JP2008002960A - Method for measuring solid carrier mobility - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To provide a method capable of measuring a solid carrier mobility in the dynamic state accurately in a short time. <P>SOLUTION: The method is characterized as follows: an actual time vibration structure by plasmon in a reflectivity change or in a transmissivity change is measured by using a pump-probe spectroscopy, relative to a solid which is a measuring object wherein phonon is not bonded to the plasmon; fitting of time domain data is performed by using expression (1) to the measured data, to thereby determine a relaxation time τ of the plasmon; and the value of the determined relaxation time τ is substituted for expression (2), to thereby determine the solid carrier mobility μ. In the expressions, A is the amplitude of vibration by the plasmon, τ is the relaxation time of the plasmon, t is a time, ω<SB>p</SB>is a frequency of the plasmon, ϕ is an initial phase of the vibration by the plasmon, e is a charge of the electron, and m* is an effective mass of the electron. There are two other procedures. <P>COPYRIGHT: (C)2008,JPO&INPIT

Description

  The present invention relates to a method for measuring solid carrier mobility. More specifically, the present invention measures the change in the refractive index of a solid in the real-time domain due to vibrations of a nonequilibrium plasmon or plasmon-like longitudinal wave optical (LO) phonon-plasmon coupled mode caused by laser pulse excitation of the solid, The present invention relates to a method for directly determining the mobility of a solid from its vibration damping time constant, and is particularly applicable as a method for evaluating electrical characteristics of nanostructure devices made of materials such as semiconductors and metals.

  In recent years, the miniaturization of devices has progressed, and nanometer-sized devices are being realized. Here, when the average free path of electrons (average distance traveled until electrons are scattered; about 10 to 50 nm for Si) becomes smaller than the device size, the region where electrons are not scattered by lattice vibration or lattice defects (that is, ballistic conduction region) )become. If this region can be realized, in a device such as a field effect transistor (FET), electrons proceed without being scattered from the source to the drain, and a transistor with a dramatically high switching speed can be realized.

  Currently, most researches on ballistic conduction take a method of reducing the device size depending on microfabrication techniques such as electron beam lithography. On the other hand, there is also a method of increasing the mean free path of electrons instead of decreasing the device size. For example, the effect of phonon scattering can be eliminated by making the temperature extremely low, or a crystal having high purity and few defects can be produced by using a technique such as molecular beam epitaxy (MBE).

  However, in order to use the device at room temperature, it is difficult to select the former extremely low temperature, and it is almost impossible to make the defect density zero even for the latter crystallinity. For this reason, nano-structuring of device size has been advanced by microfabrication technology.

What is important in the above background is the evaluation of mobility in nanostructures such as semiconductors and metals. In general, the mobility measurement of a semiconductor generally uses the Hall effect (for example, Non-Patent Document 1). This Hall effect is a phenomenon discovered by EH Hall in 1879. When a current flows through a semiconductor or metal, an electromotive force is generated in a direction perpendicular to both when a magnetic field is applied perpendicular to the current. is there. In the Hall effect experiment, the Hall coefficient is obtained from the relationship between the generated electromotive force, the applied magnetic field, and the current density, and the mobility can be derived from the Hall coefficient. However, for nanostructures, it is considered difficult to measure holes with a spatial resolution on the order of nanometers (10 ⁻⁹ m).

  Against this background, it has recently been found that the mobility of a semiconductor can also be obtained by an optical method using Raman scattering using a laser (for example, Non-Patent Documents 2 and 3). By using microscopic Raman spectroscopy (for example, Non-Patent Document 4) that can achieve a spatial resolution of about 1 μm and near-field Raman spectroscopy (for example, Non-Patent Document 5) that can achieve a spatial resolution of 1 μm or less, It has the potential to evaluate very local carrier mobility (one or several nanostructures).

In this method using Raman scattering, first, there are longitudinal wave optical (LO) phonons (hereinafter also referred to as LO phonons) and plasmons (collective vibrations of electrons), which are peculiar in polar semiconductors (such as SiC). The spectrum corresponding to the plasmon-like branch (L ₊ or L ₋ ) of the upper branch (L ₊ ) and the lower branch (L ₋ ) of the combined LO phonon-plasmon coupled mode is A plasmon damping constant γ is obtained by curve fitting.
Here, as shown in FIG. 1, the plasmon-like mode is a bare (not coupled to a phonon) of the upper branch (L ₊ ) and the lower branch (L ₋ ) of the coupled mode. This indicates a state close to a straight line E indicating the frequency of plasmons (A and B regions in FIG. 1). Regions C and D are “phonon-like” regions where the coupling mode attenuation is governed by phonon relaxation.

