Corresponding author: Aleksandr G. Belov (iadenisov@giredmet.ru)

The concentrations of conduction electrons in

Hall data for tellurium doped CZ

Giredmet JSC has been for many years developing contactless nondestructive methods for room temperature measurement of free carrier concentration in heavily doped semiconductors. Free carrier concentration measurement includes taking far infrared reflection spectra of test specimens and determination of characteristic frequencies by mathematical processing of the reflection spectra. Studies have been conducted for Pb_{1-}_{x}_{x}_{x}_{1-}_{x}

This work is a continuation of our earlier series of studies [

The band structure and electrical properties of

Noteworthy, the concentration and mobility of electrons are calculated from Hall data using simple formulas in the assumption that

The aim of this work is to calculate the concentrations and effective masses of conduction electrons in n-GaSb at 295 and 77 K.

It is well known that the conduction band of GaSb has several valleys [

Band structure of GaSb [22]

The cited work also reported formulas describing the temperature dependences of the energy gaps _{g} and EL:

where

Using Eqs. (1) and (2) one can calculate the energy gaps for 295 and 88 K:

– for

_{g295} = 0.728 eV; _{L}_{295} = 0.813 eV;

∆_{295} = _{L}_{295} – _{g295} = 0.085 eV = 85 meV;

– for

_{g77} = 0.800 eV; _{L}_{77} = 0.888 eV;

∆_{77} = _{L}_{77} – _{g77} = 0.088 eV = 88 meV.

It can be seen from these data that the gap ∆

Taking into account that for ^{–16} erg/deg is the Boltzmann constant) and for _{295}/_{77}/

We consider the statistics for conduction electrons in

where _{1} and _{1} are the concentration and effective mass of electrons in the Г-valley, respectively, _{0} = 9.11 ∙ 10^{–28} g is the mass of a free electron, _{cv} = 8,7 ∙ 10^{–8} eV ∙ cm [^{–27} erg ∙ s is the Planck constant. It is assumed that the parameter _{cv} does not depend on energy and temperature [

Equations (3) and (4) use two-parameter Fermi integrals:

where

_{0} = [1 + exp(^{–1}, (6)

η = _{F}/_{g} is the parameter describing the non-parabolic deviation of the Г-valley in the conduction band.

Unlike the Г-valley, the _{2}_{l}_{0}, the transverse one being _{2}_{t}_{0} [_{1} = _{2}_{l}_{2}_{t}

The concentration of electrons in the _{2} is described by the following relationship:

Here _{3⁄2}(η) is the single-parameter Fermi integral.

The integral ^{0}_{0}^{3⁄2}(η, β) transforms to _{3⁄2}(η) at β → 0, i.e., if the non-parabolic deviation of the band shape can be ignored. The argument of the integral _{3⁄2} in Eq. (8) is (η – 3.35) because the

As a result, Eqs. (3), (4) and (8) transform as follows for

_{1} = 2.376 ∙ 10^{17} ∙ ^{0}_{0}^{3⁄2}(η, 0.0349); (9)

_{2} = 7.900 ∙ 10^{18} ∙ _{3⁄2} (η – 3.35). (11)

Then, substituting η in the range (–4,0 ÷ +4,0) into Eqs. (9)–(11) one can calculate the above-listed parameters as follows (Table

Calculated parameters for

_{F}/ |
^{0}_{0}^{3/2} (η, 0.0349) |
^{0}_{–1}^{3/2} (η, 0.0349) |
_{1} (cm^{–3}) |
_{1}/_{0} |
_{3/2} (η – 3.35) |
_{2} (cm^{–3}) |

–4.0 | 0.02746 | 0.02341 | 6.525 ∙ 10^{15} |
0.0645 | 8.540 ∙ 10^{–4} |
6.747 ∙ 10^{15} |

–3.5 | 0.04510 | 0.03843 | 1.072 ∙ 10^{16} |
0.0646 | 1.408 ∙ 10^{–3} |
1.112 ∙ 10^{16} |

–3.0 | 0.07389 | 0.06293 | 1.756 ∙ 10^{16} |
0.0646 | 2.321 ∙ 10^{–3} |
1.833 ∙ 10^{16} |

–2.5 | 0.1206 | 0.1026 | 2.865 ∙ 10^{16} |
0.0646 | 3.825 ∙ 10^{–3} |
3.021 ∙ 10^{16} |

–2.0 | 0.1956 | 0.1662 | 4.647 ∙ 10^{16} |
0.0647 | 6.301 ∙ 10^{–3} |
4.978 ∙ 10^{16} |

–1.5 | 0.3143 | 0.2665 | 7.468 ∙ 10^{16} |
0.0649 | 1.037 ∙ 10^{–2} |
8.192 ∙ 10^{16} |

–1.0 | 0.4879 | 0.4209 | 1.188 ∙ 10^{17} |
0.0651 | 1.708 ∙ 10^{–2} |
1.349 ∙ 10^{17} |

