The invention relates to techniques for monitoring semiconductors. It is most advisable to use the proposed invention for non-contact, on-line monitoring of the parameters of deep levels (DL), surface states (SS), surface potential (charge), as well as the lifetime of minority charge carriers. Essence: surface photoEMF is excited by rectangular pulses of electromagnetic radiation, the intensity of which varies from zero to values ​​that ensure saturation mode. Radiation hits the surface of the semiconductor through a transparent capacitive electrode. The amplitude and shape of the surface photoEMF pulse is recorded using this electrode and the measuring circuit. Measurements are carried out at several different intensities of electromagnetic radiation pulses. Based on the registered characteristics, the parameters of relaxation processes are calculated, which makes it possible to determine the electrophysical parameters of the semiconductor - the concentration, energy and capture cross section of the GC and PS, as well as the surface charge, surface potential, and the lifetime of minority charge carriers. 2 salary f-ly, 1 table., 7 ill.

Drawings for RF patent 2330300

The invention relates to techniques for monitoring semiconductors. It is most appropriate to use the proposed invention to control the parameters of deep levels (DL), surface states (SS), surface potential (charge), as well as the lifetime of minority charge carriers.

There are a number of known methods for determining the parameters of semiconductors. The capacitance-voltage method is based on creating a metal-dielectric-semiconductor (MDS) structure on the controlled surface of a semiconductor, determining the dependence of the capacitance of such a structure on the voltage applied between the semiconductor and the metal, and analyzing this dependence. The method makes it possible to determine a number of semiconductor parameters - surface potential (charge), PS density, volumetric generation time of charge carriers, dopant concentration. The disadvantage of this method is the need to create such a structure, as well as the relative complexity of carrying out measurements.

There is also a known method for determining the lifetime of minority charge carriers, the essence of which is to determine the stationary value of the surface photovoltage (SPE) at several different wavelengths of electromagnetic radiation irradiating the surface of the controlled semiconductor wafer. In this case, periodic modulation of the intensity of electromagnetic radiation is used, and the stationary value of the surface photoEMF is determined from the amplitude of the fundamental harmonic of the signal of this EMF, recorded using a capacitive probe. The disadvantage of this method is its complexity (it is necessary to carry out measurements at several (up to 10) wavelengths). It should be noted that for a given shape of electromagnetic radiation pulses and their repetition frequency, the shape of the surface photoEMF signal also depends on the intensity of the electromagnetic radiation pulse. This introduces additional error and limits the scope of the method.

The closest to the proposed invention is the method for determining the electrophysical parameters of semiconductors according to RF patent No. 2080611.

When using this method, the controlled semiconductor wafer is irradiated with pulses of electromagnetic radiation. Irradiation is carried out through a transparent capacitive electrode, which is a transparent conductive pad located parallel to the surface of the semiconductor wafer. The result of irradiation is the generation of a nonequilibrium potential difference at the surface-volume barrier transition of the semiconductor. Registration of this potential difference is carried out by determining the amplitude and shape of the voltage pulses between the capacitive electrode and the volume of the semiconductor. Measurements are carried out over a range of temperatures. The parameters of the relaxation processes of establishing and dissolving a nonequilibrium potential difference are determined from the amplitude and shape of the voltage pulses, and the electrophysical parameters of the semiconductor are calculated from the dependence of the parameters of these processes on temperature. The advantage of this method is to provide non-destructive monitoring of the parameters of a semiconductor deep state with a sufficiently high sensitivity (up to 10 8 ÷ 10 9 cm -3) and high resolution (better than 10 -2 eV) without any additional technological operations. The disadvantage of this method is that to determine the parameters of the semiconductor it is necessary to cool and heat the semiconductor. This makes operational control in in-line mode impossible. In addition, it is possible to determine only the parameters of the GI.

The purpose of the invention is to provide operational control of semiconductor parameters without heating or cooling the controlled samples, as well as to obtain the ability to control the parameters of the surface potential, surface charge and volumetric lifetime of minority charge carriers. This goal is achieved by the fact that in the known method of determining the electrophysical parameters of semiconductors, including the creation of a nonequilibrium potential difference at the barrier junction surface - volume of the semiconductor by irradiating a semiconductor wafer located on a conducting table-stand with rectangular pulses of electromagnetic radiation, the quantum energy of which is higher than the generation energy threshold free charge carriers in the semiconductor wafer, through a capacitive electrode, which is a transparent conductive plate located parallel to the surface of the semiconductor wafer, registration of the mentioned nonequilibrium potential difference by determining the amplitude and shape of voltage pulses between the capacitive electrode and the stand table, calculating the parameters of the relaxation processes of establishing and resorption of the nonequilibrium potential difference at the mentioned barrier transition by the amplitude and shape of the voltage on the mentioned capacitive electrode and calculation according to the parameters of relaxation processes of the electrophysical parameters of the semiconductor, the duration of the electromagnetic radiation pulses is set longer than the time of establishment of the nonequilibrium potential difference at the barrier transition surface - volume of the semiconductor. The time interval between radiation pulses is set longer than the time for resorption of this nonequilibrium potential difference. Registration of the voltage on the said capacitive electrode is carried out by determining the amplitude and shape of the voltage pulses by means of a measuring circuit, the time constant of which, equal to the product of the capacitance between the capacitive electrode and the semiconductor wafer and the input resistance of this measuring circuit, is greater than the time of both establishment and dissolution of the mentioned nonequilibrium potential difference . Calculation of the parameters of the relaxation processes of establishing and resolving a nonequilibrium potential difference at the barrier junction surface - volume of the semiconductor is carried out at several different intensities of electromagnetic radiation pulses, varying from the minimum values ​​at which it is still possible to register a nonequilibrium potential difference at the barrier junction, to values ​​at which the signal amplitude from the capacitive electrode does not depend on the intensity of the radiation pulse. The parameters of surface states, the parameters of deep levels and the value of the surface potential are calculated from the dependence of the parameters of relaxation processes on the intensity of the radiation pulses. It is advisable to determine the dependence of the parameters of the relaxation processes of establishing and resolving a nonequilibrium potential difference at the barrier transition surface - volume of the semiconductor on the intensity of the radiation pulse when creating an electric potential difference of positive or negative polarity between the capacitive electrode and the volume of the semiconductor. This makes it possible to determine the parameters of PS and GI in a wider range of their values. In addition, it is advisable to determine the parameters of relaxation processes at two or more wavelengths of electromagnetic radiation. This makes it possible to determine the volumetric lifetime of minority charge carriers, which is calculated from the dependences of the parameters of the relaxation processes of establishment and resorption of a nonequilibrium potential difference at the barrier transition surface - volume of the semiconductor, both on the intensity and on the wavelength of the electromagnetic radiation pulse.

The proposed method for determining the electrophysical parameters of semiconductors is a further development of the method described in. The main distinctive feature of the proposed method is that the determination of the nonequilibrium potential difference at the barrier junction is carried out in the range of changes in the radiation intensity from zero to a value at which the dependence of the signal amplitude on the radiation intensity becomes saturated. Mathematical processing of the results of these measurements makes it possible to determine the parameters of the GU, PS, as well as the surface charge at room temperature. It is possible to carry out measurements at other temperatures.