  In curve fitting, since the LO phonon-plasmon coupled mode, which generally has an asymmetric spectral waveform, is targeted, the following special functions are used (for example, Non-Patent Documents 2 and 3).

Here, Γ is a phonon attenuation constant, γ is a plasmon attenuation constant, ω _T , ω _L , and ω _P are a TO phonon frequency, a LO phonon frequency, and a plasma frequency, respectively, e is an electron charge, N is an electron density, and m ^*. Is the effective mass of electrons, ε _∞ is the dielectric constant at the high frequency limit, and ε ₀ is the dielectric constant of the vacuum. The mobility is calculated by substituting the plasmon attenuation constant γ thus obtained into the following equation.

This method using Raman is effective for semiconductors with polarity, because the mobility can be obtained in a non-contact optical manner without attaching an electrode to the measurement sample or applying a magnetic field. I think that the.
"Semiconductor Physics" Baifukan, edited by Junichi Nishizawa, Nobuo Mikoshiba, 1995 Journal of Applied Physics, Vol. 78, No.3, pp.1996, 1995 Journal of Applied Physics, Vol. 90, No.10, pp.5211, 2001 Journal of Applied Physics, Vol. 78, No.5, pp.3357, 1995 Journal of Chemical Physics, Vol. 117, No.3, pp.1296, 2002

  However, the above method fits the spectrum obtained by the frequency domain measurement technique called Raman spectrum with a fairly large mathematical group (formulas (8) to (12)), and the plasmon constant, which is originally a constant in the time domain. Since the attenuation constant γ = 1 / τ corresponding to the reciprocal of the relaxation time τ is obtained, a very complicated curve fitting of the spectrum is required, and there is a problem that the data processing takes time. In the above method, the carrier mobility in a static state is measured. However, in a dynamic state, for example, carriers generated by photoexcitation are conducted from the excited surface in the depth direction of the sample. It has been desired to measure the carrier mobility in various states with high sensitivity.

  The present invention has been made in view of the circumstances as described above, and provides a solid carrier mobility measurement method capable of accurately measuring carrier mobility in a dynamic state in a short time. It is an issue.

  The solid carrier mobility measuring method of the present invention is characterized by the following in order to solve the above problems.

  First, using a pump-probe spectroscopy method, the real-time vibration structure due to plasmon in the change in reflectance or transmittance is measured for a solid that is a measurement target in which phonons and plasmons do not bind. The plasmon relaxation time τ is obtained by fitting time domain data using the following equation (1), and the value of the obtained relaxation time τ is substituted into the following equation (2) to obtain the solid carrier mobility. Find μ.

(Where A is the amplitude of plasmon oscillation, τ is the plasmon relaxation time, t is the time, ω _P is the plasmon frequency, φ is the initial phase of the plasmon oscillation, e is the charge of the electron, m ^* is (Effective mass of electrons)

Secondly, real-time vibration of a solid, which is a measurement object in which phonon and plasmon are combined, by a plasmon-like longitudinal wave optical phonon-plasmon coupled mode in reflectance change or transmittance change using pump-probe spectroscopy. By measuring the structure and fitting the time domain data to the measured data using the following equation (3), the relaxation time τ of the plasmon-like longitudinal wave optical phonon-plasmon coupled mode is obtained, and the obtained relaxation is obtained. The solid carrier mobility μ is obtained by substituting the value of time τ into the following equation (2).

