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US-12625202-B2 - Method, computer program, and system for determining respective transport properties of majority as well as minority charge carriers in a sample

US12625202B2US 12625202 B2US12625202 B2US 12625202B2US-12625202-B2

Abstract

The invention relates to method for determining respective transport properties of majority as well as minority charge carriers in a sample ( 107 ) comprising the majority and the minority charge carriers that correspond to electrons and holes or vice versa. The method particularly allows to determine the charge carrier density of the majority charge carriers and the charge carrier density of the minority charge carriers. For the method, a plurality of Hall measurement trials is performed on the sample ( 107 ), wherein during each Hall measurement trial, the sample ( 107 ) is exposed to an illumination intensity I, wherein a Hall coefficient and a conductivity are acquired from each Hall measurement trial, wherein in a first Hall measurement trial, the sample ( 107 ) is exposed to a first illumination intensity I 1 , in the range of zero to 0.02 suns, particularly wherein the first illumination intensity is zero, and a first Hall coefficient R H (I 1 ) and a first conductivity σ(I 1 ) are acquired, wherein from the first Hall coefficient and the first conductivity, a carrier mobility μ 1 is determined, wherein in a second measurement trial, the sample ( 107 ) is exposed to a second illumination intensity I 2 and a second Hall coefficient R H (I 2 ) and a second conductivity σ(I 2 ) are acquired, wherein from the second Hall coefficient and the second conductivity, a second carrier mobility μ 2 is determined, wherein the second illumination intensity I 2 is so high that a charge carrier density of electrons and a charge carrier density of holes in the sample ( 107 ) are identical, that the second Hall coefficient asymptotically approaches zero and that a second Hall mobility obtained from the product of the second Hall coefficient and the second conductivity asymptotically approaches a constant value, wherein a third carrier mobility μ 3 is determined from the first and the second carrier mobility, particularly by subtracting the second carrier mobility from the first carrier mobility if the Hall coefficient has the same sign for the first and the second illumination intensity or by adding the second carrier mobility to the first carrier mobility if the Hall coefficient changes its sign for the first and the second illumination intensity, wherein the first carrier mobility μ 1 is assigned to, particularly corresponds to a mobility of the majority charge carriers, μ 2 is assigned to, particularly corresponds the absolute value of the difference between hole and electron mobility, and the third carrier mobility μ 3 is assigned to, particularly corresponds to a mobility of the minority charge carriers in the sample ( 107 ). The invention also relates to a computer program and a system for determining respective transport properties of majority as well as minority charge carriers in a sample ( 107 ).

Inventors

  • Artem Musiienko

Assignees

  • HELMHOLTZ-ZENTRUM BERLIN FÜR MATERIALIEN UND ENERGIE GMBH

Dates

Publication Date
20260512
Application Date
20240509
Priority Date
20230516

Claims (15)

