RU2037911C1 - Method of contactless determination of concentration of free charge carriers in semiconductors - Google Patents

Abstract

FIELD: nondestructive testing. SUBSTANCE: semiconductor sample is cooled up to helium temperatures, is subjected to influence of permanent magnetic field with induction B and to SHF radiation directed perpendicular to investigated surface of sample and in parallel to induction vector

Description

 The invention relates to methods for non-destructive testing of parameters of semiconductors and semiconductor structures containing degenerate electron gas of reduced dimension, and can be used for scientific research and to determine the quality of materials used in semiconductor instrument making.

(B^-1)]^-3/2.A known contact method for determining the concentration of free charge carriers in degenerate semiconductors using the Schottky barrier [1] is based on the static Shubnikov-de Haas effect, which consists in creating electrical contacts to the sample, then the sample of the studied semiconductor is cooled to helium temperatures, placed in magnetic field B, register the oscillations of the transverse magnetoresistance of the test sample when the magnetic field changes, measure the period of oscillations of the transverse magnetoresistance values inverse to the magnetic field, and determine the concentration of free charge carriers in degenerate semiconductors using the calculation formula
n

(B ^-1 )] ^-3/2 .

 The disadvantages of this method are the destructive effect of the contacts on the surface of the investigated semiconductor, which limits the application of the method, for example, it does not allow to determine the concentration of charge carriers of an electron gas located under a layer of a dielectric and metal coating or above a conductive substrate without destruction; the impossibility of creating a Schottky barrier at the depth of the layer for many heavily doped semiconductors; the fundamental impossibility of creating contacts in the diagnosis of submicron structures.

монохроматическим когерентным излучением, энергия кванта которого меньше ширины запрещенной зоны полупроводника (h

ΔE), поляризованного так, что вектор напряженности электрического поля перпендикулярен постоянному магнитному полю

, регистрируют интенсивность I прошедшего через образец или отраженного от него излучения, по соседним максимумам зависимости второй производной ∂²I/∂В² от величины В определяют концентрацию носителей заряда по расчетной формуле
n

-

(B $\genfrac{}{}{0ex}{}{\text{-1}}{\text{N+}}$ ₁- B $\genfrac{}{}{0ex}{}{\text{-}}{\text{N}}$ ¹)^-3/2 где К 0, 1, 3, 5,
N номер максимума (номер уровня Ландау).A known non-contact method for determining the concentration of charge carriers in degenerate semiconductors [2] is based on the optical Shubnikov-de Haas effect, namely, that the sample of the semiconductor under study is cooled to helium temperatures, simultaneously exposed to it by a changing constant magnetic field B, an alternating magnetic field with amplitude b, much smaller than B, and directed perpendicular to the constant magnetic field

monochromatic coherent radiation, whose quantum energy is less than the band gap of the semiconductor (h

ΔE), polarized so that the electric field vector is perpendicular to the constant magnetic field

, the intensity I of the radiation transmitted through the sample or reflected from it is recorded, from neighboring maxima of the dependence of the second derivative ∂ ² I / ∂B ² on the value of B, the concentration of charge carriers is determined by the calculation formula
n

-

(B $\genfrac{}{}{0ex}{}{\text{-1}}{\text{N +}}$ ₁ - B $\genfrac{}{}{0ex}{}{\text{-}}{\text{N}}$ ¹ ) ^-3/2 where K 0, 1, 3, 5,
N is the maximum number (Landau level number).

The disadvantage of this method is the impossibility of determining the concentration of charge carriers in very thin layers of a semiconductor above a conductive substrate and in semiconductor structures containing a degenerate electron gas of reduced dimension. This limitation is mainly due to the fact that the frequency of optical radiation is many times greater than the frequency of collisions of electrons with atoms τ _p ^-1 . In this case, the transfer of energy from a light wave to free electrons is inefficient, since the absorption of light wave energy occurs within the same period of electron oscillations. As a result, the sensitivity of the method in the diagnosis of low-dimensional electron gas is insufficient.