(In the above equation, t is the same as in equation (1), A _{L +} is the amplitude of vibration in the plasmon-like mode of the longitudinal wave optical phonon-plasmon coupled mode, and τ (= τ _{L +} ) is the longitudinal wave optical phonon-plasmon coupled mode. Plasmon-like mode relaxation time, Ω _{L +} is the frequency of the plasmon-like mode in the longitudinal wave optical phonon-plasmon coupled mode, φ _{L +} is the initial phase of the plasmon-like mode vibration in the longitudinal wave optical phonon-plasmon coupled mode, and B is the longitudinal phase Amplitude of vibration by wave optical phonon, τ _LO is relaxation time of longitudinal optical phonon, Ω _LO is frequency of longitudinal optical phonon, φ _LO is initial phase of longitudinal optical phonon vibration, C is longitudinal optical phonon − The amplitude of vibration due to the phonon-like mode of the plasmon coupled mode, τ _L− is the phonon line of the longitudinal optical phonon-plasmon coupled mode. Mode relaxation time, Ω _L− is the frequency of the phonon-like mode of the longitudinal wave optical phonon-plasmon coupled mode, φ _L− is the initial phase of the phonon-like mode of the longitudinal wave optical phonon-plasmon coupled mode, and e is the electron Charge, m ^* is the effective mass of electrons)

  Third, real-time vibrations in a plasmon-like longitudinal-wave optical phonon-plasmon coupled mode in a change in reflectance or transmittance using a pump-probe spectroscopy for a solid that is a measurement target in which phonons and plasmons are combined. The structure is measured, and the measurement data is subjected to wavelet transform using the following equation (4) to obtain the relaxation time τ of the plasmon-like longitudinal wave optical phonon-plasmon coupled mode, and the obtained relaxation time τ Obtain the solid carrier mobility μ by substituting the value into the following formula (2).

(In the above equation, Ψ (x) is a function that represents a wave packet called a mother wavelet, x is a function of time as an input signal, b is a variable that represents a time delay of the wave packet, and a is a time axis of the wave packet. Degree of elasticity, t is time, f (t) is the transformed function)

Fourth, in the third invention, a Gabor mother wavelet obtained by transforming a Gaussian function expressed by the following equation (5) is used as a mother wavelet for wavelet transformation.

(In the above formula, σ is the full width at half maximum of the Gaussian function, x is the input signal and a function of time)
Use.

  Fifth, by using pump-probe spectroscopy for a solid, which is a measurement object in which phonon and plasmon are combined, an actual measurement using a plasmon-like longitudinal wave optical phonon-plasmon coupled mode in reflectance change or transmittance change is performed. The time oscillation structure was measured and the relaxation time τ of the plasmon-like longitudinal wave optical phonon-plasmon coupled mode was determined by performing a short-time Fourier transform on the measured data using the following formula (6). By substituting the value of the relaxation time τ into the following formula (2), the carrier mobility μ of the solid is obtained.

(In the above formula, Ψ (x) is a window function, x is an input signal and a function of time, b is a time delay, ω is a frequency, t is time, and f (t) is a transformed function)

  Sixth, in the fifth invention, a Gaussian function is used as the window function of the short-time Fourier transform, and the short-time Fourier transform is performed by the following equation (7).

(In the above formula, σ is the full width at half maximum of the Gaussian function)

  According to the present invention, the relaxation time is directly obtained by measuring the real-time oscillation waveform of the plasmon or plasmon-like LO phonon-plasmon coupled mode, which is the collective oscillation of electrons or holes, and thereby the carrier mobility of the solid. Therefore, it is possible to provide a solid carrier mobility measuring method capable of accurately measuring carrier mobility in a dynamic state in a short time.

The present invention can be applied particularly as a method for evaluating electrical characteristics of nanostructure devices made of solid materials such as semiconductors and metals. The dynamic state or process in solids, especially nanostructured solids, is a phenomenon that occurs, for example, when developing ultrafast switches with carriers using semiconductor nanostructures, and the electrical characterization of semiconductor nanodevices Therefore, mobility measurement is essential, and it is expected to be very useful for the development of ultra high-speed devices in the future. According to the present invention, carrier mobility can be evaluated under various conditions (temperature, defect density, etc.), and a terahertz (10 ¹² Hz) electronic device using high-speed electron transfer such as HEMT (High Electron Mobility Transistor). It can contribute to the transmission / reception of large-capacity data and energy saving problems such as the development of high-speed data communication and the realization of ultra-power-saving equipment.

  Hereinafter, embodiments of the present invention will be described in detail.