According to the second variant of the invention, when measuring the parameters of relaxation processes, a constant electric voltage is applied to the capacitive electrode. This makes it possible to determine the parameters of PS and GU, the activation energy of which lies in the entire range of the energy gap.

According to the third embodiment of the invention, measurements are carried out at two or more wavelengths of electromagnetic radiation, which makes it possible to determine the volumetric lifetime of minority charge carriers. Note that to determine the exact value of this parameter, it is necessary to take into account the dependence of the amplitude and shape of the surface photoEMF signal on both the wavelength and the intensity of electromagnetic radiation.

The combination of three technical solutions into one application is due to the fact that they all solve the problem of determining the electrophysical parameters of a semiconductor based on one principle - taking into account not only the amplitude, but also the shape of the surface photoEMF signal, as well as the dependence of the PFE signal on the intensity of electromagnetic radiation pulses.

In what follows, the surface photovoltage generated by rectangular radiation pulses will be called pulsed surface photovoltage (PSPE).

Figure 1 shows a functional block diagram of a device that implements the proposed method, Figure 2 shows an equivalent measurement diagram, Figure 3 shows energy diagrams of the barrier transition surface - volume of a semiconductor. Figures 4-8 show the results of measurements of surface photovoltage (SPE) on a KEF 4.5 silicon wafer. Figure 4 shows graphs of the PFE signal when the signal amplitude changes from 0 to 0.24 V; Figure 5 shows normalized graphs of the same processes. Figures 6 and 7 show graphs of increments of the trailing edge of the IPPE signal.

The device that implements the proposed method consists of a stand table 1 on which the controlled plate 2 is placed. This table is made of conductive material. The plate 2 is irradiated with electromagnetic radiation through a transparent conductive electrode 3. The radiation source is a laser LED 5, excited by a generator of rectangular current pulses of adjustable amplitude 4. Electromagnetic radiation from the LED 5 enters the light guide 6 and then through the electrode 3 to the controlled semiconductor wafer 2. Electrode 3 is connected to a constant voltage source 10. The surface photoEMF signal is collected by electrode 3 and fed through an isolation capacitor 9 to the input of a high-impedance measuring amplifier 7 and then to a recording device 8. It is advisable to use a digital oscilloscope as a recording device.

Figure 3 shows energy diagrams of the barrier transition surface - volume of the semiconductor. In this fig. E - energy, q - electron charge, V k - barrier potential difference of the surface-volume transition, V m - V value in saturation mode, E c, E v - boundaries of the conduction band and valence band, F - Fermi level, F e - quasi-Fermi level for electrons, F h - quasi-Fermi level for holes, E 0 - energy level (GU), h 1, h 2, h 3 - coordinates of the boundary of the space charge region, w 1, w 2 - values ​​of the longitudinal coordinate at E 0 = F and at E 0 =F e . Diagram “a” corresponds to an equilibrium state, “b” to a stationary nonequilibrium state, when electromagnetic radiation generates a photoEMF of magnitude Hz, “c” corresponds to the case when q·V=q·V k -E 0 ; “g” corresponds to saturation, when the zones are straightened and the photoEMF has a maximum value that does not depend on the radiation intensity. The energy ranges of the PS filled with electrons are marked with circles.

Figure 4 shows recordings of the IPPE signal for a KEF 4.5 silicon washer with a diameter of 100 mm and a thickness of 1.5 mm at various intensities of rectangular radiation pulses with a wavelength of 0.86 μm. Pulse duration - 1.2 ms.

Figure 5 shows the same signals, normalized in such a way that at the moment the radiation pulse ends, the value of the normalized IPPE signal was equal to 1.

Figure 6 shows graphs of increments of the trailing edge of the IPPE signal; continuous line - amplitude varied from 0 to 20 mV; small dotted line - from 20 to 30 mV and large dotted line - from 30 to 40 mV.

Figure 7 shows similar graphs for increments from 70 to 80 mV - continuous line, from 80 to 90 mV - small dotted line and from 90 to 100 mV - large dotted line.

The method is implemented as follows.

Let us consider the case when a homogeneous n-type semiconductor wafer is irradiated by pulses of electromagnetic radiation with duration T 0 and intensity I 0 , and the voltage of the power source 10 is zero. When the radiation is turned on, the generation of nonequilibrium charge carriers occurs, their diffusion and drift under the influence of the electric field of the barrier transition surface - volume of the semiconductor. This leads to a decrease in the barrier transition potential difference and the appearance of surface photoEMF; in this case, electrons are captured by those GU and PS that are below the quasi-Fermi level for electrons. We will neglect the processes associated with the generation of Dember's emf. Let us choose the value T 0 sufficient to establish a stationary state. At the end of the radiation pulse, the dissolution of nonequilibrium charge carriers and the depletion of the deep and polarized states located above the Fermi level occur. An equivalent circuit for measuring surface photovoltage V is shown in Fig. 2. We will choose the values ​​of the capacitance of the capacitive electrode C and the input resistance of the measuring amplifier R in such that the time constant of the measuring circuit, equal to the product R in ·(C + C 0), is greater than both the time it takes for the photoEMF to establish a stationary value from 0 to V 0 and the time resorption from V 0 to 0. Surface photoEMF measurements are made either in the mode of single radiation pulses, or with a sufficiently low repetition frequency to ensure complete resorption of the photoEMF from V 0 to 0.

To determine the value of the surface charge Q S, let us increase the radiation intensity I 0 until saturation, i.e. such a value at which V 0 does not depend on I 0 . The energy diagram for this mode is shown in Fig. 3d. The limiting value V m corresponds to the surface potential barrier V k . The surface charge Q S is determined by the relation

Where ,

n i is the equilibrium concentration of charge carriers in the intrinsic semiconductor,

Relative dielectric constant of a semiconductor,

0 - dielectric constant of vacuum,

k - Boltzmann constant,

T - absolute temperature,

n 0, p 0 - total volume concentrations of electrons and holes under thermodynamic equilibrium conditions.

To determine the parameters of GU and PS, it is necessary to determine the parameters of the relaxation processes of establishment and resorption of IPPE at different intensities of electromagnetic radiation, varying from zero to saturation. Figure 4 shows records of such processes for an n-type silicon washer. The intensity of the radiation pulse was changed in such a way that the value of the surface photovoltage in steady state at the end of the radiation pulse varied from 0.03 to 0.32 V. Figure 5 shows graphs of the same relaxation processes, normalized so that at the end of the pulse radiation their values ​​coincided. As can be seen, at different radiation intensities, not only the amplitude, but also the shape of the surface photovoltage pulses changed significantly.

Next, we will limit ourselves to considering the relaxation processes of PFE resorption. In addition, we will consider the case of a depleted space charge layer. The trailing edge of the IPFE will be represented as a sum of exponentials. In this case, the fastest process (on the order of units - tens of microseconds) corresponds to the resorption of nonequilibrium charge carriers (NCC); Let us denote by 0 the time constant of this process.