 The second disadvantage of this method is the impossibility of determining the concentration of charge carriers in layers containing a low-dimensional electron gas, due to the fact that the direction of the optical radiation is chosen perpendicular to the direction of the constant magnetic field (Voigt geometry), therefore this method is not applicable when the thickness of the investigated two-dimensional layer is less than the electron diameters orbits making up tenths of a micron.

 Closest to the proposed method is a non-contact method for determining the concentration of free charge carriers in semiconductors [3], which consists in the fact that the sample of the studied semiconductor is cooled, exposed to it by a constant magnetic field, the induction vector of which is perpendicular to the surface of the sample, is excited in the microwave radiation of a given frequency, parallel to the direction of the magnetic field, measure the intensity of the microwave radiation reflected from the sample, depending on the terms of a constant magnetic field and determine by it the concentration of charge carriers by calculation.

The disadvantages of this method. Cooling the sample to nitrogen temperatures is not sufficient to satisfy the degeneracy condition of the electron gas and for the appearance of quantum effects. The sensitivity of the method is not sufficient to measure small changes in the intensity of reflection of microwave radiation from thin layers of degenerate semiconductors. The frequency of collisions of charge carriers τ _p ^-1 at nitrogen temperatures decreases so much that it becomes less than the frequency of microwave radiation. This limits the possibility of applying the method for measuring degenerate semiconductors in thin layers. The method is not applicable when it is impossible to measure the thickness of the investigated semiconductor layer of the sample. The method is not applicable for measurements in very thin layers when Fabry-Perot microwave resonance is not possible.

 The technical result of the proposed method is the possibility of measuring thin layers of degenerate semiconductors.

которого перпендикулярен поверхности образца, возбуждают в образце СВЧ-излучение заданной частоты и направленное параллельно направлению магнитного поля, измеряют интенсивность отраженного от образца СВЧ-излучения в зависимости от величины постоянного магнитного поля и определяют по ней концентрацию носителей заряда расчетным путем, согласно изобретению образец охлаждают до гелиевых температур, дополнительно воздействуют переменным магнитным полем, изменяющимся со звуковой частотой ω и амплитудой b << B, направленным параллельно постоянному магнитному полю, частоту СВЧ-излучения выбирают меньше частоты столкновения носителей с атомами, а их концентрацию определяют по формуле
n

(

-

где е заряд электрона, Кл;
ℏ постоянная Планка, Дж с;
М число долин в зоне проводимости полупроводника;
B_N и B_N+1 значения индукции постоянного магнитного поля, соответствующие двум соседним экстремумам N и N+1, Т.This result is achieved by the fact that by the method of non-contact determination of the concentration of free charge carriers in semiconductors, which consists in the fact that the test sample is cooled, it is exposed to a constant magnetic field, the induction vector

which is perpendicular to the surface of the sample, excite microwave radiation of a given frequency in the sample and directed parallel to the direction of the magnetic field, measure the intensity of microwave radiation reflected from the sample depending on the magnitude of the constant magnetic field and determine the concentration of charge carriers by calculation, according to the invention, the sample is cooled to helium temperatures are additionally affected by an alternating magnetic field, changing with sound frequency ω and amplitude b << B, directed in parallel along a constant magnetic field, the frequency of microwave radiation is chosen less than the frequency of collision of carriers with atoms, and their concentration is determined by the formula
n

(

-

where e is the electron charge, C;
ℏ Planck constant, J s;
M is the number of valleys in the conduction band of the semiconductor;
B _N and B _{N + 1 are the} values of the induction of a constant magnetic field corresponding to two adjacent extrema of N and N + 1, T.

 The physical nature of the proposed method is as follows.

