CN102636258A - Optical test method for electron spin diffusion transport kinetics based on pure phase liquid crystal spatial light modulator - Google Patents

Abstract

The invention relates to an optical test method for electron spin diffusion transport kinetics based on a pure phase liquid crystal spatial light modulator, wherein a one-dimensional pure phase liquid crystal spatial light modulator is combined with a time resolution pumping-detecting device, and the principle of an experimental device is indicated in the abstract figures. The one-dimensional pure phase liquid crystal spatial light modulator 7 is arranged in an overlapped area of a pumping light 2 and a detecting light 3, and is used for modulating circular polarization degrees of the cross sections of both the pumping light 2 and the detecting light 3 into the one-dimensional space period distribution. A modulation light field formed like this is imaged onto a sample 9 through an imaging lens 8. The modulated pumping light stimulates an electron spin polarization transient grating in the sample, and the modulated detecting light measures the decay kinetics of the electron spin polarization transient grating. An analytic model for the electron spin diffusion transport kinetics of the circular polarization one-dimensional sinusoidal and square wave type space period modulation light fields is provided. The method disclosed by the invention has the advantages of simple experimental device and operation, capability of testing the spin diffusion transport kinetics with the absorption saturation effect, and high sensitivity.

Description

A kind of electron spin diffusion based on pure phase type LCD space light modulator transports the dynamics optical test method

Technical field

The present invention relates to a kind of electron spin diffusion and transport the dynamics optical test method.Be characterized in utilizing the polarization state of direct modulated pumping of pure phase type LCD space light modulator (POLCSLM) and detecting light beam to distribute, produce the modulation light field that the circular polarization one dimension cycle changes, excite its decay dynamics of transient state electron spin grating and detection respectively.Transport the kinetic measurement field in the semiconductor electronic self-rotation diffusion and have significant application value.

Background technology

Ultra-large semiconductor microactuator electronics integrated device has suffered from the difficult restriction with quantum size effect of heat radiation at present, so present CPU makes can not be towards more high integration development, and can only develop towards many cpu bounds.How breaking through such restriction, develop ultra-large nano-electron integrated device, is present international research focus.The semiconductor spintronics just for this reason target be born, its target is to develop low-power consumption, high-speed nano electron device, solves the difficulty that present ultra-large microelectronics integrated device runs into, and realizes the more ultra-large integrated manufacturing of high integration.Based on the electron spin coherent state, even possibly realize semiconductor quantum logical and calculating device.Yet, realizing the semiconductor spin electric device, one of the basic problem that need research and solve is the related basic problem of electron spin transport process in the semiconductor.To this problem; Theoretical research at present is more, and experimental study is quite few, but the experimental studies results of minority has been found important new phenomenon; As the coulomb that spins pulls effect, Einstein relation and is false etc., demonstrates to comprise abundant physics new phenomenon in the spin transport dynamics.The main cause that experimental study is few is not have at present simple highly sensitive experiment test technology.Unique experiment test technology of report is a transient state spin optical grating diffraction mensuration at present.The experiment test system of this method is very complicated, needs four bundle laser, and is arranged in special geometry.Wherein two bundles are as pump light, and their polarized orthogonal, superposition synthesize the light field that produces a circular polarization one dimension cycle variation, and this light field vitalizing semiconductor sample then can produce the electron-spin polarization grating, i.e. transient state spin grating.And in addition in the two-beam, the detection light of a branch of decay dynamics diffractometry as transient state spin grating, the reference light that another Shu Zuowei diffracted signal optical heterodyne amplifies.So not only device is complicated, operation easier is also very big.On the other hand, diffracted signal also quite a little less than, detection sensitivity is not high yet, even used the optical heterodyne amplifying technique.In addition, transient state spin grating is excited by the synthetic light field of two-beam in this technology, and the phase differential fluctuation that receives easily between the two-beam is disturbed, and causes noise.These problems have hindered extensively carrying out of spin transport dynamic experiment research.Thereby, development experimental provision and simple to operate, the high spin transport of detection sensitivity test new technology is very important, can promote the research of spin transport dynamic experiment extensively, carry out in a deep going way.The present invention will develop such new method just.