Let us first consider the case without GI. We will divide the energy interval in the energy diagram at x=0 from the bottom of the conduction band to the Fermi level into N smaller intervals of width E each. The average PS energy at each of these intervals is equal to

.

Here i is the number of the interval (counting from the Fermi level). Relaxation process of PFE resorption at i=1, when , is described by the relation

In the case of the i-th interval

Here A 0i is the amplitude of the relaxation process of resorption of the NNS; And sj, sj are the amplitude and time constant of the relaxation process of depletion at the jth energy interval. Note that V i (0)=V 0 . It's obvious that

Relations (2)-(4) are approximate, obtained under the assumption that the resorption of nonequilibrium charge carriers, as well as the depletion of the PS, occurs according to an exponential dependence.

Let us denote by N si the average PS density in the i-th interval (i.e., the average number of PSs per unit illuminated area of ​​the semiconductor and per unit energy range). Then

where C si is the differential capacity of the space charge corresponding to the i-th interval (per unit area). The value of C si is determined by the relation

In ratio (6)

The average PS density corresponding to the energy E i,

To determine N si, it is necessary to record the trailing edge signals of the PFE Vi+1 and Vi, corresponding to the energy values ​​E i+1 and E i, calculate the difference Vi+1 -V i, decompose this difference into exponentials and determine A si. The value of C si can be calculated from the value of V k . The PS capture cross section corresponding to the energy E i - si can be calculated from the relation:

where is the average thermal speed of electrons; N 0 is the effective density of states in the conduction band of the semiconductor.

Note that the time constants of the relaxation processes associated with the recharging of the PS - si depend on the energy of the PS; As the PS energy approaches the bottom of the conduction band, they decrease. This leads to an increase in the total duration of the resorption process of PFE with a decrease in the amplitude of PFE (see Figs. 5, 6).

where A li and l are the amplitude of the relaxation process of depletion of the deep state and its time constant.

The relaxation processes associated with the deep state do not change the time constant with changes in the intensity of the radiation pulse. This allows us to distinguish them from relaxation processes associated with PS. At the same time, starting from a certain value i, when E i >(qV k -E 0) and A li =A li+1, the term with the exponent e -t/ l disappears in relation (4"). This can be used to determine the value of E 0 .

Note that as the radiation intensity increases from zero to saturation, the time constant of the relaxation process of resorption of the NNS remains unchanged.

Let us denote by Q li the value of the resorption charge from the GU per unit area of ​​the illuminated surface of the semiconductor. Amplitude A li is related to Q l1 by the relation:

On the other side

where N l is the volumetric concentration of GU. B shows that w 2 -w 1 =h 2 -h 1. In the case of a depleted space charge layer, the electrostatic potential varies according to a parabolic dependence (as in the case of the Schottky barrier). For n type semiconductor

Substituting (11) and (12) into (10) and (10) into (9) we get

Relationship (13) allows you to determine the concentration of GU. The capture cross section of the GU - l can be calculated for known E 0 and l from the relation:

where g is the coefficient of degeneration of the state.

Thus, having determined the parameters of the IPPE relaxation processes when the radiation intensity changes from zero to saturation, it is possible to determine the following electrophysical parameters of the semiconductor: surface potential Vk, surface charge Qs, density Nsi and capture cross section of Si PS, as well as concentration Nl, energy E 0 and capture cross section l GU.

In the case of two or more BLs in relations (2"), (3") and (4"), additional exponentials will appear with a time constant that does not depend on the radiation intensity, but the algorithm for determining the BL and PS parameters will not change significantly.

Above, we considered the case when zero voltage was applied to the capacitive electrode from voltage source 10, and the surface potential was determined only by the properties of the semiconductor surface. When a voltage is applied from a source 10 of positive or negative polarity, an additional charge is induced on the surface of the semiconductor, and the energy diagram shifts down or up. This makes it possible to determine the parameters of the PS and GU in a larger energy range throughout the entire bandgap.

Let us next consider the algorithm for determining the volumetric lifetime of the nonconductive charge mc. As is known, mc is related to the diffuse wavelength L p. To calculate this parameter, it is necessary to determine the dependence of the IPPE signal on the intensity of the radiation pulse, at least at two wavelengths. Then you should select two values ​​of radiation intensity I 01 and I 02, corresponding to two wavelengths 1 and 2, at which the stationary IPPE values ​​V 01 and V 02 are equal to each other. The diffuse length is determined by the expression:

where 1 =c/ 1; 2 =c/ 2 ; 1 and 2 - absorption coefficients of electromagnetic radiation at wavelengths 1 and 2; h is Planck's constant, c is the speed of light in vacuum.

The volumetric lifetime of the NNZ is determined by the relation:

where D is the diffusion coefficient.

Note that analysis of the dependence of the amplitude and shape of the IPPE signal on the radiation intensity will allow optimizing the values ​​of I01 and I02, which provide the smallest error in determining mc.

As an example, we present the results of a study using the proposed method of a KEF 4.5 silicon washer with a diameter of 100 mm and a thickness of 700 microns. Figure 4 shows IPPE recordings on this washer with a radiation pulse duration of 1 ms, a wavelength of 0.87 μm, and a radiation source power of 200 mW. The diameter of the irradiated area was 3 mm. The time constant of the measuring circuit was 0.3 s. Figure 5 shows normalized IPPE graphs. The normalization is performed in such a way that the beginnings of the decline in the surface photovoltage impulse graphs coincide. It can be seen that the shape of the pulse depends significantly on the radiation intensity. As it increases, the steepness of both the leading and trailing edges increases, which indicates an increase in the contribution of faster relaxation processes. The surface potential of the washer under study was 0.24 V, which corresponds to the surface charge Q S = 2.9·10 -7 K/cm 2 (1.8·10 12 charged particles per sq.cm).

Figure 7 shows graphs of increments in the trailing edge of the IPPE as the signal amplitude increases from 0 to 20 mV - continuous line, from 20 to 30 mV - small dotted line, from 30 to 40 mV - large dotted line; Fig. 8 shows the same graphs when the signal amplitude increases from 70 to 80 mV - a continuous line, from 80 to 90 mV - a small dotted line, from 90 to 100 mV - a large dotted line. Processing of the measurement results consisted of decomposing the graphs of the increments of the trailing edge of the IPPE signal into exponentials using a standard nonlinear regression program. The calculation results are shown in Table 1. In this table, E is the middle of the energy range for which the calculation was carried out. The exponents obtained as a result of calculations are divided into four groups. The first group includes exponentials with a time constant of no more than 10 μs. This gives grounds to associate them with the processes of resorption of nonequilibrium charge carriers. The second group includes exponentials with a time constant of the order of several tens of microseconds, the third - several hundred microseconds, and the fourth - of the order of several milliseconds. These three groups of exponentials are most likely associated with the depletion of PSs lying in the energy range from 0 to 0.24 eV. Note that the same energy corresponds to several exponentials with significantly different time constants. This indicates that PSs with different capture cross sections correspond to the same energy, i.e. of different physical nature.