  1. 1 . A method for determining respective transport properties of majority as well as minority charge carriers in a sample ( 107 ) comprising the majority and the minority charge carriers that correspond to electrons and holes or vice versa, wherein a plurality of Hall measurement trials is performed on the sample ( 107 ), wherein during each Hall measurement trial, the sample ( 107 ) is exposed to an illumination intensity I, wherein a Hall coefficient and a conductivity are acquired from each Hall measurement trial, wherein in a first Hall measurement trial, the sample ( 107 ) is exposed to a first illumination intensity I 1 , in the range of zero to 0.02 suns, particularly wherein the first illumination intensity is zero, and a first Hall coefficient R H (I 1 ) and a first conductivity σ(I 1 ) are acquired, wherein from the first Hall coefficient and the first conductivity, a carrier mobility μ 1 is determined, wherein in a second measurement trial, the sample ( 107 ) is exposed to a second illumination intensity I 2 and a second Hall coefficient R H (I 2 ) and, a second conductivity σ(I 2 ) are acquired, wherein from the second Hall coefficient and the second conductivity, a second carrier mobility μ 2 is determined, wherein the second illumination intensity I 2 is so high that the charge carrier density of electrons and a charge carrier density of holes in the sample ( 107 ) are identical, that the second Hall coefficient asymptotically approaches zero and that a second Hall mobility obtained from the product of the second Hall coefficient and the second conductivity asymptotically approaches a constant value, wherein a third carrier mobility μ 3 is determined from the first and the second carrier mobility, particularly by subtracting the second carrier mobility from the first carrier mobility if the Hall coefficient has the same sign for the first and the second illumination intensity or by adding the second carrier mobility to the first carrier mobility if the Hall coefficient changes its sign for the first and the second illumination intensity, wherein the first carrier mobility μ 1 is assigned to, particularly corresponds to a mobility of the majority charge carriers, μ 2 is assigned to, particularly corresponds to the absolute value of the difference between hole and electron mobility, and the third carrier mobility μ 3 is assigned to, particularly corresponds to a mobility of the minority charge carriers in the sample ( 107 ).
  2. 2 . The method according to claim 1 , wherein if R H (I 1 )>0, the majority charge carriers are holes and the minority charge carriers are electrons and wherein if R H (I 1 )<0, the majority charge carriers are electrons and the minority charge carriers are holes.
  3. 3 . The method according to claim 1 , wherein one or more intermediate Hall measurement trials are executed, wherein for each of the intermediate measurement trials the sample ( 107 ) is exposed to a different illumination intensity, wherein the illumination intensity of each intermediate measurement trial lies between the first illumination intensity I 1 and the second illumination intensity I 2 , wherein an intermediate Hall coefficient, and an intermediate conductivity is determined for each intermediate measurement trial, particularly wherein the Hall coefficient R H and the conductivity σ is determined as a function of the illumination intensities I from the plurality of Hall measurement trials.
  4. 4 . The method according to claim 2 , wherein a minority charge carrier density n 3 is determined, particularly as a function of the illumination intensity I, from the conductivity σ, a Hall coefficient R H as well as the first and the third mobility μ 1 , μ 3 , particularly from n 3 = σ ⁡ ( μ 1 - R H ⁢ σ ) ( q ⁡ ( μ 1 ⁢ μ 3 + μ 3 ⁢ μ 3 ) ) if holes are the majority charge carriers and from n 3 = σ ⁡ ( μ 3 + R H ⁢ σ ) ( q ⁡ ( μ 1 ⁢ μ 3 + μ 1 ⁢ μ 1 ) ) if electrons are the majority charge carriers, with q being the elementary charge.
  5. 5 . The method according to claim 4 , wherein a majority charge carrier density n 1 is determined, particularly as a function of the illumination intensity I, from the conductivity σ, the minority charge carrier density n 3 , as well as the first and the third mobility μ 1,3 , particularly from n 1 = ( σ q - n 3 ⁢ μ 3 ) / μ 1 .
  6. 6 . The method according to claim 1 , wherein at least one of the following transport properties is determined as follows, particularly as a function of the illumination intensity I: a majority and/or a minority charge carrier lifetime τ 1,3 based on the majority charge carrier density n 1 and/or the minority charge carrier density n 3 well as a generation rate G, particularly from τ 1,3 (G)=[n 1,3 (G)−n 1,3 (G=0)]/G, with the generation rate G=Iλ10 −9 /hc10 3 d, and λ the wavelength of electromagnetic radiation providing the illumination and d a thickness of the sample ( 107 ); a majority and/or a minority charge carrier diffusion coefficient D 1,3 from the majority charge carrier mobility μ 1 and/or the minority charge carrier mobility μ 3 as well as a temperature T of the sample ( 107 ), particularly from D 1,3 =μ 1,3 k B T/q; a majority and/or a minority charge carrier diffusion length L 1,3 based on the majority and/or the minority charge carrier lifetime τ 1,3 as well as the majority and/or the minority charge carrier diffusion coefficient D 1,3 , particularly from L 1,3 =√{square root over (D 1,3 τ 1,3 )}; a Quasi-Fermi level splitting QFLS 1,3 of the majority and the minority charge carriers, based on the sample ( 107 ) temperature T as well as the majority and the minority charge carrier density n 1,3 , particularly from QFLS 1,3 =k B T ln(n 1,3 /n int ), a total Quasi-Fermi level splitting QFLS total based on the sample ( 107 ) temperature T, the majority and the minority charge carrier density n 1,3 , particularly from QFLS total =k B T ln(n 1 n 3 /n int ) where n int =√{square root over (n 1 (I 1 =0)n 3 (I 1 =0))} is an intrinsic carrier density, and an ideality factor η = q ⁡ ( QFLS B - Q ⁢ F ⁢ L ⁢ S A ) ln ⁢ ( G B / G A ) ⁢ k b ⁢ T , wherein QFLS B,A are total Quasi Fermi level splitting determined at two different generation rates G A,B , respectively, particularly wherein the two generation rates G A,B differ from each other by at least 25%.