Summary of the invention

The present invention has developed a kind of simple electron spin diffusion and has transported the dynamics optical test method.Its pure phase type LCD space light modulator of combination (commercially produced product) and traditional pumping-detection experimental provision.In other words, only need in traditional pumping-detection experimental provision, increase a pure phase type LCD space light modulator, can realize the measurement that electron spin transports.The experimental provision principle is as shown in Figure 1.7 is pure phase type LCD space light modulator among the figure.Obviously, this device is extremely simple, has only a branch of pump light 2 and a branch of detection light 3.What the more important thing is that the present invention measures is the saturated variation of absorption of transient state spin grating.As shown in Figure 1, detector 10 is directly measured the optical power change of surveying after light 3 sees through sample 9, rather than resembles the diffracted signal intensity of surveying after light passes through sample of measuring of diffractometry method of present report.The experimental study report has been arranged, and the detection optical transmission variable power that same transient grating causes is its more than 200 times of detection diffraction of light power that cause.So, use the absorption saturation effect to measure and can improve measurement sensitivity more than 200 times.Therefore, the present invention has not only simplified the experimental provision structure and has reduced the experimental implementation difficulty, and has greatly improved detection sensitivity.

Optical principle of the present invention is as shown in Figure 1.Its key feature is to introduce spatial light modulator 7.It is an One-Dimensional Pure phase type liquid crystal array or claims spatial light modulator.Each unit or pixel all are controllable phase retardation plates (birefringent plate) in the one-dimensional array, and the phase-delay quantity of different pixels can independently be provided with.If phase-delay quantity between 0-2 π, the changing with its positional cycle property of each pixel in the one-dimensional array liquid crystal is set; Then can realize the one dimension cycle change modulates of polarization state on its transmitted light xsect, be equivalent in the experimental technique of present report with the synthetic light field of two bundle cross polarization pump light superpositions.

Time resolution pumping-detection device 1 output line polarized pump light pulse 2 and detecting optical pulses 3 shown in Figure 1, traditional, through behind 1/2 wave plate 4 and 5, scioptics 6 focus on the pure phase type LCD space light modulator 7 more respectively; If rotate the position angle of 1/2 wave plate 4 and 5; The polarization direction that makes pump light 2 and survey light 3 all with 7 soon or slow axis become miter angle; And the phase-delay quantity that each pixel of 7 is set is with the changing between 0-2 π of its positional cycle property, then sees through 7 pump light 2 and surveys light 3 and become the light field with the circular polarization one-dimensional space periodic modulation of rotation direction.Otherwise if the position angle 90 of rotating wave plate 5 is spent again, then 2 and 3 become orhtogonal linear polarizaiton, and they see through the light field of the circular polarization one-dimensional space periodic modulation of 7 back changeabout rotation directions.Imaging lens 8 with 7 imaging (can dwindle or amplify) to sample 9; The result; The pump light field excitation sample 9 of circular polarization one-dimensional space periodic modulation; In sample, produce the variation of electron-spin polarization degree one-dimensional space cycle, but the uniform electron population in total electron concentration space, promptly so-called transient state spin grating (TSG).And the circular polarization one dimension of identical or opposite rotation direction is surveyed saturated absorption variation of transient state circular dichroism or the transmission variation that light 3 is measured TSG with the cycle spatial modulation, can reflect that then electron spin transports dynamics.What be worth to stress is to survey light 3 just to have and pump light 2 synperiodic circular polarization spatial modulation, can measure electron spin and transport dynamics.Otherwise,, then insensitive to spin transport dynamics if survey the circularly polarized light of light 3 for the non-space modulation.To embodiment of the present invention be described with two instances below.

Description of drawings

Fig. 1 electron spin diffusion transports dynamic experiment proving installation schematic diagram

Among Fig. 1,1 is traditional time resolution pumping-detection device, comprises femto-second laser, non-colinear both arms interferometer and optical delay line; 2 and 3 are respectively the light beam from two arms of both arms interferometer in 1, and 2 are better than 3, pretend to be pump light, and 3 a little less than, as surveying light; 4 and 5 is 1/2 wave plate, controls 2 and 3 polarization direction respectively; 6 is condenser lens, focuses on same point on 7 with 2 and 3; 7 is One-Dimensional Pure phase type LCD space light modulator (1D-POLCSLM), and modulation is through its light beam 2 and the space distribution of the polarization state on 3 the xsect; 8 is imaging lens, and imaging 7 is to sample 9, and magnification can be provided with as required; 10 is photodetector, measure to survey light 3 and sees through in the sample 9 the transmission power variable quantity of being responded to by behind the 2 transient state spin gratings that excite.

