I wanna say something that could be helpful or maybe not. You judge:

In literature, there is a misconception between lattice mismatch between film and subst. and strain in the thin films. The strain can come from several factors:

1- in plane lattice mismatch/misfit (accurately: at growth temperature, not at room temp!)

2- intrinsic strain due to coalesce of grains and islands during film growth. The islands like to form a close surface to minimize their surface energy, however, this causes an induction of tensile strain. Therefore, the growth rate and temp. (growth conditions in general) affects the strain regardless of lattice mismatch!

3- If you do your growth at elevated temperautres, clearly you coold down your films to room temp. which that will induce an extra strain due to the difference between the thermal expansion coefficients of films and substr.

However, you can exprimentally measure the strain in your films using X-ray diffractions/reciprocal space mapping. You can use PL/Raman/CL spectroscopy in the case of semiconductors. Also, you can use some optical means to measure the substrate curvature/bowing and then calculate the strain and stress of your films!

I hope it was helpful for you!

Good luck

(as-af)/as; as-substrate lattice parameter, af- film lattice parameter. If your growth is along say (001) on substrate (001), then you will use the c- lattice parameter for both. Multiply the formula by 100, you get the % of mismatch. -Cheers

Thank you. Yes, you have shown me a method i have mentioned in the question, however, i have noticed it is likely that the mismatch is not determined by the relation for a expitaxial film

The formula you and Adam wrote is the correct one: however, it is true that you can find slightly different notations in the literature. As long as you use always the same one in calculations, things should not be different. Therefore, I recommend you define clearly in your works what definition of mismatch you consider and you stay coherent to that.

Thank Seravali very much！You have pointed out a problem always to be noticed in paper, the necessary method did not stated so making the paper to be unreadable.

I wanna say something that could be helpful or maybe not. You judge:

In literature, there is a misconception between lattice mismatch between film and subst. and strain in the thin films. The strain can come from several factors:

1- in plane lattice mismatch/misfit (accurately: at growth temperature, not at room temp!)

2- intrinsic strain due to coalesce of grains and islands during film growth. The islands like to form a close surface to minimize their surface energy, however, this causes an induction of tensile strain. Therefore, the growth rate and temp. (growth conditions in general) affects the strain regardless of lattice mismatch!

3- If you do your growth at elevated temperautres, clearly you coold down your films to room temp. which that will induce an extra strain due to the difference between the thermal expansion coefficients of films and substr.

However, you can exprimentally measure the strain in your films using X-ray diffractions/reciprocal space mapping. You can use PL/Raman/CL spectroscopy in the case of semiconductors. Also, you can use some optical means to measure the substrate curvature/bowing and then calculate the strain and stress of your films!

I hope it was helpful for you!

Good luck

Thank you very much!

My field is on Mathematics where I have studied the lattice theory with related to POSET in Discrete Mathematics.

Kamran is very right, growth induced effects are always there.

‘Lattice mismatch’ in literature identifies w.r.t substrate. Strain is what has resulted in the film. Its up to you to compare with substrate/ conditions (including thinckness variation) or with bulk of the same sample (provided single crystal data is available or the bulk structural data is taken after several calcinations so that there is no further change in XRD).

In general (exceptions may be there) close lattice matching (with substrate) is necessary for epitaxial growth. Otherwise, highly oriented or textured films are obtained. In such cases stain can be discussed w.r.t the bulk of the material.

Hope this helps

Thanks！

Well in my opinion, there is very less lattice mismatch in case of epitaxial thin films. Because epitaxial growth of thin films takes place matching with the orientation of substrate. So there is very less possibility of any lattice mismatch for epitaxial thin films.

Acturally, lattice mismatch is a initial parameter showing the lattice constant difference between the film and the substrate when you chosing a proper substrate. Ideally, after epitaixy, there is lattice mismatch on the interface when a proper substrate was used, however, lattice strain so lattice stress is being on the interface because of the lattice mismatch existing before epitaxy. Therefore, lattice mismatch is yet a useful parameter to evaluate the effect of lattice constant difference on the film properties.

