends on the temperature, the graph of the LN Ni dependence from 1 / t must be expressed by the straight line. Danior semiconductor. At low temperatures, it is possible to neglect the number of electron transitions from the valence zone to the conductivity zone and consider only the transition of electrons from donor levels into the conductivity zone. Temperature dependence The concentration of free electrons of the donor semiconductor at relatively low temperatures and partial ionization of impurity atoms is expressed by the ratio: where Na is the number of levels (atoms) of the donor impurity in a single semiconductor unit (the end of the donor impurity) E A-depth of a donor impurity. From (10) follows This is an area of \u200b\u200bweak ionization of impurities. It is indicated by the number 1 in fig. 6, which shows the change in the concentration N with a temperature for a donor semiconductor. Fig. 6. At a higher temperature KT\u003e E A, when all electrons with donor levels can go to the C-zone. The ending of electrons in the conduction zone becomes equal to the ending of the donor impurity N \u003d Na. This temperature range at which the impurity ionization occurs, is the name of the impurity exhaustion region and in Fig. 6 marked digit 2. With a further increase in temperature, the ionization of the main substance atoms begin. The ending of the electrons of the C-zone will increase already due to the transitions of electrons from the valence in the C-zone, the non-core carriers of charge-hole appear in the valence zone. When the Fermi level reaches the middle of the prohibited zone, then n \u003d p \u003d ni and the semiconductor from the impurity proceeds to its own (region 3. Fig. 6). Acceptor semiconductor. At low temperatures, you can neglect the transition of electrons from V to the C-zone and consider only the transition of electrons from the valence zone to acceptor levels. In this case, the temperature dependence of the concentration of free holes is expressed as: Where Na is an acceptance of acceptor impurity, Acceptor impurity activation energy. From (12) follows With increasing temperature, all acceptor levels are filled with electrons moving from the V-zone. When KT\u003e E A, the impurity is exhausted, the concentration of holes in the V-zone is equal to the concentration of the acceptor impurity Na. With a further increase in temperature, there are more and more of its own carriers due to the transition of an electron from V to the C-zone and at some temperature the conductivity of the semiconductor is transformed into its own. Temperature dependence of carrier mobility. Mobility of charge carriers m, numerically equal to the velocity of the carriers, purchased by them under the action of an electrical field of single tension: The mobility of electrons, M, and holes, MPs are different due to the difference in the effective masses and the time of the free mileage of the electron and the hole, which depends on the electron scattering mechanism and holes in the semiconductor crystal lattice. You can select several charge carrier scattering mechanisms: on thermal oscillations of the atoms of the crystal lattice; on ionized impurities (impurity ions); on neutral impurities (impurity atoms); on lattice defects (vacancies, point defects, dislocations, crystallite boundaries, etc.); on charge carriers. In view of the smallness of the defecation of defects and charge carriers 4) and 5), scattering species are usually neglected. In the case of the scattering of charge carriers (waves of these carriers) on the heat oscillations of the lattice, the mobility caused by this type of scattering decreases with increasing temperature by law Scattering on heat oscillations The lattice prevails at high temperatures. In the case of the scattering of carriers on ionized impurities, mobility grows with a temperature: This scattering mechanism prevails at low temperatures. If both mechanisms 1) and 2 are involved in the scattering of carriers) and they are independent, then the temperature dependence of M can be represented as: where a and b techniques proportionality. The scattering of carriers on neutral impurities does not depend on the temperature or on the energy of the carriers and affects at very low temperatures, when the heat oscillations of the lattice do not play a noticeable role and the degree of ionization of the impurities of the Mala. Temperature dependence. Considering the dependence of the concentration and mobility of charge carriers from temperature, the specific electrical conductivity of its own semiconductor can be written as: The multiplier varies slowly with the temperature, while the multiplier strongly depends on the temperature when E \u003e\u003e KT. Therefore, for not too high temperatures, we can assume that And expression (18) replace easier Consider the behavior of the semiconductor when moving from low temperatures to high. In a donor or acceptor semiconductor, conductivity at low temperatures is impurity. Since the temperature is low, then the ionized impurities and the scattering on neutral atoms prevails, at which M does not change with the temperature. Therefore, the temperature dependence will be determined by the dependence of the concentration on temperature. For the electrical conductivity of the donor semiconductor according to (2.4) and (2.5) you can record Accordingly, for the electrical conductivity of the acceptor semiconductor. Obviously, if equations (14) and (15) construct graphically in the coordinates of LN and 1 / T, then from the slopes of these dependencies (Fig. 7), you can determine the ionization energy of a donor or acceptor impurity: We will raise the temperature and fall into the depletion area of \u200b\u200bthe impurity (Fig. 6. Region 2), in which the end of the main carriers remains constant and the conductivity changes due to the change in mobility M with a temperature. On the section 2 of the curve Ln (1 / T) (Fig. 7) And the electrical conductivity grows slightly with the temperature, because The scattering on the ions of the impurity prevails, at which M ~ T3 / 2. Next, with increasing temperature, the electrical conductivity decreases, because Scattering on heat oscillations of the lattice prevails, at which M ~ T3 / 2 (section 3, Fig. 7). Finally, at sufficiently high temperatures, the conductivity of the semiconductor becomes its own, and in these conditions you can determine the width of the forbidden semiconductor zone Where k \u003d 1.38 * 10-23 j / k \u003d 8.6 * 10-5 eV / k Fig. 7. a) - own semiconductor, b) - impurity semiconductor. Instructions for laboratory work "Study of the temperature dependence of electrical conductivity of metals and semiconductors" Electric furnace serves for heating the samples. The temperature of the samples is measured by the temperature meter sensor on the measuring device. The dependence of metal resistance on temperature can be found according to the formula: In the semiconductor, the same dependence is following: Operating procedure: E \u003d 2K (Ln (R) / (1 / T)) where k \u003d 1.38 * 10-23 j / k, T-temperature in Kelvin, R (Ohm) -to resistance. 3.3 Determine the slope of the linear part of the graph and calculate the temperature coefficient for the metal and compare it with a tabular value.
In semiconductors with different concentrations of impurities. N PR1.