–0.5 | 0.7738 | 0.6512 | 1.839 ∙ 10^{17} |
0.0654 | 2.808 ∙ 10^{–2} |
2.218 ∙ 10^{17} |

0 | 1.173 | 0.9805 | 2.787 ∙ 10^{17} |
0.0658 | 4.607 ∙ 10^{–2} |
3.640 ∙ 10^{17} |

0.5 | 1.725 | 1.430 | 4.098 ∙ 10^{17} |
0.0664 | 7.537 ∙ 10^{–2} |
5.954 ∙ 10^{17} |

1.0 | 2.455 | 2.012 | 5.833 ∙ 10^{17} |
0.0671 | 0.1227 | 9.693 ∙ 10^{17} |

1.5 | 3.379 | 2.732 | 8.028 ∙ 10^{17} |
0.0680 | 0.1983 | 1.566 ∙ 10^{18} |

2.0 | 4.506 | 3.588 | 1.071 ∙ 10^{18} |
0.0691 | 0.3169 | 2.504 ∙ 10^{18} |

3.0 | 7.378 | 5.669 | 1.753 ∙ 10^{18} |
0.0716 | 0.7655 | 6.047 ∙ 10^{18} |

4.0 | 11.07 | 8.178 | 2.631 ∙ 10^{18} |
0.0745 | 1.653 | 1.306 ∙ 10^{19} |

As can be seen from Table _{2} values are always greater than _{1}, this difference increasing with an increase in the reduced Fermi level. For example, for η = –4, _{1} ≈ _{2}, whereas for η = 2, _{2}/_{1} = 2.34 (see Table _{1} ~ 10^{18} cm^{–3} most conduction electrons are concentrated in the

We now consider the statistics for electrons at the liquid nitrogen temperature (_{g} = 0.00829 and ∆

_{1} = 3.649 ∙ 10^{16} ∙ ^{0}_{0}^{3⁄2}(η, 0.00829); (12)

_{2} = 1.053 ∙ 10^{18} ∙ _{3⁄2}(η – 13.3). (14)

The results of calculations with Eqs. (12)–(14) are summarized in Table

Calculated parameters for

η = _{F}/ |
^{0}_{0}^{3/2} (η, 0.00829) |
^{0}_{–1}^{3/2} (η, 0.00829) |
_{1} (cm^{–3}) |
_{1}/_{0} |
_{3/2} (η – 13,3) |
_{2} (cm^{–3}) |

+2 | 3.927 | 3.702 | 1.433 ∙ 10^{17} |
0.0648 | 1.645 ∙ 10^{–5} |
1.732 ∙ 10^{13} |

+5 | 12.63 | 11.51 | 4.609 ∙ 10^{17} |
0.0663 | 3.303 ∙ 10^{–4} |
3.478 ∙ 10^{14} |

+8 | 25.58 | 22.39 | 9.335 ∙ 10^{17} |
0.0690 | 6.624 ∙ 10^{–3} |
6.975 ∙ 10^{15} |

+10 | 36.28 | 30.91 | 1.324 ∙ 10^{18} |
0.0709 | 0.04840 | 5.097 ∙ 10^{16} |

+13.3 | 57.38 | 46.78 | 2.094 ∙ 10^{18} |
0.0741 | 1.017 | 1.071 ∙ 10^{18} |

The value η = +13.3 corresponds to the Fermi level position at the bottom of the

It can be seen from Table _{1} is always greater than _{2} (conduction electrons are mostly concentrated in the Г-valley). Only for _{1} ≈ 2 ∙ 10^{18} cm^{–3} the _{2} values prove to be of the same order of magnitude.

The concentration and mobility of electrons in

ρ = (^{–1}; (15)

Here ρ is the electrical resistivity (Ohm ∙ cm), ^{3}/C); μ is the electron mobility (cm^{2}/(V ∙ s)); ^{–19} C is the electron charge (taken by absolute value).

This approach is justified at low temperatures (

where μ_{1} and μ_{2} are the electron mobilities in the Г- and

Introducing the dimensionless parameter _{1}/µ_{2} describing the relation of the electron mobilities in the Г- and

The values _{1} and _{2} are interrelated (see Eqs. (3) and (8)) but the parameter _{1} and _{2} by the order of magnitude.

We studied

Electrical measurements were conducted for cylindrical reference wafers cut from the top and bottom sections of the single crystal, grounded with M14 powder and etch-polished for damaged layer removal. The wafers were then cut into 10–15 mm sized specimens. The specimen thickness was

Results of electrica measurements

Specimen # | ρ (Ohm ∙ cm) | |^{3}/C) |
^{–3}) |
μ = |^{2}/(V ∙ s)) |
_{77}/_{295} |
||