  In the present invention, the plasmon or plasmon-like LO phonon-plasmon coupled mode attenuation constant γ is not obtained in the frequency domain as in the conventional method, but the plasmon or plasmon-like LO phonon-plasmon coupled mode directly in the time domain. It is a completely new idea to determine the relaxation time τ of metal and semiconductors (both polar and non-polar) bulk crystals and thin films, and the mobility in solid nanostructures including metals and semiconductors. is there.

In the present invention, in order to obtain the relaxation time τ of the plasmon or plasmon-like LO phonon-plasmon coupled mode, a general time-resolved measurement method called pump-probe spectroscopy (for example, time-resolved reflectance change measurement method or time Measurement of real-time vibration structure by plasmon or plasmon-like LO phonon-plasmon coupled mode in reflectivity (or transmissivity) change using a method for measuring a change in decomposed transmissivity (see JP 2002-214137 A). By doing. However, there are roughly two types of observation means for obtaining the relaxation time τ of the plasmon or plasmon-like LO phonon-plasmon coupled mode.

  The first is a method of acquiring a signal by naked plasmon by measuring a change in reflectance (or transmittance). In this case, since the plasmon frequency is given by a well-known relational expression (formula (13)), the plasmon frequency increases in proportion to the carrier density of the sample. Accordingly, the higher the carrier density, the higher the plasmon frequency and the higher the time resolution. This technique is effective in a state where plasmons do not couple with other elementary excitations (especially LO phonons). Examples of such materials include metals and nonpolar semiconductors (Si, Ge, etc.) that are nonpolar substances and plasmons do not bind to LO phonons.

The second is a method that positively utilizes the coupling between plasmons and LO phonons in polar substances (GaAs, InP, SiC, etc.). In this case, as in the case of Raman scattering, a real-time signal corresponding to the LO phonon-plasmon coupled mode is acquired. It should be noted that the LO phonon-plasmon coupled mode also has a phonon-like mode (mode decay time is determined by phonon decay time) and plasmon-like mode (mode decay time is determined by plasmon decay time). Therefore, if the decay time is obtained by observing a plasmon-like mode, it can be approximately regarded as the decay time of plasmon (τ = τ _{L +} ). Here, the plasmon-like mode is a bare (phonon and phonon) of the upper branch (L ₊ ) and the lower branch (L ₋ ) of the coupled mode as shown in FIG. This indicates a state close to a straight line E indicating the frequency of plasmons (not coupled). That is, it refers to the area A or B in FIG. Regions C and D are “phonon-like” regions where the coupling mode attenuation is governed by phonon relaxation.

In any case, the decay time of vibration due to plasmons observed in the real-time region must be obtained. In the first case, the data is simply a damped vibration.

The plasmon relaxation time τ can be obtained by fitting with a damped harmonic vibration equation such as Here, A is the amplitude of the plasmon vibration, τ is the plasmon relaxation time, ω _P is the plasmon frequency, φ is the initial phase of the plasmon vibration, and t is the time.

  However, in the second case, the plasmon-like LO phonon-plasmon coupled mode signal is an LO phonon signal observed at the same time (this LO phonon is mainly present in the depletion layer near the surface). In some cases, a simple expression such as the above expression (1) cannot be fitted.

  Therefore, the following three methods are taken as solution means. First, the following equation obtained by adding the above equation (1) for all modes that can be observed:

To fit the time domain vibration waveform. Here, A _{L +} in the first term is based on a plasmon-like mode of the longitudinal wave optical phonon-plasmon coupling mode (for example, in N-type GaAs, N> 1 × 10 ¹⁸ cm ⁻³ corresponds to L ₊ of A in FIG. 1). Amplitude of vibration, τ (= τ _{L +} )
Is the relaxation time of the L ₊ mode, Ω _{L +} is the frequency of the L ₊ mode, φ _{L +} is the initial phase of the vibration of the L ₊ mode, B in the second term is the amplitude of the vibration due to the LO phonon, and τ _LO is the relaxation of the LO phonon. Time, Ω _LO is the LO phonon frequency, and φ _LO is the initial phase of the LO phonon vibration. C in the third term is the amplitude of vibration due to the phonon-like mode of the LO phonon-plasmon coupled mode (for example, N> 1 × 10 ¹⁸ cm ⁻³ in n-type GaAs corresponds to L ₋ in FIG. 1), τ _L− is the relaxation time of the L ₋ mode, Ω _L− is the frequency of the L ₋ mode, and φ _L− is the initial phase of the vibration of the L ₋ mode.