  7. 7 . The method according to claim 3 , wherein, if the Hall coefficient R H (I) and the conductivity σ(I) are altered by a parasitic conductivity σ s , such that an acquired Hall mobility μ H (I)==R H (I)σ(I) comprises a peak value max(μ H (I)) with max(μ H (I))>μ 1 (I=0), a corrected majority charge carrier density at the first illumination intensity, particularly at zero intensity, n 1,corr (I=0) accounting for parasitic contributions σ s to the first conductivity σ(I 1 ) is determined based on the first conductivity σ(I 1 ) and μ H (I), particularly from n 1,corr (I 1 =0)=μ H (I 1 )σ(I 1 )/q(max(μ H (I))) 2 , and/or wherein a corrected first mobility μ 1,corr is determined based on μ H (I), particularly from μ 1,corr =max(μ H (I)).
  8. 8 . The method according to claim 7 , wherein the parasitic conductivity σ s is determined based on the first conductivity σ(I 1 ), the corrected first mobility μ 1,corr and the corrected majority charge carrier density n 1,corr (I 1 ), particularly from σ s =σ(I 1 )−qμ 1,corr n 1,corr (I 1 ).
  9. 9 . The method according to claim 1 , wherein, if the determined μ 1 and μ 3 differ from each other by less than 15%, and/or the acquired Hall coefficient R H (I) and the acquired conductivity σ(I) are affected by parasitic conductivity and/or the acquired Hall coefficient R H (I) comprises a signal-to-noise ratio of less than 1, μ 2 is determined based on respective Hall mobilities μ H (I C ), μ H (I D ) and conductivities σ(I C ), σ(I D ) acquired at two different illumination intensities I C and I D between I 1 and I 2 , respectively, wherein I D exceeds I C by at least 25%, μ 2 = ❘ "\[LeftBracketingBar]" ( μ H ( I D ) ⁢ σ ⁡ ( I D ) - μ H ( I C ) ⁢ σ ⁡ ( I C ) ) / ( σ ⁡ ( I D ) - σ ⁡ ( I C ) ) ❘ "\[RightBracketingBar]" = ❘ "\[LeftBracketingBar]" β ❘ "\[RightBracketingBar]" , wherein μ 3 is determined by adding the first and the second charge carrier mobility if β<0 and (μ H (I D )<μ H (I C )), or if β>0 and μ H (I D )>μ H (I C ) and wherein μ 3 is determined by subtracting the second charge carrier mobility from the first charge carrier mobility, if β>0 and μ H (I D )<μ H (I C ).
  10. 10 . The method according to claim 4 , wherein, if the acquired Hall coefficient R H (I) comprises a signal-to-noise ratio of less than 1 and if the majority and the minority charge carriers are both photogenerated by the same amount, Δn 1 (I)=Δn 3 (I), Δn 1 and Δn 3 are determined based on σ as well as μ 1 and μ 3 , particularly from Δ ⁢ n 1 ( I ) = Δ ⁢ n 3 ( I ) = ( σ ⁡ ( I ) - σ ⁡ ( I = 0 ) ) / ( e ⁡ ( μ 1 + μ 3 ) ) .
  11. 11 . The method according to claim 4 , wherein, if the acquired Hall coefficient R H (I) comprises a signal-to-noise ratio of less than 1 and if only the majority charge carriers are photogenerated, such that only n 1 (and not n 3 ) changes as a function of the illumination intensity, n 1 is determined based on σ(I) and μ 1 , particularly from n 1 = σ ⁡ ( I ) / ( e ⁡ ( μ 1 ) ) .
  12. 12 . The method according to claim 1 , wherein the illumination is provided by means of a continuous electromagnetic wave, particularly using a light emitting diode (LED) or a laser diode (LD).
  13. 13 . The method according to claim 1 , wherein the sample ( 107 ) is comprised by a semiconductor device or a component of the semiconductor device, such as a solar cell, an interface, a transistor, a photodetector and/or a diode, wherein the respective transport properties of the majority as well as minority charge carriers of the sample ( 107 ) comprised by the semiconductor device are characterized.
  14. 14 . A computer program comprising instructions which, when the program is executed by a computer ( 214 ), cause the computer ( 214 ) to carry out the method according to claim 1 .
  15. 15 . A system ( 100 ) for determining respective transport properties of majority and minority charge carriers in a sample ( 107 ) comprising the majority and the minority charge carriers that correspond to electrons and holes or vice versa, using the method according to claim 1 , wherein the system comprises: a magnet ( 104 ) configured to expose the sample ( 107 ) to a magnetic field, a current source ( 205 ) configured to apply an electrical current through the sample ( 107 ), a source of electromagnetic radiation ( 109 ) configured to expose the sample ( 107 ) to electromagnetic radiation with different illumination intensities, at least one measurement unit ( 200 ) configured to acquire a Hall coefficient and a conductivity both at a first illumination intensity I 1 in the range of zero to 0.02 suns, particularly at zero illumination, as well as at a second illumination intensity I 2 that is so high that a charge carrier density of electrons and a charge carrier density of holes in the sample ( 107 ) are identical, that the second Hall coefficient asymptotically approaches zero at the second illumination intensity I 2 and that a second Hall mobility obtained from the product of the second Hall coefficient and the second conductivity asymptotically approaches a constant value, and a processing unit, particularly wherein the processing unit is part of said computer ( 214 ), wherein the processing unit is configured to determine: a first mobility μ 1 acquired at the first illumination intensity as well as a second mobility μ 2 acquired at the second illumination intensity, and a third mobility μ 3 from the first and the second mobility, particularly by subtracting the second mobility from the first mobility, if the Hall coefficient has the same sign for the first and the second illumination intensity or by adding the second mobility with the first mobility, if the Hall coefficient changes its sign for the first and the second illumination intensity, wherein the first mobility μ 1 corresponds to a mobility of the majority charge carriers μ 2 corresponds to the absolute value of the difference between hole and electron mobility and the third mobility μ 3 corresponds to a mobility of the minority charge carriers in the sample ( 107 ).