Table 1
E, meV	A 1, mV	1 , μs	A 2, mV	2 , μs	2 ×10 20, cm 2	N S2 ×10 12, cm -2 V -1	A 3, mV	3 , μs	3 ×10 20, cm 2	N S3 ×10 -12, cm -2 B -1	A 4, mV	4 , ms	4 ×10 20, cm 2	N S3 ×10 -12, cm -2 V -1
10,5	-	-	2,2	23	52	3	19	360	3,3	26,5	-	-	-	-
30,4	-	-	1,9	17	33	4,05	9,6	210	2,7	20,2	6,6	1,9	0,29	13,9
49,5	2	2,1	5,6	47	5,6	7,87	12	290	0,91	16,8	-	-	-	-
70,4	4	9,8	5,3	34	3,5	4,32	13	340	0,35	10,8	-	-	-	-
92,1	5,3	2,2	4,6	27	1,9	2,64	6,5	140	0,36	3,71	7,2	0,7	0,076	4,1
112	-	-	3	39	0,61	1,32	9,5	270	0,09	4,15	-	-	-	-
128	1,4	6,6	-	-	-	-	4,2	110	0,11	1,89	4,9	0,5	0,024	2,12
145	6,6	2,2	4,9	38	0,17	1,02	-	-	-	-	8,6	0,7	0,01	1,79
167	-	-	-	-	-	-	-	-	-	-	-	-	-	-
187	12	2,1	-	-	-	-	2,6	100	0,013	0,3	7,7	0,8	0,002	0,87
209	11	4,6	-	-	-	-	-	-	-	-	12	0,8	0,007	0,76
228	22	6	-	-	-	-	-	-	-	-	-	-	-	-

Literature

1. Pavlov L.P. Methods for measuring parameters of semiconductor materials. M.: Higher School, 1987. 239 p.

2. ASTM Standard F 391-96. Standard Test Methods for Minority Carrier Diffusion Length in Extrinsic Semiconductors by Measurement of Steady-State Surface Photovoltage.

3. Rusakov N.V., Kravchenko L.N., Podshivalov V.N. Method for determining the electrophysical parameters of semiconductors. RF patent No. 2080611.

4. Rzhanov A.V. Electronic processes on the surface of semiconductors. M.: Nauka, 1971, 480 p.

5. Berman L.S., Lebedev A.A. Capacitive spectroscopy of deep centers in semiconductors. - L.: Science, Leningrad branch, 1981.

CLAIM

1. A method for determining the electrical parameters of semiconductors, including the creation of a nonequilibrium potential difference at the barrier transition surface - volume of the semiconductor by irradiating a semiconductor wafer located on a conductive table-stand with rectangular pulses of electromagnetic radiation, the quantum energy of which is higher than the energy threshold for the generation of free charge carriers in the semiconductor wafer , through a capacitive electrode, which is a transparent conductive plate located parallel to the surface of the semiconductor wafer, recording the said nonequilibrium potential difference by determining the amplitude and shape of the voltage pulses between the capacitive electrode and the mentioned stand table, calculating the parameters of the relaxation processes of establishing and resolving the nonequilibrium potential difference on the mentioned barrier transition by the amplitude and shape of the voltage pulses on the mentioned capacitive electrode and the calculation of the parameters of relaxation processes of the electrophysical parameters of the semiconductor, characterized in that the duration of the electromagnetic radiation pulses is set longer than the time for establishing a nonequilibrium potential difference at the barrier transition surface - volume of the semiconductor, and the time interval between radiation pulses set more time for resorption of this non-equilibrium potential difference, while recording the voltage on the said capacitive electrode is carried out by determining the amplitude and shape of the voltage pulses by means of a measuring circuit, the time constant of which is equal to the product of the capacitance between the capacitive electrode and the semiconductor wafer by the input resistance of this measuring circuit , more time for both establishment and resorption of the mentioned nonequilibrium potential difference, registration of the nonequilibrium potential difference at the barrier transition surface - volume of the semiconductor and calculation of the parameters of the relaxation processes of establishment and resolution of the nonequilibrium potential difference at the barrier transition surface - volume of the semiconductor is carried out at several different intensities of electromagnetic pulses radiation, increasing from the minimum values ​​at which it is still possible to register a nonequilibrium potential difference at the barrier junction, to at least such values ​​at which it is possible to confidently register that an increase in the voltage amplitude between the capacitive electrode and the stand-table with an increase in radiation intensity by a fixed value is less than the signal amplitude corresponding to the radiation intensity, numerically equal to this fixed value, and the parameters of surface states, parameters of deep levels and the value of the surface potential are calculated from the dependence of the parameters of relaxation processes on the intensity of the radiation pulses.

2. The method according to claim 1, characterized in that the parameters of the relaxation processes of establishing and resolving a nonequilibrium potential difference between the volume and surface of the semiconductor are determined by creating an electrical potential difference between the capacitive electrode and the stand-table, and the parameters of deep levels and surface states are calculated from the dependencies relaxation processes on the intensity of radiation pulses, as well as on the polarity and magnitude of the mentioned potential difference between the capacitive electrode and the stand.

3. The method according to claim 1, characterized in that the dependence of the parameters of the relaxation processes of establishing and resolving a nonequilibrium potential difference at the barrier transition surface - volume of the semiconductor on the intensity of the radiation pulse is determined at two or more wavelengths of electromagnetic radiation, and the volumetric lifetime of minority charge carriers are calculated from the dependences of the parameters of the relaxation processes of establishing and resolving a nonequilibrium potential difference at the barrier transition surface - volume of the semiconductor, both on the intensity of the electromagnetic radiation pulse and on the wavelength of this radiation.

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Introduction

Physical processes in semiconductors and their properties

1 Proprietary semiconductors

2 Electronic semiconductor

3 Hole semiconductor

4 Energy diagrams of semiconductors

5 Major and minority charge carriers

6 Temperature dependence of charge carrier concentration

7 Donor and acceptor semiconductors

8 Dependence of electron concentration on Fermi level energy

9 Position of the Fermi level and concentration of free charge carriers in intrinsic semiconductors

Calculation of temperature dependences of electrophysical parameters of semiconductors

1 Approximate calculation of the dependence of hole concentration on temperature

1.1 Calculation of average temperature

1.2 Calculation of the effective mass of electron and hole

1.3 Calculation of the effective density of states in the valence and conduction bands

1.4 Temperature calculation

1.5 Low temperature region

1.6 Average temperature region

1.7 High temperature region

2 Analytical calculation of the dependence of the concentration of free charge carriers and the position of the Fermi level on temperature

2.1 Finding exact values

2.2 Low temperature region (exact values)

2.3 Average temperature region (exact values)

2.4 High temperature region (exact values)

Conclusion

List of sources used

Appendix A Calculation program

Appendix B Dependency Graphs

Abstract The explanatory note contains 54 pages of typewritten text, includes 2 appendices, 14 figures, and a list of used sources of 10 titles.

Key words: intrinsic semiconductor, acceptor-type semiconductor, effective mass, effective mass ratio, effective density of states, Fermi level, band gap, charge carriers, charge carrier concentration, acceptor impurity, impurity ionization energy, impurity depletion region, intrinsic conductivity region, intrinsic conductivity region, weak ionization region, hole concentration, exact values.