If the vibration waveform cannot be fit well with the above equation (3), then a unique method called wavelet transform is used ("Wavelet Beginners Guide" Susumu Sugawara (Tokyo Denki University Press, 1995) and JP (See 2003-296301). This is a method that has been established mathematically, but is rarely used in the fields of physics and measurement. Specifically, for a given function f (t),

Defined by Here, Ψ (x) is called a mother wavelet (x is a function of time with an input signal), and is a function representing a wave packet. b is a variable representing the time delay of the wave packet, and a is a variable representing the degree of stretch of the wave packet with respect to the time axis, that is, the time width of the envelope of the wave packet. The signal intensity after conversion becomes a two-dimensional contour map in the [b, 1 / a] plane, that is, the [time, frequency] plane. By performing such conversion, signal components buried at different frequencies on the same time axis can be extracted separately. Examples of the function type of the mother wavelet include a Gaussian function. Thereby, it is possible to obtain a time change of a signal in a plasmon-like LO phonon-plasmon coupled mode having a frequency different from that of the LO phonon. The relaxation time τ of the plasmon-like mode obtained in this way is obtained by finding the peak frequency of the plasmon-like mode on the wavelet signal and using the exponential function e ^{−t / τ} to determine the relaxation time of this frequency component in the time delay direction. Fit and ask.

  On the other hand, without using the wavelet transform, using a short-time Fourier transform using a window function Ψ (x) (see “Wavelet Beginners Guide” Susumu Sugawara (Tokyo Denki University Press, 1995) etc.) It is also possible to acquire the relaxation time τ of the plasmon-like mode. In this case,

Is applied to the function f (t) to be converted to obtain short-time Fourier spectra for various time delays b. Since this spectrum attenuates with the delay time, the relaxation time τ of the plasmon-like mode is obtained from the attenuation. In the above formula, ω is a frequency.

  The relaxation time τ obtained as described above is not the above formula (14), but the following general formula (2) (“Semiconductor Physics” Baifukan, edited by Junichi Nishizawa, Nobuo Mikoshiba, 1995) reference)

The mobility μ can be directly obtained by substituting into.

  Thus, in frequency domain spectroscopy, the derivation of mobility requiring a very complex data fit, according to the present invention, is the relaxation time τ of a direct plasmon or plasmon-like LO phonon-plasmon coupled mode in the time domain. Can be done very simply.

Next, the present invention will be described in more detail with reference to examples.
<Example>
FIG. 2 shows the principle of the solid carrier mobility measuring method according to the present invention. Depending on the type of plasmon signal of the sample measured in the time-resolved reflectance (transmittance) measuring device A by pump-probe spectroscopy using a femtosecond pulse laser as a light source, three types of data processing (B, C, D) is conceivable. The measuring apparatus A includes a reflection type and a transmission type.

First, FIG. 2B shows a case where the vibration waveform in the time domain can be easily fitted by the above equation (1). Since simple damped harmonic vibration can be well reproduced by the above equation (1), the relaxation time τ of plasmon vibration can be obtained by this fitting. The obtained relaxation time τ is immediately substituted into the above equation (2), and the mobility μ is obtained.

Second, as shown in FIG. 2C, the signal of time-resolved reflectance (transmittance) is not a simple damped harmonic oscillation, but a plurality of damped harmonics such as a plasmon-like LO phonon-plasmon coupled mode and LO phonon. This is a case where the vibration can be well reproduced by the sum of vibrations (formula (3) above). In this case, the relaxation time τ of the obtained plasmon-like LO phonon-plasmon coupled mode (L ₊ ) can be obtained by this fitting. The obtained relaxation time τ is immediately substituted into the above equation (2), and the mobility μ is obtained.

  The third is a case where the vibration structure is very complicated as shown in FIG. 2D, and the data cannot be fitted by the above formula (1) or the above formula (3). In this case, wavelet transform (the above formula (4)) or short-time Fourier transform (the above formula (6)) is used.