Description

CROSS-REFERENCE TO RELATED APPLICATIONS This application claims the right of priority of European Patent Application No. EP 23173681.0, filed May 16, 2023, the contents of which are incorporated by reference herein in its entirety. FIELD The present invention concerns a method, a computer program as well as a system for determining respective transport properties of majority as well as minority charge carriers in a sample. BACKGROUND OF THE INVENTION The development of novel and improvement of existing semiconductor and semi-insulating materials relies on the precise knowledge of free-charge transport properties. In particular, knowledge of minority and majority charge carrier diffusion lengths, lifetimes, mobilities, and carrier densities allows for tailoring the design of semiconductor devices and control effectiveness of solar cells, transistors, detectors, sensors, and LEDs. At the moment, the detection of minority and majority charge properties, i.e. of electrons and holes or vice versa, is challenging due to the limitation of the experimental methods. The methods, such as time-resolved photoluminescence (trPL), can detect a lifetime for only a minority of carriers, which control the fast decay component of the signal. The additional drawback of the commonly used methods is that they probe charge carriers' properties in a transient regime which is not matched with device or material operation conditions in a steady state. Therefore, there is a demand for probing the charge carrier properties, that include separate holes and electron carrier densities, lifetimes, diffusion length, and mobility, particularly under steady state conditions, in order to characterize semiconductor materials and -devices. The Hall effect, which is a reaction of the material on the magnetic field, was discovered by Edwin Hall in 1879. The Hall effect voltage is proportional to the strength of the magnetic field and the applied electrical current. Hall described the reaction of gold films on the magnetic field and considered only one carrier type—the electron. In the twentieth century, it was demonstrated that a combination of light and the Hall effect could be used to access electron and hole signals in semiconductors due to the unique feature of the Hall effect signal consisting of contributions of both holes and electrons. On the other hand, this approach had limiting implications due to the unknown value of electron and hole mobility which have not allowed resolute carrier transport parameters. Recently, several studies used a single carrier regime assuming an equal charge carrier density of holes and electrons to assess the properties of carriers in general. Methods for measurements of photogenerated charge carriers by the Hall method were presented (U.S. Pat. No. 10,197,640 B2, US2021262982A1), wherein the charge carrier density of holes and electrons and their lifetimes were assumed to be equal for any light intensity. However, particularly in semiconductors, the carrier densities, lifetimes, and diffusion lengths of electrons and holes are not equal due to the presence of traps. Assuming the same charge carrier density of holes and electrons therefore leads to incorrect charge transport description. SUMMARY OF THE INVENTION Based on this, it is subject of the present invention to provide a method, a computer program as well as a system that allows reliable determination of transport properties of majority as well as minority charge carriers, respectively. This task is solved by a method for determining respective transport properties of majority as well as minority charge carriers with the features of claim 1 as well as a corresponding computer program with the features of claim 14 and a corresponding system with the features of claim 15. BRIEF DESCRIPTION OF THE DRAWINGS In the following, examples of the present invention and its preferred embodiments are described with reference to the accompanying drawings. FIG. 1 shows a schematic overview of an embodiment of the system according to the third aspect of the invention; FIG. 2 shows an embodiment of the method and the computer program according to the first and the second aspect of the invention, wherein an AC magnetic field is applied; FIG. 3 shows an embodiment of the method and the computer program according to the first and the second aspect of the invention, wherein a DC magnetic field is applied; FIG. 4 shows a flowchart of an embodiment of the method and the computer program according to the first and the second aspect of the invention; FIG. 5 shows a flowchart of another embodiment of the method and the computer program according to the first and the second aspect of the invention; FIG. 6 shows a flowchart of a correction algorithm used to determine respective corrected values of the majority charge carrier mobility, the majority charge carrier density, and minority charge carrier mobility in case of parasitic conductivity and/or for low signal to noise ratio;