The purpose of the work: to calculate the temperature dependence of the concentration of free charge carriers in an acceptor-type semiconductor, as well as to plot this dependence in coordinates: ln n = F(1/T). Determine and plot graphically the dependence of the Fermi level energy on temperature, and calculate the temperatures of transition to intrinsic conductivity and impurity depletion.

Objectives: use this course work as the basis of a foundation of knowledge about the physics of semiconductors, as well as develop your technical horizons to improve your professional suitability.

2. Calculation of temperature dependences of electrophysical parameters of semiconductors

In order to calculate the necessary parameters, I entered the necessary values, such as:

Electron charge

Atomic rest mass

Donor level ionization energy

Electron masses along the main axes of the ellipsoids

Masses of holes along the main axes of the ellipsoids

Number of valleys in the conduction zone

Number of valleys in the valence band

Donor atom concentration

Boltzmann's constant

Bandgap width

Temperature

Planck's constant

This was followed by the need to convert them to the SI system. Now that all the data is in front of us, we can begin with an approximate calculation of the dependence of electron concentration on temperature.

2.1 Approximate calculation of the dependence of electron concentration on temperature

To begin with, I found the average temperature and effective mass of electrons and holes, which are then necessary to calculate the effective density of states in the valence and conduction bands and

3) calculation of the heat capacity cn and the amount of heat of the process q; 4) calculation of the work of volume change l and the external work of the process l`. 5) calculation of changes in thermodynamic functions: a) internal energy, b) enthalpy, c) entropy...

Analysis of the polytropic process of a mixture of ideal gases

The relationship for pressures and volumes in the initial and final states follows from (10); relationships for temperatures and pressures or temperatures and volumes can be obtained...

Analysis of modes and selection of main parameters of the power transmission system

For each standard section of a line of a given type and voltage, we will plot the dependence of the reduced costs on the power З=f(Р). This expression is, according to a parabola of the form 3=A+BI2nb, where Inb=. According to the probability of filling these levels with electrons. Integration must be carried out from the lower (E c) to the uppermost level of the conduction band, i.e.

(1.1.5)

where is the effective density of states in the conduction band, the energy of which is reduced to the bottom of the conduction band.

Similarly, for the equilibrium concentration of holes in any non-degenerate semiconductor we obtain:

(1.1.6)

where is the effective density of states in the valence band, the energy of which is reduced to the top of the valence band (E v).

Taking into account (1.1.1) for the intrinsic semiconductor we have:

From here, by taking logarithms it is easy to find the position of the Fermi level:

(1.1.8)

Taking into account the closeness of the values ​​of N V and N c , we come to the conclusion that in an intrinsic semiconductor the Fermi level is located approximately in the middle of the band gap (see Figure 1.1.1):

Where bandgap width.

For a graphical representation of the temperature dependence, expression (1.1.9) can be conveniently represented as:

(1.1.11)

The product N C N V is a weak function of temperature; therefore, the dependence of the logarithm of the charge carrier concentration on the inverse temperature is close to linear, and the slope of the straight line characterizes the band gap of the semiconductor. For example, Figure 1.1.2 shows the temperature change in the intrinsic concentration of charge carriers in silicon and germanium.

Figure 1.1.2 - Temperature dependence of the intrinsic concentration of charge carriers in silicon and germanium

The mechanism of intrinsic electrical conductivity of covalent semiconductors is explained in Figure 1.1.3

Figure 1.1.3 - Schematic representation of the intrinsic electrical conductivity of a semiconductor

Silicon and germanium, being elements of group IV of the periodic system, crystallize in the structure of diamond. In this structure, each atom is in a tetrahedral environment of four nearest neighbors, with which it interacts through covalent bonds. Four valence electrons of any atom go towards the formation of four covalent bonds. All chemical bonds turn out to be closed and completely saturated. The states of bound electrons correspond to energy levels in the valence band. In fact, the flat mesh in Figure 1.1.3 is a projection of the crystal lattice onto the (100) plane.

Valence electrons that carry out chemical bonds cannot be removed from their atoms without significant energy expenditure. The energy costs for breaking a bond and releasing an electron are quantified by the band gap. Atoms that have lost electrons turn into positively charged ions, and the unfilled valence bond contains an energetic vacancy for electrons, i.e., it manifests itself as a hole. A positively charged ion can borrow an electron from any neighboring atom, causing a hole to move around the crystal. The resulting electrons and conduction holes wander randomly around the lattice until they recombine when they meet.

Under the influence of an external electric field, the movement of charge carriers becomes directional. In this case, the movement of a hole to the negative pole of the source can be represented as a relay race of valence electrons from one atom to another in the direction opposite to the field.

The considered case of intrinsic electrical conductivity is of theoretical interest, since it allows us to evaluate the potential capabilities of the material. The operation of most semiconductor devices is disrupted when their own electrical conductivity appears.

1.2 Electronic semiconductor

Electronic semiconductor or type semiconductor n(from the Latin negative - negative) is a semiconductor whose crystal lattice (Figure 1.3) in addition to the main (tetravalent) atoms contains impurity pentavalent atoms, called donors. In such a crystal lattice, four valence electrons of an impurity atom are occupied in covalent bonds, and the fifth (“extra”) electron cannot enter into a normal covalent bond and is easily separated from the impurity atom, becoming a free charge carrier. In this case, the impurity atom turns into a positive ion. At room temperature, almost all impurity atoms are ionized. Along with the ionization of impurity atoms, thermal generation occurs in an electronic semiconductor, as a result of which free electrons and holes are formed, but the concentration of electrons and holes resulting from generation is significantly less than the concentration of free electrons formed during the ionization of impurity atoms, because the energy required to break covalent bonds is significantly greater than the energy spent on ionization of impurity atoms. The electron concentration in an electronic semiconductor is denoted by n n, and the hole concentration by p n. In this case, electrons are the majority charge carriers, and holes are minority carriers.

3 Hole semiconductor

A hole semiconductor or p-type semiconductor (from the Latin positive) is a semiconductor whose crystal lattice contains impurity trivalent atoms called acceptors. In such a crystal lattice, one of the covalent bonds remains unfilled. A free bond of an impurity atom can be filled by an electron that leaves one of the neighboring bonds. In this case, the impurity atom turns into a negative ion, and a hole appears in the place where the electron left. In a hole semiconductor, as well as in an electronic one, thermal generation of charge carriers occurs, but their concentration is many times less than the concentration of holes formed as a result ionization of acceptors. The concentration of holes in a hole semiconductor is denoted p p, they are the main charge carriers, and the electron concentration is denoted n p, they are minority charge carriers.