FIG. 3 shows a very complicated vibration structure in which data cannot be fitted by either the above formula (1) or the above formula (3), and a plasmon-like LO phonon-plasmon using the wavelet transform (the above formula (4)). It is a figure which shows the analysis result of the Example which calculated | required relaxation time of coupling mode (L _<+> ). FIG. 3A is an enlarged view of D in FIG. 2 and shows a vibration waveform in the time domain. This time waveform is wavelet transformed, but here Gabor's mother wavelet with a modified Gaussian function as the mother wavelet:

Provides a much smoother spectrum in the [time, frequency] plane than with other mother wavelets, such as Meyer's mother wavelet, and therefore more clearly LO phonon-plasmon coupled modes and LO phonons. Can be distinguished. In Equation (5), σ is the full width at half maximum (FWHM) of the Gaussian function. A time-frequency plot of the signal converted using the Gabor mother wavelet is shown in FIG.
B). In the wavelet transform spectrum of FIG. 3B, the plasmon-like LO phonon-plasmon coupled mode (L ₊ ) having a peak near about 26 THz and the LO phonon having a peak near about 7 THz are very much in the time-frequency domain. It can be seen that they are well separated. Such separation is very difficult in the real-time region as shown in FIG. It can be seen from FIG. 3B that the relaxation time τ of the plasmon-like LO phonon-plasmon coupled mode (L ₊ ) is approximately 0.2 ps (picoseconds). In order to obtain τ more accurately, it is only necessary to extract the time change of the intensity at the peak frequency position in the plasmon-like LO phonon-plasmon coupled mode (L ₊ ) in FIG. The result is shown in FIG. FIG. 3C is a diagram in which the peak intensity of the plasmon-like mode (L ₊ ) is plotted against the delay time (b) on the wavelet transform spectrum.

From this, it is immediately understood that the relaxation time τ of the plasmon-like LO phonon-plasmon coupled mode (L ₊ ) is about 0.2 ps (picosecond). Therefore, the mobility μ is determined to be about 5250 cm ² / Vs from the above formula (2). However, here, an electron charge e = 1.60219 × 10 ⁻¹⁹ C and an effective mass of electron m ^* = 0.067 m (m = 9.110956 × 10 ⁻³¹ kg) were used.

FIG. 4 shows a very complicated vibration structure, and when the data cannot be fitted by the above formula (4) or the above formula (6) (in the case of D in FIG. 2 or FIG. 3A), short-time Fourier transform ( It is a figure which shows the analysis result of the Example which calculated | required the relaxation time of plasmon-like LO phonon-plasmon coupling mode (L _<+> ) using the said Formula (6).

  The window function in the short-time Fourier transform (the above formula (6)) was a Gaussian function. This is because a much smoother short-time spectrum can be obtained than when using other window functions such as a rectangular function, and therefore, the longitudinal wave optical (LO) phonon-plasmon coupling mode and LO phonon can be more clearly distinguished. is there. That is, when the window function is made a Gaussian function by the short-time Fourier transform (the above formula (6)),

It becomes. Where σ is the full width at half maximum of the Gaussian function (Full width at half maximum = FWHM). FIG. 4A shows a spectrum every 0.036 ps obtained by performing a short-time Fourier transform on the time-domain vibration waveform data of D of FIG. It can be seen that the plasmon-like LO phonon-plasmon coupled mode (L ₊ ) having a peak in the vicinity of about 26 THz and the LO phonon having a peak in the vicinity of about 7 THz are very well separated and observed in the frequency domain. Such separation is very difficult in the real-time region as shown in FIG. In FIG. 4B, the time change of the intensity at the peak frequency position in the plasmon-like LO phonon-plasmon coupled mode (L ₊ ) is plotted with respect to the delay time b. From this, it is immediately understood that the relaxation time τ of the plasmon-like LO phonon-plasmon coupled mode (L ₊ ) is about 0.2 ps (picosecond). Therefore, the mobility μ is determined to be about 5250 cm ² / Vs from the above formula (2). However, here, an electron charge e = 1.60219 × 10 ⁻¹⁹ C and an effective mass of electron m ^* = 0.067 m (m = 9.110956 × 10 ⁻³¹ kg) were used.