It is known that in semiconductor structures with superlattices, or in heterostructures, for example, of the type AlGaAs / GaAs, energy bands bend. This leads to the formation of discrete energy levels E ₀ , E ₁ , E _i , in a thin transition layer at the interfaces of two semiconductors: a metal semiconductor or a dielectric semiconductor. If the Fermi energy is E _F > E _i , then the energy levels E <E _i are filled with electrons. When the thickness of the transition layer is less than the mean free path of the electron or the de Broglie wavelength of the electron, the motion of the electron occurs mainly in the plane of this layer.

, где S площадь образца. В расчете на единицу поверхности слоя, т.е. при S 1, ΔK_xΔK_y π ². Число ячеек для электронов, имеющих волновое число К, равно KdK/2π Число состояний с учетом принципа Паули и числа долин М в зоне проводимости полупроводника
dz

.We define the density function of quantum states for a two-dimensional layer. The cell size of the phase plane in this case ΔK _x ΔK _y

where S is the sample area. Based on the unit surface of the layer, i.e. at S 1, ΔK _x ΔK _y π ² . The number of cells for electrons having a wave number K is KdK / 2π. The number of states, taking into account the Pauli principle and the number of valleys M in the conduction band of a semiconductor
dz

.

.Replacing KdK m * dE / ℏ ² , we obtain
dz

.

.The surface density of quantum states is independent of energy:

.

dE
n_i=

dE

(E_F- E_i).When E _F > E _{i, the} probability of filling the level of E _i is equal to one. Then the surface electron concentration is determined by integrating the expression
dn

dE
n _i =

dE

(E _F - E _i ).

)ℏω_c, где N номер уровня Ландау;
ω_c циклотронная частота вращения электрона в магнитном поле, равная еВ/m*.It is known that in quantizing magnetic fields the density of electronic states has a resonant character with maxima near the Landau levels:
E _N = (N +

) ℏω _c , where N is the Landau level number;
ω _c cyclotron frequency of rotation of an electron in a magnetic field, equal to eV / m *.

 When the Fermi level coincides with the Landau level, during the interaction of a microwave electromagnetic wave with a two-dimensional electron gas, a maximum of the high-frequency conductivity of the semiconductor is observed, due to the maximum density of quantum electronic states. This is due to oscillations in the intensity of microwave radiation reflected from the sample, which are proportional to changes in the conductivity of the sample under study.

(N +

)B_N
n

(N+1)+

B_N+1;
n

(B $\genfrac{}{}{0ex}{}{\text{-1}}{\text{N+}}$ ₁- B $\genfrac{}{}{0ex}{}{\text{-}}{\text{N}}$ ¹)^-1.
Амплитуда отраженного СВЧ-излучения зависит от индукции поля В. Индукция поля В с помощью модуляционных катушек изменялась на величину
ΔB_~ b sin ωt.From the condition E _F E _i E _N it follows
n

(N +

) B _N
n

(N + 1) +

B _{N + 1} ;
n

(B $\genfrac{}{}{0ex}{}{\text{-1}}{\text{N +}}$ ₁ - B $\genfrac{}{}{0ex}{}{\text{-}}{\text{N}}$ ¹ ) ^-1 .
The amplitude of the reflected microwave radiation depends on the induction of the field B. The induction of the field B using modulation coils changed by
ΔB _~ b sin ωt.

ΔB_~+
Использование модуляционной методики измерения резонансных явлений после детектирования отраженного СВЧ-излучения диодом детектором позволяет выделять с помощью селективного детектора изменения амплитуды отраженного СВЧ-излучения со звуковой частотой ω и регистрировать с помощью самописца изменение интенсивности отраженного СВЧ-излучения ∂I/∂В в зависимости от В.Then the amplitude of the reflected microwave radiation
A = A (B) A (B _o + ΔB _~ ) A (B _o ) +

ΔB _~ +
Using a modulation technique for measuring resonance phenomena after the reflected microwave radiation is detected by a diode detector, it is possible to select, with a selective detector, changes in the amplitude of reflected microwave radiation with sound frequency ω and to record using a recorder the change in the intensity of reflected microwave radiation ∂I / ∂В depending on В .