4 Energy diagrams of semiconductors

According to the concepts of quantum physics, electrons in an atom can take on strictly defined energy values ​​or, as they say, occupy certain energy levels. Moreover, according to the Pauli principle, two electrons cannot be in the same energy state at the same time. A solid, such as a semiconductor crystal, consists of many atoms that strongly interact with each other due to small interatomic distances. Therefore, instead of a set of allowed discrete energy levels characteristic of an individual atom, a solid body is characterized by a set of allowed energy bands consisting of a large number of closely spaced energy levels. Allowed energy bands are separated by energy intervals that electrons cannot possess and which are called band gaps. At absolute zero temperature, electrons fill several lower energy bands. The top of the allowed bands filled with electrons is called the valence band, and the next unfilled band is called the conduction band. In semiconductors, the valence band and conduction band are separated by a band gap. When a substance is heated, electrons are given additional energy, and they move from energy levels of the valence band to higher energy levels of the conduction band. In conductors, insignificant energy is required to make such transitions, therefore conductors are characterized by a high concentration of free electrons (about 10 22 cm -3). In semiconductors, in order for electrons to move from the valence band to the conduction band, they must be given an energy of at least the band gap. This is the energy that is necessary to break covalent bonds. In Fig. Figure 1.4.1 shows energy diagrams of intrinsic electron and hole semiconductors, on which E C denotes the lower boundary of the conduction band, and E V the upper boundary of the valence band. Band gap E z = E c - E v . In silicon it is 1.1 eV, in germanium it is 0.7 eV.

Figure 1.4.1 Energy diagrams of intrinsic electron and hole semiconductors

From the point of view of band theory, the generation of free charge carriers should be understood as the transition of electrons from the valence band to the conduction band (Fig. 1.4.1a). As a result of such transitions, free energy levels appear in the valence band, the absence of electrons on which should be interpreted as the presence of fictitious charges on them - holes. The transition of electrons from the conduction band to the valence band should be interpreted as a recombination of mobile charge carriers. The wider the band gap, the fewer electrons are able to cross it. This explains the higher concentration of electrons and holes in germanium compared to silicon.

In an electronic semiconductor (Fig. 1.4.1, b), due to the presence of pentavalent impurities within the band gap, allowed energy levels E D appear near the bottom of the conduction band. Since one impurity atom accounts for approximately 10 6 atoms of the main substance, the impurity atoms practically do not interact with each other. Therefore, impurity levels do not form an energy band and are depicted as one local energy level E D, on which there are “extra” electrons of impurity atoms that are not occupied in covalent bonds. the energy interval E and = E c -E D is called the ionization energy. The value of this energy for various pentavalent impurities lies in the range from 0.01 to 0.05 eV, so “extra” electrons easily move into the conduction band.

In a hole semiconductor, the introduction of trivalent impurities leads to the appearance of allowed levels E A (Fig. 1.4.1, c), which are filled with electrons passing to it from the valence band, as a result of which holes are formed; the transition of electrons from the valence band to the conduction band requires large energy costs than the transition to acceptor levels, so the electron concentration n p turns out to be less than the concentration n i , and the hole concentration p p can be considered approximately equal to the concentration of acceptors N A.

5 Major and minority charge carriers

Charge carriers whose concentration is higher in a given semiconductor are called majority, and charge carriers whose concentration is lower are called minority. Thus, in an l-type semiconductor, electrons are the majority carriers, and holes are minority carriers; In a p-type semiconductor, holes are the majority carriers and electrons are the minority carriers.

When the concentration of impurities in a semiconductor changes, the position of the Fermi level and the concentration of charge carriers of both signs, i.e., electrons and holes, change. However, the product of the concentrations of electrons and holes in a non-degenerate semiconductor at a given temperature under conditions of thermodynamic equilibrium is a constant value that does not depend on the impurity content. Indeed, from (1.1.3) and (1.1.6) we have:

where is the intrinsic concentration of charge carriers at a given temperature.

If, for example, in an n-type semiconductor the donor concentration is increased, then the number of electrons passing per unit time from impurity levels to the conduction band will increase. Accordingly, the rate of charge carrier recombination will increase and the equilibrium concentration of holes will decrease. Expression

often called the “effective mass” relationship for charge carriers. With its help, you can always find the concentration of minority charge carriers if the concentration of majority charge carriers is known.

1.6 Temperature dependence of charge carrier concentration

Elements of electron statistics. Over a wide temperature range and for different impurity contents, there are temperature dependences of the concentration of charge carriers in an n-type semiconductor, shown in Figure 1.6.1

Figure 1.6.1 - Typical dependences of the concentration of charge carriers in a semiconductor on temperature at various concentrations of donor impurity:

Let us consider the nature of the curve corresponding to a relatively low concentration of donors N d1. In the region of low temperatures, the increase in electron concentration when the semiconductor is heated is due to an increase in the degree of ionization of donors (section of the curve between points 1 to 4). Each ionized donor can be considered as a center that has captured a hole. Considering that the total number of energy states at donor levels per unit volume is N d1, for the concentration of ionized donors we write:

(1.6.1)

where E d1 is the position of the donor level on the energy scale.

At low temperatures, the concentration of ionized donors is equal to the electron concentration:

It follows that

and correspondingly

Where

From expression (1.6.4) it follows that the slope of the straight line in section 1-4 of Figure 1.6.1 characterizes the ionization energy of impurities. In the process of further heating at a certain temperature corresponding to point 4, all electrons from impurity levels are transferred to the conduction band. In this case, the probability of ionization of the semiconductor’s own atoms is still negligible. Therefore, in a fairly wide temperature range (section 4-6), the concentration of charge carriers remains constant and almost equal to the concentration of donors. This region of the temperature dependence is usually called the region of impurity depletion.

At relatively high temperatures (the section of the curve beyond point 6), the transfer of electrons across the band gap begins to play a dominant role, i.e., a transition occurs to the region of intrinsic electrical conductivity, where the concentration of electrons is equal to the concentration of holes, and the steepness of the curve is determined by the band gap of the semiconductor.

For most impurity semiconductors, the temperature of transition to intrinsic electrical conductivity significantly exceeds room temperature. Thus, for n-type germanium with a donor concentration, the temperature is approximately 450 K. The value depends on the impurity concentration and the band gap of the semiconductor.

As the impurity concentration increases, the sections of the curves corresponding to the impurity electrical conductivity shift upward. The reason for this shift is easy to understand using formula (1.6.4). In addition, it must be taken into account that as the concentration of impurity atoms increases, the distance between them decreases. This leads to stronger interaction between the electron shells of impurity atoms and the splitting of discrete energy levels into impurity bands. Accordingly, the ionization energy of impurities decreases. Due to the stated reason. The higher the concentration of impurities, the higher the temperature of their depletion.

At a sufficiently high concentration of donors (), their ionization energy becomes zero, since the resulting impurity band is overlapped by the conduction band. Such a semiconductor is degenerate. The temperature dependence of the charge carrier concentration in this case is characterized by a broken line with two straight segments 3-8 and 8-9. The electron concentration in a degenerate l-type semiconductor is constant over the entire range of impurity electrical conductivity. A degenerate semiconductor is capable of conducting electric current even at very low temperatures. The listed properties make degenerate semiconductors similar to metals. Therefore, they are sometimes called semimetals.