Finally, FIG. 5 shows the measurement results of the example in the plasmon-like LO phonon-plasmon coupling mode (L ₊ ) measured with an actual sample n-type GaAs single crystal. A short-time Fourier transform (the above formula (5)) was used for the analysis of the time domain vibration. FIG. 5A shows a signal actually obtained by the time-resolved reflectance change measurement. FIG. 5B is a diagram showing a spectrum obtained by subjecting the time domain vibration waveform of FIG. 5A to a short-time Fourier transform, and the interval of the delay time b for each spectrum is variable (0.05 to 1.0 ps). FIG. 5C is a diagram in which the peak intensity of the plasmon-like mode (L ₊ ) is plotted against the delay time (b) on the spectrum diagram 5B subjected to the short-time Fourier transform. From the intensity time dependence of the plasmon-like LO phonon-plasmon coupled mode (L ₊ ) in FIG. 5C, it can be seen that the relaxation time τ of the L ₊ mode is approximately 0.2 ps (picoseconds). Therefore, the mobility μ is determined to be about 5250 cm ² / Vs from the above formula (2). However, here, an electron charge e = 1.60219 × 10 ⁻¹⁹ C and an effective mass of electron m ^* = 0.067 m (m = 9.110956 × 10 ⁻³¹ kg) were used. The obtained value of the mobility μ is a value substantially close to the experimental value μ = 8800 cm ² / Vs obtained by the Hall measurement at room temperature.

Or 2-5 results of Example described the plasmon-like LO phonon - not for limiting the L ₊ as plasmon coupling mode, the plasmon-like LO phonon - as plasmon coupling mode L _- mode observed in the Gives similar results. Thus, according to the present invention, it has been shown that the semiconductor mobility μ, which has been obtained by fitting the Raman spectrum obtained in the frequency domain so far with a complicated mathematical formula, can be easily obtained in the real time domain. It was done.

Further, the solid sample to be measured of the present invention is not limited to the semiconductor single crystal as in the embodiment where the measurement results of FIG. 5 are obtained, but is a metal single crystal, a thin film, a semiconductor superlattice, or a semiconductor quantum well structure. All solids having plasmons such as metal / semiconductor nanocrystals (such as quantum dots) and nanowires can be used, and the present invention is effective as a method for evaluating these electrical characteristics (mobility).

LO phonon - shows the carrier density dependence of the frequency of _- branching on the plasmon coupling mode (L ₊₎ and branched lower (L). It is explanatory drawing of the principle of the solid carrier mobility measuring method by this invention. It is a figure which shows the analysis result of the Example which calculated | required the relaxation time of the plasmon-like LO phonon-plasmon coupling mode (L _<+> ) using the wavelet transformation among the methods of this invention, (A) is D of FIG. (B) is a diagram showing a wavelet transform spectrum, (C) is a wavelet transform spectrum, and the peak intensity of plasmon-like mode (L ₊ ) is expressed as a delay time (L ₊ ). It is the figure plotted with respect to b). It is a figure which shows the analysis result of the Example which calculated | required the relaxation time of the plasmon-like LO phonon-plasmon coupling mode (L _<+> ) using short-time Fourier transform among the methods of this invention, (A) is FIG. The figure which shows the spectrum which carried out the short-time Fourier transform of the time-domain vibration waveform of D of (D), (B) is the spectrum which carried out the short-time Fourier transform, and the peak intensity of plasmon-like mode (L _<+> ) is shown as delay time ( It is the figure plotted with respect to b). The figure which shows the measurement result of the Example which calculated | required the relaxation time of plasmon-like LO phonon-plasmon coupling mode (L _<+> ) experimentally in n-type GaAs using short-time Fourier transform among the methods of this invention. Yes, (A) is a signal obtained by time-resolved reflectance measurement, (B) is a diagram showing a spectrum obtained by short-time Fourier transform of the time-domain vibration waveform of (A), and (C) is a spectrum obtained by short-time Fourier transform. FIG. 5B is a diagram in which the peak intensity of the plasmon-like mode (L ₊ ) is plotted against the delay time (b).