Embodiment

According to above-mentioned principle, the present invention practical implementation two instances, produce circular polarization one dimension sinusoidal pattern and square wave type space periodic modulation light field respectively, and be used for electron spin and spread and transport kinetic measurement, obtain the electron spin coefficient of diffusion.

Instance one circular polarization one dimension sinusoidal pattern space periodic modulation light field produces with the electron spin diffusion and transports kinetic measurement

If the xy coordinate is positioned on the 1D-POLCSLM plane, the y axle is parallel to the long limit of pixel, and the x axle is described the one dimension position distribution of pixel.The phase-delay quantity that each pixel is set through controller satisfies following relation:

$\mathrm{\φ}\left(i\right)=\frac{2\mathrm{\π}}{\mathrm{\Λ}}\mathrm{RMod}\left(\frac{i}{\mathrm{\Λ}}\right)---\left(1\right)$

Function R Mod () expression two numbers are divided by in the formula, round remainder, and the generation cycle is Λ, and amplitude is the sawtooth wave of 2 π.Introducing this function is to consider that the phase-delay quantity of liquid crystal is limited, so, only use the phase-delay quantity of 0-2 π scope in this instance; I=0, l ..., M-1 is the pixel location numbering of 1D-POLCSLM; M is the number of total picture element of 1D-POLCSLM array; Λ is an integer, expression modulation period is unit with the width Delta x of pixel, and promptly actual space periodic is Λ * Δ x.

So, see through 1D-POLCSLM after, the one-dimensional space periodic modulation of light beam 2 and 3 circular polarization is:

P(i)＝sin(φ(i)) (2)

When A＞＞1, promptly comprise many modulation pixels a modulation period, be equivalent to close sampling.At this moment, the discretize effect of phase modulation can be ignored, and is approximately the continuous function of x coordinate.Equation (1) and (2) can be approximated to be following x continuous function:

$\mathrm{\φ}\left(x\right)=\frac{2\mathrm{\π}}{\mathrm{\Λ}*\mathrm{\Δx}}\mathrm{RMod}\left(\frac{x}{\mathrm{\Λ}*\mathrm{\Δx}}\right)---\left(3\right)$

$P\left(x\right)=\sin \left(\mathrm{\φ}\left(x\right)\right)\≈\sin \left(\frac{2\mathrm{\π}}{\mathrm{\Λ}*\mathrm{\Δx}}x\right)---\left(4\right)$

If incident ray polarized light intensity is I ₀, receive 7 the modulation after, the spatial modulation of the left and right rounding polarized component intensity of transmitted light can be expressed as:

$I_{\mathrm{\σ}}^{\±}\left(x\right)=\frac{I_{0,\mathrm{\σ}}}{2}[1\±P\left(x\right)]=\frac{I_{0,\mathrm{\σ}}}{2}[1\±\sin \left(\frac{2\mathrm{\π}}{\mathrm{\Λ}*\mathrm{\Δx}}x\right)]---\left(5\right)$

Left end subscript " ± " expression " right side/left side " rounding polarized light in the formula; σ=2 or 3 are represented pump beam 2 and detected beam 3 respectively." ± " of equation right ends number corresponding relation is for get or following symbol simultaneously.Down together.

The circular polarization sinusoidal pattern space periodic light modulated that equation (5) is described is by in 8 imaging to 9, and then the modulation light intensity on the sample front surface is:

$I_{\mathrm{\σ}}^{\±}\left(x\right)=\frac{I_{0,\mathrm{\σ}}}{2Q^2}[1\±\sin \left(\frac{2\mathrm{\π}}{\mathrm{\Λ}*\mathrm{\Δ}{x}^{\′}}\right)]---\left(6\right)$

Q is the magnification of imaging lens 8 in the formula, Δ x '=Q* Δ x.

Modulated pump light; excited sample; The electron concentration that produces electron-spin polarization degree Sine Modulated distributes, and can be expressed as:

$n_{\±}\left(x\right)=\frac{n_0}{2}[1\±\mathrm{\Δ}n\sin \left(\frac{2\mathrm{\π}}{\mathrm{\Λ}*\mathrm{\Δ}{x}^{\′}}\right)]---\left(7\right)$

Left end subscript " ± " expression electron-spin polarization " up/down " in the formula; n ₀Being light activated total electron concentration, is that the space is uniform; Δ n is the electronic spin polarization, and value is 0～1, and the band structure and the interband that depend on sample excite optical selection rule.