Position of the Fermi level. The Fermi level is one of the main parameters characterizing the electron gas in semiconductors. The position of the Fermi level in a non-degenerate semiconductor at low temperatures can be found by taking the logarithm of equation (1.6.2):

it follows that

(1.6.6)

As can be seen, at very low temperatures the Fermi level in an n-type semiconductor lies midway between the bottom of the conduction band and the donor level. As the temperature increases, the probability of filling donor states decreases, and the Fermi level moves down. At high temperatures, the properties of the semiconductor are close to its own, and the Fermi level rushes to the middle of the band gap, as shown in Figure 1.6.2, a.

Figure 1.6.2, a - Temperature change in the position of the Fermi level in an n-type impurity semiconductor

All the considered patterns are similarly manifested in p-type semiconductors. The temperature dependence of the Fermi level for a hole semiconductor is shown in Fig. 1.6.2, b.

Figure 1.6.3 shows the temperature dependence of the concentration of free electrons for an n-type semiconductor doped with a donor impurity with a concentration

Figure 1.6.2, b - Temperature change in the position of the Fermi level in a p-type impurity semiconductor

Figure 1.6.3 - Temperature dependence of electron concentration in an n-type semiconductor

As can be seen from Figure 1.6.3, there are three temperature ranges in which the change in charge carrier concentration is of a different nature. Let us consider the physical processes that determine the n(T) dependence. Region I (temperature range from T=0 K to T S). With increasing temperature, the concentration of free electrons increases due to the ionization of semiconductor atoms and impurity atoms. But to ionize a semiconductor atom, it is necessary to impart an energy to the electron that is no less than Eg, therefore, in the considered region of low temperatures, the intrinsic concentration of charge carriers is negligible. An n-type semiconductor contains a donor impurity, which gives the energy level E D in the band gap. Therefore, the increase in electron concentration in the temperature range under consideration occurs mainly due to the ionization of donor impurity atoms. Region I is called the region of weak ionization or the freeze-out region. The boundary of this interval on the high temperature side is the impurity depletion temperature T S . If we qualitatively analyze the relationship between the impurity depletion temperature and the depth of the impurity level (E C -E D) and the impurity concentration N d, it will become clear that T S is proportional to the indicated value

Region II (temperature range from T S to T I). With a further increase in temperature, the number of ionized impurity atoms and, accordingly, the concentration of free electrons in the conduction band increase. Finally, the impurity is completely depleted, after which the concentration of free electrons remains almost constant and equal to Nd, since the entire impurity is completely ionized and cannot serve as a source of further increase in the number of free electrons, therefore this region is called the impurity depletion region. Temperature T I is the temperature of transition from impurity electrical conductivity to intrinsic conductivity.

Region 3 (temperature range greater than T I). As the temperature in this region increases, the electron concentration increases due to the ionization of semiconductor atoms, and intrinsic electrical conductivity occurs. The temperature T I of the transition from impurity electrical conductivity to intrinsic conductivity is proportional to the band gap and the concentration of the donor impurity

(1.6.8)

In the region of weak ionization

(1.6.9)

In the region of impurity depletion

In the region of intrinsic conductivity

where n i is the intrinsic concentration of charge carriers in the semiconductor, defined as

(1.6.11)

here N C and N V are the effective density of states in the conduction band and valence band, respectively

(1.6.12)

(1.6.13)

7 Donor and acceptor semiconductors

Donor semiconductors are obtained by adding elements to a semiconductor from which an electron can easily be “torn off”. For example, if pentavalent arsenic (or phosphorus) is added to tetravalent silicon (or germanium), the latter uses its 4 valence electrons to create 4 valence bonds in the crystal lattice, and the fifth electron turns out to be “extra”; such an electron easily breaks away from the atom and begins move relatively freely around the crystal. In this case, an excess of free electrons is formed in the crystal. We should not forget about the formation of electron-hole pairs, as was considered in the case of a pure semiconductor, however, this requires much more energy, and therefore the probability of such a process at room temperatures is quite small

In the language of band theory, the appearance of “easily detached” electrons corresponds to the appearance of donor levels in the band gap near the lower edge of the conduction band, as shown in Figure 1.7.1

Figure 1.7.1 - Diagram of electronic states of a donor semiconductor

At temperatures on the order of room temperature, the main contribution to the conductivity of the semiconductor will come from electrons that have transferred to the conduction band from donor levels, while the probability of electrons transferring from the valence band will be very small. electron acceptor semiconductor temperature

As the temperature increases, a significant portion of the electrons from a small number of donor levels will move to the conduction band; in addition, the probability of the transition of electrons from the valence band to the conduction band will become significant. Since the number of levels in the valence band is much greater than the number of impurity levels, then with increasing temperature the difference in the increasing concentrations of electrons and holes will become less noticeable; they will differ by a small amount - the concentration of donor levels. The donor character of the semiconductor will be less and less pronounced. And finally, with an even greater increase in temperature, the concentration of charge carriers in the semiconductor will become very large, and the donor semiconductor will become similar to a pure semiconductor, and then to a conductor whose conduction band contains many electrons.

The Fermi level in a donor semiconductor shifts up the energy scale, and this shift is greater at low temperatures, when the concentration of free electrons significantly exceeds the number of holes. With increasing temperature, when the donor character of the semiconductor becomes less and less pronounced, the Fermi level shifts to the middle part of the band gap, as in a pure semiconductor.

Acceptor semiconductors are obtained by adding elements to the semiconductor that easily “take away” an electron from the atoms of the semiconductor. For example, if trivalent indium is added to tetravalent silicon (or germanium), the latter will use its three valence electrons to create three valence bonds in the crystal lattice, and the fourth bond will be without an electron. An electron from a neighboring bond can move to this empty space, and then a hole will appear in the crystal. This is shown in Figure 1.7.2

In this case, an excess of holes is formed in the crystal. We should not forget about the formation of electron-hole pairs, as was considered in the case of a pure semiconductor, but the probability of this process at room temperatures is quite low.

Figure 1.7.2 - Formation and movement of electrons and holes in semiconductors

In the language of band theory, the transition of an electron from a full covalent bond to a bond with a missing electron corresponds to the appearance of acceptor levels in the band gap near the lower edge of the conduction band. The diagram of such a state is shown in Figure 1.7.3

Figure 1.7.3 - Diagram of electronic states of an acceptor semiconductor

For such a transition from the valence band to the acceptor level (in this case, the electron simply moves from one covalent bond to almost the same another bond) requires less energy than for the transition from the valence band to the conduction band (Figure 1.7.3), that is, for "complete departure" of an electron from a covalent bond.

At temperatures on the order of room temperature, the main contribution to the conductivity of the semiconductor will come from holes formed in the valence band after the transition of valence electrons to acceptor levels, while the probability of the transition of electrons from the valence band to the conduction band will be very small.

As the temperature increases, a significant part of the small number of acceptor levels will be occupied by electrons. In addition, the probability of electrons moving from the valence band to the conduction band will become significant. Since the number of levels in the valence band is much greater than the number of impurity levels, then with increasing temperature the difference in the increasing concentrations of electrons and holes will become less noticeable, since they differ by a small amount - the concentration of acceptor levels. The acceptor character of the semiconductor will be less and less pronounced. And finally, with an even greater increase in temperature, the concentration of charge carriers in the semiconductor will become very large, and the acceptor semiconductor will become similar, first to a pure semiconductor, and then to a conductor.