Claims (6)

Using a pump-probe spectroscopy method, the real-time vibration structure due to plasmons in the change in reflectance or transmittance is measured for a solid that is a measurement target in which phonons and plasmons do not bind. 1) is used to obtain the plasmon relaxation time τ by fitting time domain data, and the value of the obtained relaxation time τ is substituted into the following equation (2) to obtain the solid carrier mobility μ. A solid carrier mobility measuring method characterized by the above.
(Where A is the amplitude of plasmon oscillation, τ is the plasmon relaxation time, t is the time, ω _P is the plasmon frequency, φ is the initial phase of the plasmon oscillation, e is the charge of the electron, m ^* is (Effective mass of electrons) Using pump-probe spectroscopy, we measure the real-time vibration structure of the plasmon-like longitudinal-wave optical phonon-plasmon coupling mode in the reflectance change or transmittance change for the solid that is the object of measurement where phonon and plasmon are combined. Then, the relaxation time τ of the plasmon-like longitudinal wave optical phonon-plasmon coupled mode is obtained by fitting time domain data to the measured data using the following equation (3), and the value of the obtained relaxation time τ The solid carrier mobility μ is obtained by substituting into the following formula (2).
(In the above equation, t is the same as in equation (1), A _{L +} is the amplitude of vibration in the plasmon-like mode of the longitudinal wave optical phonon-plasmon coupled mode, and τ (= τ _{L +} ) is the longitudinal wave optical phonon-plasmon coupled mode. Plasmon-like mode relaxation time, Ω _{L +} is the frequency of plasmon-like mode in longitudinal optical phonon-plasmon coupled mode, φ _{L +} is the initial phase of plasmon-like mode vibration in longitudinal wave optical phonon-plasmon coupled mode, and B is longitudinal Amplitude of vibration by wave optical phonon, τ _LO is relaxation time of longitudinal optical phonon, Ω _LO is frequency of longitudinal optical phonon, φ _LO is initial phase of longitudinal optical phonon vibration, C is longitudinal optical phonon − The amplitude of vibration due to the phonon-like mode of the plasmon coupled mode, τ _L− is the phonon line of the longitudinal optical phonon-plasmon coupled mode. Mode relaxation time, Ω _L− is the frequency of the phonon-like mode of the longitudinal wave optical phonon-plasmon coupled mode, φ _L− is the initial phase of the phonon-like mode of the longitudinal wave optical phonon-plasmon coupled mode, and e is the electron Charge, m ^* is the effective mass of electrons) Using pump-probe spectroscopy, we measure the real-time vibration structure of the plasmon-like longitudinal-wave optical phonon-plasmon coupling mode in the reflectance change or transmittance change for the solid that is the object of measurement where phonon and plasmon are combined. The measurement data is subjected to wavelet transform using the following equation (4) to obtain the relaxation time τ of the plasmon-like longitudinal wave optical phonon-plasmon coupled mode, and the value of the obtained relaxation time τ is expressed by the following equation: A solid carrier mobility measuring method, wherein the solid carrier mobility μ is obtained by substituting in (2).
(In the above equation, Ψ (x) is a function that represents a wave packet called a mother wavelet, x is a function of time as an input signal, b is a variable that represents a time delay of the wave packet, and a is a time axis of the wave packet. Degree of elasticity, t is time, f (t) is the transformed function) 4. The solid carrier mobility measuring method according to claim 3, wherein a Gabor mother wavelet obtained by transforming a Gaussian function represented by the following equation (5) is used as a mother wavelet for wavelet transformation.
(In the above formula, σ is the full width at half maximum of the Gaussian function, x is the input signal and a function of time)
A method for obtaining a carrier mobility μ of a solid, characterized in that Using pump-probe spectroscopy, we measure the real-time vibration structure of the plasmon-like longitudinal-wave optical phonon-plasmon coupling mode in the reflectance change or transmittance change for the solid that is the object of measurement where phonon and plasmon are combined. The measurement data is subjected to short-time Fourier transform using the following equation (6) to obtain the relaxation time τ of the longitudinal optical phonon-plasmon coupled mode, and the value of the obtained relaxation time τ is represented by the following equation ( A solid carrier mobility measuring method, wherein the solid carrier mobility μ is obtained by substituting in 2).
(In the above formula, Ψ (x) is a window function, x is an input signal and a function of time, b is a time delay, ω is a frequency, t is time, and f (t) is a transformed function) 6. The solid carrier mobility measuring method according to claim 5, wherein a Gaussian function is used as a window function of the short-time Fourier transform, and the short-time Fourier transform is performed by the following equation (7).
(In the above formula, σ is the full width at half maximum of the Gaussian function)