The electronic spin polarization distribution is:

$S\left(x\right)=n_{+}\left(x\right)-n_{-}\left(x\right)=n_0\mathrm{\Δ}n\sin \left(\frac{2\mathrm{\π}}{\mathrm{\Λ}*\mathrm{\Δ}{x}^{\′}}x\right)---\left(8\right)$

Equation (8) is so-called transient state spin grating (TSG).Obviously, the present invention has promptly produced TSG with the single bundle of 1D-POLCSLM modulation pump light, and is simpler than the two-beam pump light superposition field generation TSG of present report.

The transient state spin polarization space distribution that equation (7) and (8) are described is by the moment injection of a pump light pulse excitation, because electronics-hole-recombination, spin relaxation and diffusion transport effect, they will develop in time, and described by following equation:

$n_{\±}(x,t)=\frac{n_0}{2}{e}^{-t/{\mathrm{\τ}}_r}[1\±\mathrm{\Δ}n\sin \left(\frac{2\mathrm{\π}}{\mathrm{\Λ}\′}x\right)e^{-({(2\mathrm{\π}/\mathrm{\Λ}\′)}^2{D}_s+2/{\mathrm{\τ}}_s)t}]---\left(9\right)$

Or

$S(x,t)=n_0\mathrm{\Δ}n\sin \left(\frac{2\mathrm{\π}}{\mathrm{\Λ}\′}x\right)e^{-[{(2\mathrm{\π}/\mathrm{\Λ}\′)}^2{D}_s+2/{\mathrm{\τ}}_s+l/{\mathrm{\τ}}_r]t}---\left(10\right)$

Λ ' in the formula=Λ * Δ x ', D _sBe the electron spin coefficient of diffusion, its reflection spin diffusion transports dynamics just; τ _rAnd τ _sBe respectively electronics-hole-recombination life-span and electron spin relaxation life-span.

How measuring the dynamic process of describing in equation (9) or (10) is another important content of the present invention.As shown in Figure 1; The detection light 3 that has identical circular polarization one-dimensional space periodic modulation with pump light 2;

(shown in equation (6)); See through the TSG in the sample 9, the transmission power variable quantity of detection light 3 of being measured the TSG induction by detector 10 is with the variation of time delay.This is different from the slotted line polarization of reporting in the world at present fully and surveys light 3 through the variation of the diffracted signal power behind the TSG with time delay.The detection sensitivity of detection method of the present invention greatly improves.

The transmission power change kinetics that the detection light that detector 10 measures sees through behind the TSG can be expressed as:

$\mathrm{\ΔP}\left(t\right)=A{e}^{-t/{\mathrm{\τ}}_r}\{1+B{e}^{-[{(2\mathrm{\π}/\mathrm{\Λ}\′)}^2{D}_s+2/{\mathrm{\τ}}_s]t}\}---\left(11\right)$

A and B are fitting parameter in the formula, 0＜B＜1, the band structure and the optical excitation interband transition selection rule of dependence sample.

Because the electronics of sample-hole-recombination life-span τ _rWith electron spin relaxation life-span τ _sCan independent measurement, be known, the method for testing report of existing multiple maturation.So,, just can obtain description electron spin diffusion and transport dynamic (dynamical) electron spin diffusion coefficient D with the dynamics data that this instance of equation (11) optimization match modulator approach experiment test arrives _s

Instance two circular polarization one dimension square wave type space periodics modulation light field produces with the electron spin diffusion and transports kinetic measurement

When the one-dimensional space of circular polarization of light degree little modulation period, a few pixel only, like Λ＜10 o'clock, the continuous approximation modulation that equation (3) and (4) are described just is false.Instance one described spin transport kinetics of diffusion is no longer set up.Thereby for minor cycle modulation, instance of the present invention develops a kind of circular polarization one dimension square wave type space periodic modulation light field and produces, and is used for the electron spin diffusion and transports kinetic measurement.

The periodic phase that spatial light modulator 7 is set is following:

$\mathrm{\φ}\left(x\right)=\underset{q=-M/2}{\overset{M/2}{\mathrm{\Σ}}}\frac{\mathrm{\π}}{2}\mathrm{rect}\left(\frac{x+\mathrm{\Λ\Δx}/2-2\mathrm{q\Λ\Δx}}{\mathrm{\Λ\Δx}}\right)+\frac{3\mathrm{\π}}{2}\mathrm{rect}\left(\frac{x-\mathrm{\Λ\Δx}/2-2\mathrm{q\Λ\Δx}}{\mathrm{\Λ\Δx}}\right)---\left(12\right)$

Function definition in the formula:

all the other symbolic significances are with instance one.