The lattice mismatch actually relate to the orientation relationship, or else you need to "guess" which directions is possible to mach together. The latter usually the case when you try to predict the orientation relationship. Anyway, there are two ways, the first one is directly calculate the misfit value, the second one is applying graphic plot to show the match/mismatch.

(1)For two crystals (1 and 2) at a given orientation relationship, the misfit between two vectors from crystal 1 and 2 can be defined as deta x = x1 - x2. In simple but not strict manner, x1 and x2 connect the nearest atoms from crystal 1 and 2 respectively. For defining the more useful quantity, i.e. misfit strain, which is defined as |deta x|/|x1| with crystal 1 as reference cystal, x1 and x2 should be close to the origin so that they do not cross a dislocation.

(2)The graphic method is plot the atoms from the two crystals in a plane and denote the atoms with misfit smaller than certain value. This can be done by the software PTCLab, see http://sourceforge.net/projects/tclab/

I have seen that authors use different formula for lattice mismatch calculation.

can someone please tell me the exact formula for lattice mismatch calculation?

I am using GaAs(111) substrate and a thin film grown has lattice parameters a= 4.26 and c=30.45 A. the film is grown along c-axis.

Thank you

How to calculate lattice mismatch in thin film? - ResearchGate. Available from: https://www.researchgate.net/post/How_to_calculate_lattice_mismatch_in_thin_film [accessed Aug 21, 2016].

Now it is clear there are two formulae often adopted to calculate the lattice mismatch. If we set the lattice constant of the substrate plane is a, and the lattice constant of the epitaxially grown film matached on the susbtrate lattice as b, one formula is (b-a)/a *100%, and another one is (b-a)/b*100%.   In my own opinion, the second one is right and the first one is wrong, since the film lattice will be modulated by the substrate lattice constraint, but not inversely.

"However, you can exprimentally measure the strain in your films using X-ray diffractions/reciprocal space mapping. You can use PL/Raman/CL spectroscopy in the case of semiconductors. Also, you can use some optical means to measure the substrate curvature/bowing and then calculate the strain and stress of your films!",  - Kamran Forghani

CORRECT!

Despite the expectation of "epitaxial growth of thin films takes place matching with the orientation of substrate", the films grow in a manner to minimize the free energy and are controlled by the growth conditions as well as rate. Our interest is in precisely and quantitatively characterizing the in situ Nanostructural stain conditions in simple and complex epitaxial materials systems both homo-epitaxial and heteroepitaxial. A review of the attached RG discussions may help further the thought process here.

Besides the various strains reviewed thus far, interface strains are important as well. The use of "buffer layers" to ameliorate this condition is an important aspect to consider.

Real Time NDE XRD is potentially the best method to characterize the in situ Nanostructural lattice strain state for both epi and substrate simultaneously. It would make sense to try and corroborate similar examinations with conventional alternative tools such as suggested earlier ("PL/Raman/CL spectroscopy in the case of semiconductors") besides SEM, TEM, EPD etc..

BRAGG XRD MICROSCOPY: RSM combined with real time XRD Topography will allow one to characterize the "misfit" in terms of 2θ/ω, Φ or χ (2Theta/Omega, Phi or Chi) as well as dislocation density ρ=β2/9b2 (Hirsch et al). Where β=Integral Breadth and b=Burgers Vector. XRR, X-ray Reflectivity, will allow quantification of film thickness and surface "flatness".