In the field of a microwave wave at ν <τ _p ^-1 , an increase in the time of transfer of wave energy to an electron occurs, since it is within the same period of electron oscillations that the energy of an electromagnetic wave is absorbed. In this case, the sensitivity of the proposed method, determined by the signal-to-noise ratio, increases in comparison with the prototype by about 10 ⁴ times.

The accuracy of determining the surface concentration of charge carriers depends only on the accuracy of measuring the value of the induction of a constant magnetic field corresponding to the extreme values of ∂I / ∂В, and does not exceed 0.5%
New in relation to the prototype in the proposed method are the application of the Shubnikov-de Haas microwave effect when the sample is cooled to helium temperatures, with additional exposure to the sample with an alternating magnetic field, changing with sound frequency ω and amplitude b << B, parallel to the constant magnetic field B, and the choice of the frequency of microwave radiation is less than the collision frequency τ _p ^{-1 of} free charge carriers with semiconductor atoms.

 In FIG. 1 shows a functional diagram of a device that implements the proposed method; figure 2 shows graphs of the dependence of the derivative ∂I / ∂B on the induction of a constant magnetic field for two AlGaAs / GaAs semiconductor heterostructures (curve a for the first sample and curve b for the second sample).

 The device (figure 1) contains a source 1 of probing microwave radiation, a source 2 of a constant magnetic field, modulation coils 3 and 4, creating an alternating magnetic field, a cylindrical resonator 5, a circulator 6, rectangular waveguides 7, 13, a microwave detector 8, a generator 9 audio frequency, amplifier-detector 10, recorder 11, magnetic field sensor 12, semiconductor sample 14.

 The microwave radiation source 1 may, for example, be a generator on a serial Gunn diode type AA728B. The constant magnetic field source 2 may be a superconducting solenoid. As a microwave detector 8 can be used, for example, a device based on a Schottky diode type AA123A.

 The rectangular waveguides 7, 13 may, for example, be serial 3.4 x 7.2 mm waveguides connected to the serial circulator 6. The cylindrical resonator 5 may, for example, have an internal diameter of 6.1 mm. The amplifier-detector 10 contains a selective amplifier and a synchronous detector (not shown) and can be performed, for example, on the basis of a universal device such as UNIPAN 232 B. As an audio frequency generator 9, for example, a device of the type GZ-33 can be used.

 The recorder 11 is a two-coordinate recorder, for example, ENDIMOO2. As the sensor 12 of the magnetic field can be used, for example, a Hall sensor calibrated by a device Ш1-1, operating on the basis of nuclear magnetic resonance. Modulation coils 3, 4 can, for example, have an internal diameter of 10 mm and contain 960 turns of PEL wire 0.2 mm.

 The probe microwave radiation source 1 is connected to a rectangular waveguide 7, which is connected to a circulator 6. The latter is connected to a cylindrical resonator 5 and, using a rectangular waveguide 13, to a microwave detector 8. The test sample 14 is mounted at the end of a cylindrical resonator 5 in the working volume of source 2 a constant magnetic field and together with a cylindrical resonator 5, a source 2 of a constant magnetic field, modulation coils 3 and 4 and a magnetic field sensor 12 is placed in a helium cryostat (not shown). Modulation coils 3 and 4 are connected to the first output of the sound frequency generator 9, the second output of which is connected to the reference input of the amplifier-detector 10, which is the reference input of the synchronous detector, the information input of which is connected to the output of the selective amplifier.

 The output of the microwave detector 8 is connected to the information input of the amplifier-detector 10, which is the input of a selective amplifier.

 The output of the amplifier-detector 10, which is the output of the synchronous detector, is connected to the input Y of the recorder 11, the input X of which is connected to the output of the magnetic field sensor 12 located in the working volume of the constant magnetic field source 2.

 The method is as follows.