Obviously, it is 2 Λ Δ x or 2 a Λ pixel that equation (12) is described one-period, and the half period amplitude is a pi/2, and the half period amplitude is the square wave type periodic function of 3 pi/2s in addition.Pump light 2 with survey light 3 through after 7, will become circular polarization for+1 with-1 square wave type periodic modulation light field.

If incident ray polarized light is by force I ₀, then modulating light intensity through 7 circular polarization one dimension square wave type space periodic can be expressed as:

$I_{\mathrm{\σ}}^{\±}\left(x\right)=I_{0,\mathrm{\σ}}\underset{q=-M/2}{\overset{M/2}{\mathrm{\Σ}}}\mathrm{rect}\left(\frac{x\±\mathrm{\Λ\Δx}/2-2\mathrm{q\Λ\Δx}}{\mathrm{\Λ\Δx}}\right)---\left(13\right)$

With in 7 imaging to 9, then the modulation light intensity on the sample front surface is through imaging lens 8:

$I_{\mathrm{\σ}}^{\±}\left(x\right)=\frac{I_{0,\mathrm{\σ}}}{Q^2}\underset{q=-M/2}{\overset{M/2}{\mathrm{\Σ}}}\mathrm{rect}\left(\frac{x\±\mathrm{\Λ\Δx}\′/2-2\mathrm{q\Λ\Δx}\′}{\mathrm{\Λ\Δx}\′}\right)---\left(14\right)$

The square wave type periodic modulation pumping light field that equation (14) is described;

excited sample 9 produces the electron concentration distribution that the electron-spin polarization cycle changes:

$n_{\±}\left(x\right)=\frac{n_0}{2}\underset{q=M/2}{\overset{M/2}{\mathrm{\Σ}}}[1\±\mathrm{\Δn}*\mathrm{rect}\left(\frac{x+\mathrm{\Λ\Δ}{x}^{\′}/2-2\mathrm{q\Λ\Δ}{x}^{\′}}{\mathrm{\Λ\Δ}{x}^{\′}}\right)\stackrel{-}{+}\mathrm{\Δn}*\mathrm{rect}\left(\frac{x-\mathrm{\Λ\Δ}{x}^{\′}/2-2\mathrm{q\Λ\Δ}{x}^{\′}}{\mathrm{\Λ\Δ}{x}^{\′}}\right)]---\left(15\right)$

The corresponding relation that left end subscript " ± " number and right-hand member are far calculated symbol " ± " and

in the formula is for get or symbol down simultaneously.

The electron-spin polarization period profile that equation (15) is described excites by single pumping pulse is instantaneous, i.e. TSG.It can be owing to electronics-hole-recombination, and spin relaxation and diffusion transport and develop in time.Temporal evolution is separated and can be expressed as by the control of spin diffusion equation:

$n_{\±}(x,t)=\frac{n_0}{2}{e}^{-t/{\mathrm{\τ}}_r}\left\{1\stackrel{-}{+}\frac{4\mathrm{\Δn}}{\mathrm{\π}}\underset{m=0}{\overset{\∞}{\mathrm{\Σ}}}\frac{1}{2m+1}\sin \left(\frac{(2m+1)2\mathrm{\π}}{{\mathrm{\Λ}}^{\′}}x\right)e^{-[{((2m+1)2\mathrm{\π}/{\mathrm{\Λ}}^{\′})}^2{D}_s+2/{\mathrm{\τ}}_s]t}\right\}---\left(16\right)$

Λ ' in the formula=2 Λ * Δ x ' is the modulation period of light field.

Light is surveyed in the same cycle square wave type circular polarization modulation that equation (14) is described; After saw through TSG, the temporal evolution that the transmission power that TSG responded to that is measured by detector 10 changes was:

$\mathrm{\ΔP}\left(t\right)=A{e}^{-t/{\mathrm{\τ}}_r}\{1+B\underset{m=0}{\overset{\∞}{\mathrm{\Σ}}}\frac{1}{{(2m+1)}^2}{e}^{-[{((2m+1)2\mathrm{\π}/\mathrm{\Λ}\′)}^2{D}_s+2/{\mathrm{\τ}}_s]t}\}---\left(17\right)$

A and B are fitting parameter in the formula, 0＜B＜1, the band structure and the optical excitation interband transition selection rule of dependence sample.