It can be shown that the Fermi level in an acceptor semiconductor shifts down the energy scale, and this shift is greater at low temperatures, when the hole concentration significantly exceeds the concentration of free electrons. With increasing temperature, when the acceptor character of the semiconductor becomes less and less pronounced, the Fermi level shifts to the middle part of the band gap, as in a pure semiconductor.

So, with a gradual increase in temperature, a gradual transformation of both the donor and acceptor semiconductor into a semiconductor similar to the pure one is observed, and then into a semiconductor similar in conductivity to the conductor. This is the cause of failure due to overheating of semiconductor devices consisting of several regions of donor and acceptor type semiconductors. As the temperature increases, the differences between the regions gradually disappear and eventually the semiconductor device turns into a monolithic piece of highly conductive semiconductor.

8 Dependence of electron concentration on Fermi level energy

Concentration of electrons in the conduction band from donor impurities

is determined by the position of the Fermi level and is found from the expression relating it to the Fermi level,

where EF is the Fermi level energy;

Ec is the energy corresponding to the bottom of the conduction band;

k-Boltzmann constant;

T-absolute temperature;

h-Planck's constant;

mn is the effective mass of the electron.

To plot the dependence of the electron concentration in the conduction band n on the Fermi level, it is necessary to substitute into the equation

1.9 Position of the Fermi level and concentration of free charge carriers in intrinsic semiconductors

In an intrinsic pure semiconductor, the position of the Fermi level can be found from the condition that the number of electrons in the conduction band is equal to the number of holes in the valence band: the subscript i hereinafter denotes that it belongs to the intrinsic semiconductor.

Condition (1.9.1) leads to the fact that the Fermi level should be located approximately in the middle of the band gap. If the Fermi level is located closer to the conduction band, then in such a semiconductor there will be many more electrons than holes, since the degree of filling f(E) of states at the bottom of the conduction band is significantly greater than the degree of their unfilling (1-f(E)) at the ceiling of the valence band. The distribution function f(E) for this case is presented in Figure 1.9.1, a, where for convenience of comparison with energy diagrams the energy axis is directed upward.

Figure 1.9.1, a - Distribution functions f(E) in n-type semiconductors ().

Figure 1.9.2, b - Distribution functions f(E) in n-type semiconductors ()

On the contrary, if you place the Fermi level near the valence band, then there will be many more holes in the valence band than electrons in the conduction band. This distribution function is presented in Figure 1.9.2,b

Thus, it is possible to ensure equality in the numbers of electrons and holes only if the Fermi level is located in the middle of the band gap. In general, however, the Fermi level in an intrinsic semiconductor is located only approximately in the middle of the band gap. The point is that the density of states functions in the conduction band and valence band may differ from each other. For example, if then, according to 1.9.2, b and 1.9.3, the density of states in the conduction band is higher than in the valence band (at the same distance of the considered energy intervals from the edges of the corresponding bands).

Figure 1.9.3 - Dependence of the concentration of free current carriers on the temperature in the intrinsic semiconductor

In this case, to equalize the concentrations of electrons and holes, the Fermi level should be lowered slightly towards the valence band. On the contrary, if then the Fermi level should be slightly above the middle of the band gap.

We obtain the exact value of the Fermi energy in the intrinsic semiconductor

When the Fermi level is located in the middle of the band gap, with increasing temperature it shifts to the zone where the density of states is lower. This dependence is shown in Figure 1.9.4

Figure 1.9.4 - Dependence of the Fermi energy (dash-dotted line) on the temperature in the intrinsic semiconductor

Since it is usually not very different from , then for semiconductors with a bandgap of eV or more can be considered. For narrow-gap semiconductors, the shift of the Fermi level from the middle of the bandgap cannot be ignored.

Let us find the concentration of free current carriers in the intrinsic semiconductor:

From (1.6.4) it is clear that the equilibrium concentration of current carriers in the intrinsic semiconductor is determined by the band gap and the temperature of the semiconductor, and the dependence of , on T is very sharp. Thus, a decrease from 1.12 eV (silicon) to 0.08 eV (gray tin) leads at room temperature to an increase by 9 orders of magnitude; An increase in the temperature of germanium from 100 to 600 K increases by 17 orders of magnitude. Let's take the logarithm of expression (1.9.4)

(1.9.5)

Since it depends on temperature much more weakly than according to the power law, the graph of the dependence on is approximately a straight line with an angular coefficient of .

For narrow-gap semiconductors at elevated temperatures, the Fermi level may be too close (closer than () to one of the bands or even both bands. In this case, it is impossible to use expressions for non-degenerate gases of electrons and holes and equation (1.9.5) should solve numerically.

2. Calculation of temperature dependences of electrophysical parameters of semiconductors

Electron charge

Atomic rest mass

Donor level ionization energy

- electron masses along the main axes of the ellipsoids

- masses of holes along the main axes of the ellipsoids

Number of valleys in the conduction zone

Number of valleys in the valence band

Donor atom concentration

- Boltzmann constant

Bandgap width

- temperature

Planck's constant

This was followed by the need to convert them to the SI system. Now that all the data is in front of us, we can begin with an approximate calculation of the dependence of electron concentration on temperature.

2.1 Approximate calculation of the dependence of electron concentration on temperature

To begin with, I found the average temperature and effective mass of electrons and holes and , which are then necessary to calculate the effective density of states in the valence and conduction bands and

2 Analytical calculation of the dependence of the concentration of free charge carriers and the position of the Fermi level on temperature

Analytical calculation allows you to find the exact values ​​of the required parameters.

First, I found the exact values ​​of the intrinsic conduction temperature and impurity depletion temperature using the root function, which is used to find the zeros of the function.

2.1 Finding exact values ​​and

To construct the graph of hole concentration versus temperature for various regions, presented in Appendix B, I used the exact concentration values ​​​​in the same three regions.

2.2 Low temperature range T11 (exact values)

This work is the foundation for acquiring high professional skills and is intended for a wide range of readers - engineers, scientists and junior and senior students of technical universities.

The necessary information provided in my course work serves to attract young enterprising students to active creative work in this developing direction.

List of sources used

1. Pasynkov V.V., Sorokin V.S. Materials of electronic technology 3rd ed. - St. Petersburg: Lan Publishing House, 2001. - pp. 91-101.

. Epifanov G.I., Moma Yu.A. Physical principles of design and technology of REA and EVA. - M.: Sov. Radio, 1979. - 350s.

. Pavlov P.V., Khokhlov A.F. Solid state physics. Textbook for universities. - M.: Higher. school, 2000.-384 pp. (16 copies).

. Bonch-Bruevich V.L., Kalashnikov S.G. Physics of Semiconductors, 1977 - pp. 167-200.

. Anselm A.I. Introduction to the theory of semiconductors. From 225-231.

Appendix B (mandatory)

Dependency graphs

Figure 1 - Graph of charge carrier concentration versus temperature (approximate values)

Figure 2 - Graph of charge carrier concentration versus temperature (exact values)

Figure 3 - Graph of the position of the Fermi levels versus temperature