СВЧ-волну типа Е_01n, (n 80), частота которой меньше частоты столкновений электронов с атомами и циклотронной частоты вращения электронов в магнитном поле и составляет 38,6 ГГц. Для увеличения значения сигнал/шум с помощью генератора 9 звуковой частоты и модуляционных катушек 3 и 4 создают слабое переменное магнитное поле, частота которого составляет 30-389 Гц. Величина индукции В магнитного поля, создаваемого источником 2, непрерывно изменяется от 0 до 7 Т, при этом сигнал с выхода датчика 12 магнитного поля, размещенного в рабочем объеме источника 2, поступает на вход самописца 11. Точность измерения значения индукции В магнитного поля определяется используемым датчиком магнитного поля и в данном случае составляет 0,02% На вход Y самописца 11 поступает выходной сигнал с СВЧ-детектора 8, пропорциональный изменению проводимости образца 14, предварительно прошедший через цилиндрический резонатор 5, циркулятор 6, прямоугольный волновод 13 и детектируемый СВЧ-детектором 8, усиленный и выпрямленный усилителем-детектором 10. При этом на опорный вход усилителя-детектора поступает сигнал звуковой частоты генератора 9. В результате самописец 11 выдает график зависимости производной интенсивности отраженного от образца 14 СВЧ-излучения от индукции В магнитного поля, в данном случае для двух образцов AlGaAs/GaAs (фиг.2). По графикам фиг.2 определяют период осцилляции Δ(B^-1) интенсивности отраженного СВЧ-излучения и по расчетной формуле вычисляют концентрацию свободных носителей заряда в исследуемом образце, в данном случае для первого образца n₁ 8,86 ˙10¹¹см^-2, а для второго образца n₂ 1,01 ˙10¹² см^-2. Точность измерений концентрации носителей заряда в исследуемых образцах составляет ≈0,5%
Предлагаемый способ в отличие от прототипа, во-первых, основан на СВЧ-эффекте Шубникова-де Гааза, который проявляется в результате охлаждения образца до гелиевых температур, во-вторых, является высокочувствительным в результате дополнительного воздействия на образец переменным магнитным полем, изменяющимся со звуковой частотой ω и амплитудой b<< B, направленным параллельно постоянному магнитному полю

и направлению СВЧ-излучения, в-третьих, применяется по частоте столкновений носителей заряда во много раз больше частоты СВЧ-излучения, что приводит к увеличению чувствительности и дает возможность измерять осцилляции интенсивности СВЧ-излучения, отраженного от тонких слоев вырожденных полупроводников, в-четвертых, применим, когда невозможно измерить толщину исследуемого полупроводникового слоя, так как значение концентрации свободных носителей заряда, определяемое этим способом, не зависит от толщины слоя и частоты СВЧ-излучения, а зависит лишь от значения индукции магнитного поля В, в-пятых, применим для определения концентрации свободных носителей заряда в очень тонких слоях вырожденных полупроводников, когда толщина слоя в 10⁵ раз меньше, чем в источнике, принятом за прототип. Это позволило увеличить чувствительность способа по сравнению с прототипом примерно в 10⁴ раз и осуществить возможность измерения в тонких слоях вырожденных полупроводников. Например, этим способом можно определять концентрацию свободных носителей заряда в двумерных слоях толщиной

80

, а также в субмикронных структурах, представляющих мезаструктуры размерами вплоть до величин, сравнимых с длиной волны де Бройля электрона ≈80-500 мкм.From the source 1 of the probe microwave radiation to the sample 14 through a rectangular waveguide 7, the circulator 6 and the cylindrical resonator 5 are sent parallel to the induction vector of a constant magnetic field