Because the electronics of sample-hole-recombination life-span τ _rWith electron spin relaxation life-span τ _sCan independent measurement, be known, the method for testing report of existing multiple maturation.So, with this instance of equation (17) optimization match modulator approach experiment test to spin diffusion transport dynamics data, just can obtain and describe electron spin and spread and transport dynamic (dynamical) electron spin diffusion coefficient D _s

Amplitude that it should be noted that higher order term in equation (17) formula is by 1/ (2m+1) ²Decay, so, decay very soon.When using equation (17) optimization match experimental dynamics data, only need get that first few items just can obtain enough precision in the sum term, need not calculate infinite item and.

Claims (2)

1. the electron spin diffusion based on pure phase type LCD space light modulator transports the dynamics optical test method; Combination One-Dimensional Pure phase type LCD space light modulator and time resolution pumping-detection device is characterized in that using pure phase type LCD space light modulator while modulated pumping to become identical one-dimensional space period profile with the circular polarization of surveying hot spot; The pump light excited sample of being modulated; Produce the electron-spin polarization transient grating, and the detection light transmission spin polarization transient grating of being modulated is measured the spin diffusion and is transported dynamics; Its transmission power variable quantity has reflected then that with changing time delay the electron spin diffusion transports dynamics; Pixel unit yardstick when the modulation period of circular polarization much larger than One-Dimensional Pure phase type LCD space light modulator; And when the periodic phase of One-Dimensional Pure phase type LCD space light modulator being set, surveying in the light transmission sample transmission power variable quantity behind the transient state spin grating and describe by equation (2) with changing time delay by equation (1);

$\mathrm{\φ}\left(i\right)=\frac{2\mathrm{\π}}{\mathrm{\Λ}*\mathrm{\Δx}}\mathrm{RMod}\left(\frac{i}{\mathrm{\Λ}*\mathrm{\Δx}}\right)---\left(1\right)$

Function R Mod () expression two numbers are divided by in the formula, round remainder; Λ is the number of picture elements of modulation period; Δ x is the yardstick of pixel unit;

$\mathrm{\ΔP}\left(t\right)=A{e}^{-t/{\mathrm{\τ}}_r}\{1+B{e}^{-[{(2\mathrm{\π}/\mathrm{\Λ}\′)}^2{D}_s+2/{\mathrm{\τ}}_s]t}\}---\left(2\right)$

Λ ' is the cycle of transient state spin grating, τ in the formula _rAnd τ _sBe respectively electronics-hole-recombination life-span and electron spin relaxation life-span, available other known method independent measurement is known; A, B and D _sBe fitting parameter, the spin diffusion that obtains with equation (2) optimization match experiment measuring transports the dynamics data acquisition, and D _sThe electron spin coefficient of diffusion that will measure just.

2. a kind of electron spin diffusion based on pure phase type LCD space light modulator described in the claim 1 transports the dynamics optical test method; Be merely several pixel units when the modulation period of circular polarization; When by equation (3) periodic phase of One-Dimensional Pure phase type LCD space light modulator being set, the transmission power variable quantity in the detection light transmission sample behind the transient state spin grating is described by equation (4) with changing time delay;

$\mathrm{\φ}\left(x\right)=\underset{q=-M/2}{\overset{M/2}{\mathrm{\Σ}}}\frac{\mathrm{\π}}{2}\mathrm{rect}\left(\frac{x+\mathrm{\Λ\Δx}/2-2\mathrm{q\Λ\Δx}}{\mathrm{\Λ\Δx}}\right)+\frac{3\mathrm{\π}}{2}\mathrm{rect}\left(\frac{x-\mathrm{\Λ\Δx}/2-2\mathrm{q\Λ\Δx}}{\mathrm{\Λ\Δx}}\right)---\left(3\right)$

Function definition in the formula, ; All the other symbolic significances are with (1) formula;

$\mathrm{\ΔP}\left(t\right)=A{e}^{-t/{\mathrm{\τ}}_r}\{1+B\underset{m=0}{\overset{\∞}{\mathrm{\Σ}}}\frac{1}{{(2m+1)}^2}{e}^{-[{((2m+1)2\mathrm{\π}/{\mathrm{\Λ}}^{\′})}^2{D}_s+2/{\mathrm{\τ}}_s]t}\}---\left(4\right)$

M is an integer in the formula, as the summation variable; All the other symbolic significances are with (2) formula.The spin diffusion that obtains with equation (4) optimization match experiment measuring transports dynamics data, but electron gain spin diffusion coefficient D then _s