BTW it may be insufficient to measure "miss-match" or "misfit" at one point (as is done with conventional XRD RSM's) on the 2D "wafer". We observe that it is a function of the Topographic location of the sampled VOXEL on the 2D sampling surface of the wafer. It may also be wise to consider several parallel & non-parallel (hkl) reflections to fully characterize the 3D strain matrix including any prevailing twists and tilts :-)

https://www.researchgate.net/post/What_is_the_best_way_to_interpret_REAL_TIME_3D_X_Y_o-2TH_o_ph_ch_reciprocal_space_relative_intensity_data_from_HRXRD_rocking_curve_analyses?_tpcectx=profile_questions

https://www.researchgate.net/post/Is_it_possible_to_precisely_estimate_the_epitaxial_film_thickness_with_XRD_rocking_curve_analysis_2kW_Power_Sealed_Tube_Source?_tpcectx=profile_questions

https://www.researchgate.net/post/Can_XRR_X-Ray_Reflectivity_be_implemented_in_the_Topographic_Mode_with_spatial_resolution_over_the_sample_surface_Is_there_value_for_such_analyses?_tpcectx=profile_questions

https://www.linkedin.com/pulse/nano-structural-meaning-xrd-data-converting-bragg-condition-ananth?trk=mp-reader-card

https://www.flickr.com/photos/85210325@N04/28643920586/

When the strain is produced by the embedding of a piece of a given material, e.g. a quantum dot, into another material, it is usually assumed that the coordinates are fixed to the external matrix and the dot is treated as expanded/compressed external material with different elastic constants.

Therefore, the expansion/compression, produced by the change of lattice constant in the dot region, brings a fictitious strain in the dot that must be calculated as eps0_ii = (a_m - a_d)/a_m where a_m, a_d are the matrix and dot lattice constants, respectively.

But then the system is allowed to relax by departing from this fictitious configuration, thus producing a deformation field and the associated strain defined by the tensor eps_ij. Once the equilibrium is reached, with the above definition of eps0_ii , the elastic energy must be (written in latex symbols):

\begin{equation}

E=\frac{1}{2}\int dV \sum_{ijkl}C_{ijkl}(\mathbf{r}) \left[\varepsilon_{ij}(\mathbf{r})+ \varepsilon_{ii}^{(0)}(\mathbf{r})\delta_{ij}\right] \left[\varepsilon_{kl}(\mathbf{r})+ \varepsilon_{kk}^{(0)}(\mathbf{r})\delta_{kl}\right],

\end{equation}

Some times, as e.g. in comsol I believe, eps0_i is defined with opposite sign. It means that in the above formula eps0_ii is preceded by a negative sign (instead of a positive one as in the above formula!). For details on the theory and calculation details of strain see supplement material of the paper Strain in free standing CdSe/CdS core-shell nanorods, J. Appl. Phys. 111, 014303 (2012) (doi: 10.1063/1.3673256).

Best regards,

Josep

Experimental observations with Bragg XRD Microscopy indicate both "twist" (in-plane) and "roll" (out-of-plane) of the epitaxial lattice in thin films as well besides "expanded/compressed external material with different elastic constants". In other words, the new deposited lattice may be accommodating the differential by "twist" and "roll" of the new lattice as well.

Any comments?

https://www.flickr.com/photos/85210325@N04/11487549683/in/album-72157645018820696/

But the above mentioned formulae are failed to find out the the lattice mismatch of GaN and InN Wurtzite structure

In a research paper lattice parameters of GaN is

a=0.3232nm and c= 0.5269nm   c/a =1.630   B.G=1.596 eV

for InN

a=0.3628 nm and c=0.5870nm  c/a=1.618    B.G=0.001eV

But it is reported that lattice mismatch of GaN and InN is 10.0% for a and 9.3% for c.

how to find these lattice mismatch?

have any specific formula?

(0.3628-0.3232)/0.3628=10.9%

is there any book that explains about lattice mismatch and formula to calculate lattice mismatch? I apologize for asking about this case because I am new in this field and want to learn it

That might help you, Alfiatur Rahmah

You can follow

https://www.researchgate.net/post/How_to_calculate_lattice_mismatch_in_thin_film[accessed Aug 21, 2016].