A microwave wave of type E _01n , (n 80), whose frequency is less than the frequency of collisions of electrons with atoms and the cyclotron frequency of rotation of electrons in a magnetic field and is 38.6 GHz. To increase the signal-to-noise value using a sound generator 9 and modulation coils 3 and 4, a weak alternating magnetic field is created, the frequency of which is 30-389 Hz. The magnitude of the induction B of the magnetic field generated by the source 2, continuously varies from 0 to 7 T, while the signal from the output of the sensor 12 of the magnetic field located in the working volume of the source 2 is fed to the input of the recorder 11. The accuracy of the measurement of the value of the induction B of the magnetic field is determined by the used a magnetic field sensor and in this case is 0.02%. An input signal from the microwave detector 8 is proportional to the change in the conductivity of sample 14, previously passed through a cylindrical resonator 5, to the input Y of the recorder 11 hector 6, a rectangular waveguide 13 and detected by a microwave detector 8, amplified and rectified by an amplifier-detector 10. At the same time, the sound frequency of the generator 9 is supplied to the reference input of the amplifier-detector. As a result, the recorder 11 gives a graph of the dependence of the derivative of the intensity reflected from the microwave sample 14 -radiation from induction In a magnetic field, in this case, for two samples of AlGaAs / GaAs (figure 2). The graphs of figure 2 determine the oscillation period Δ (B ^-1 ) the intensity of the reflected microwave radiation and the calculation formula calculates the concentration of free charge carriers in the test sample, in this case, for the first sample n ₁ 8.86 × ^{10 11} cm ^-2 , and for the second sample n ₂ 1.01 ˙10 ¹² cm ^-2 . The accuracy of measurements of the concentration of charge carriers in the samples under study is ≈0.5%
The proposed method, in contrast to the prototype, is, firstly, based on the Shubnikov-de Haas microwave effect, which manifests itself as a result of cooling the sample to helium temperatures, and secondly, it is highly sensitive as a result of additional exposure to the sample by an alternating magnetic field, changing with sound frequency ω and amplitude b << B, parallel to the constant magnetic field

and the direction of microwave radiation, thirdly, the frequency of collisions of charge carriers is applied many times higher than the frequency of microwave radiation, which leads to an increase in sensitivity and makes it possible to measure oscillations in the intensity of microwave radiation reflected from thin layers of degenerate semiconductors, fourthly , applicable when it is impossible to measure the thickness of the investigated semiconductor layer, since the concentration of free charge carriers determined by this method does not depend on the thickness of the layer and the frequency of microwave radiation, and Avis only on the value of magnetic field induction B, fifthly, suitable for determining the concentration of free charge carriers in very thin layers degenerate semiconductors, when the thickness of the layer 10 ^{is 5} times smaller than in the source adopted as a prototype. This made it possible to increase the sensitivity of the method compared to the prototype by about 10 ⁴ times and to make it possible to measure degenerate semiconductors in thin layers. For example, this method can determine the concentration of free charge carriers in two-dimensional layers with a thickness

80

as well as in submicron structures representing mesastructures with sizes up to values comparable to the electron de Broglie wavelength ≈80-500 μm.

Claims (1)

METHOD FOR NON-CONTACT DETERMINATION OF THE CONCENTRATION OF FREE CHARGE CARRIERS IN SEMICONDUCTORS, including cooling of the sample while it is exposed to a constant magnetic field, induction vector

which is perpendicular to the surface of the sample, irradiating the sample with microwave radiation of a given frequency in a direction parallel to the induction vector

magnetic field, measuring the intensity of the microwave radiation reflected from the sample depending on the magnitude of the induction B of a constant magnetic field and determining the concentration of charge carriers by calculation, characterized in that the sample is cooled to helium temperatures, additionally exposed to an alternating magnetic field that varies with the sound frequency W, having an induction amplitude much smaller than B, and an induction vector parallel to the vector

a constant magnetic field, and the frequency of microwave radiation is chosen lower than the frequency of collision of carriers with atoms of the semiconductor, and the concentration n is determined by the formula:

where e is the electron charge, C;

Planck's constant, J · s;
M is the number of valleys in the conduction band of the semiconductor;
B _{n + 1} and B _{n the} values of the induction of a constant magnetic field corresponding to two adjacent extrema of the measured dependence, T.

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