1 | 0.55 | 295 | 6.20 ∙ 10^{–3} |
18.1 | 3.45 ∙ 10^{17} |
2.9 ∙ 10^{3} |
1.74 |

77 | 2.02 ∙ 10^{–3} |
10.4 | 6.01 ∙ 10^{17} |
5.2 ∙ 10^{3} |
|||

2 | 1.99 | 295 | 5.26 ∙ 10^{–3} |
14.9 | 4.19 ∙ 10^{17} |
2.8 ∙ 10^{3} |
1.65 |

77 | 1.72 ∙ 10^{–3} |
8.66 | 7.23 ∙ 10^{17} |
5.0 ∙ 10^{3} |
|||

3 | 0.45 | 295 | 4.56 ∙ 10^{–3} |
13.9 | 4.50 ∙ 10^{17} |
3.1 ∙ 10^{3} |
1.65 |

77 | 1.36 ∙ 10^{–3} |
8.41 | 7.43 ∙ 10^{17} |
6.2 ∙ 10^{3} |
|||

4 | 2.12 | 295 | 4.75 ∙ 10^{–3} |
13.6 | 4.60 ∙ 10^{17} |
2.9 ∙ 10^{3} |
1.68 |

77 | 1.51 ∙ 10^{–3} |
8.07 | 7.74 ∙ 10^{17} |
5.3 ∙ 10^{3} |
|||

5 | 0.50 | 295 | 3.04 ∙ 10^{–3} |
8.10 | 7.72 ∙ 10^{17} |
2.7 ∙ 10^{3} |
1.37 |

77 | 9.13 ∙ 10^{–4} |
5.89 | 1.06 ∙ 10^{18} |
6.5 ∙ 10^{3} |
|||

6 | 0.94 | 295 | 3.16 ∙ 10^{–3} |
7.47 | 8.37 ∙ 10^{17} |
2.4 ∙ 10^{3} |
1.46 |

77 | 9.53 ∙ 10^{–4} |
5.11 | 1.22 ∙ 10^{18} |
5.4 ∙ 10^{3} |
|||

7 | 1.36 | 295 | 2.32 ∙ 10^{–3} |
6.09 | 1.03 ∙ 10^{18} |
2.6 ∙ 10^{3} |
1.25 |

77 | 7.17 ∙ 10^{–4} |
4.83 | 1.29 ∙ 10^{18} |
6.7 ∙ 10^{3} |
|||

8 | 2.04 | 295 | 1.71 ∙ 10^{–3} |
4.46 | 1.40 ∙ 10^{18} |
2.6 ∙ 10^{3} |
1.10 |

77 | 5.68 ∙ 10^{–4} |
4.07 | 1.54 ∙ 10^{18} |
7.2 ∙ 10^{3} |

Two test specimens were placed at opposite sides of a two-side specimen holder, and the conductor wires were soldered to the respective contact pads of the specimen holder. The specimen holder with the specimens was placed into a foam plastic cryostat installed between the poles of an electric magnet. The cryostat was filled with liquid nitrogen to a level to cover the specimens. The measurements were run using the conventional four-probe setup (the Van der Pau method).

The electrical resistivity of the specimens was measured without a magnetic field, and the Hall measurements were run at the magnetic field induction _{spec} = 200 mA.

The results of the electrical measurements for the

As can be seen from Table _{77}/_{295} ratio decreasing with an increase in _{77}. It was noted above that only the data for

Using Tables _{1} и _{2} for _{1} for _{77} = 1.29 ∙ 10^{18} cm^{–3}, and ^{18} cm^{–3}. Accepting that _{1} remains the same at _{F}/_{1} ≈ 1.07 ∙ 10^{18} cm^{–3} and _{2} ≈ 2.50 ∙ 10^{18} cm^{–3}. Accordingly, (_{1} + _{2})_{295} ≈ 3.57 ∙ 10^{18} cm^{–3}, which is _{295} = 1.29 ∙ 10^{18} cm^{–3}. Thus the apparent increase in the electron concentration upon specimen cooling is actually not the case.

The conductivity and Hall effect were analyzed as a function of temperature and pressure earlier [_{1} = 0.047_{0} which is considerably less than the calculation data presented in Tables

Furthermore, the overall electron concentration in the Г- and _{D} introduced into the specimen which is completely ionized in the entire experimental temperature range [_{D}. It was shown [

Finally the value

The first estimate _{1} = 0.047_{0} was obtained [_{0} [_{0} [_{0} at _{0} at _{1} are noticeably greater than the above experimental ones. This fact causes changes to the entire free carrier statistics. However, this question requires a separate study.

It was reported [

employed in the Kane theory [_{cv} = 8.73 ∙ 10^{–8} eV ∙ cm which is close to the value _{cv} = 8.7 ∙ 10^{–8} eV ∙ cm [_{cv} is assumed to be energy- and temperature-independent. This opinion is confronted by other authors [_{cv} is (8.28–8.35) ∙ 10^{–8} eV ∙ cm.

We plan a special study to determine the parameter

Statistics of conduction electrons in the Г- (_{1}) and _{2}) of the conduction band of GaSb at 295 and 77 K were calculated.

We show that at _{1} and _{2} is impossible in this case.

We also show that at

Room and LN temperature electrical measurements were carried out for a series of

_{1-}

_{x}

_{x}

_{x}

_{1-}

_{x}

^{III}

^{V}semiconductors in high electric fields.