Defining uniaxial strain in specified region of graphene
I want to create the graphene strained superlattice-like structure, having uniaxial strain defined at the specified region. Please, someone can suggest to me how to define the strained region and interface between unstrained and strained graphene regions. Thanks in Advance Shrushti
Hi Shrushti, You just need to parametrize the hoppings. See https://arxiv.org/abs/1003.5179. Best, Antonio Em qui., 15 de jul. de 2021 às 09:19, <shrushti.tapar07@gmail.com> escreveu:
I want to create the graphene strained superlattice-like structure, having uniaxial strain defined at the specified region. Please, someone can suggest to me how to define the strained region and interface between unstrained and strained graphene regions. Thanks in Advance Shrushti
-- Antonio Lucas Rigotti Manesco PhD fellow - University of São Paulo, Brazil
Dear Shrushti, it is straightforward to deal with strained graphene. In the region you want to make strain, just set the function which modifies the site position (this mainly depends on the amount and type of the strain). Second, once you have shifted the site position you need to shift the hopping as illustrated in the file provided by Mr. Antonio. here you find a simple script about how to shift the site position in the case of uniaxial strain (adjusted to you case) def Triaxial_transform(pos, center, I): """ I is a parameter that lets us to control the intensity of the strain""" x, y = pos cx, cy = center[0], center[1] # the center of the strain r=sqrt((x-cx)**2+(y-cy)**2) if r<sigma: ux = 2*I * x*y uy = I * (x**2 - y**2) return x + ux, y + uy else: return x, y Then use this position to set your hopping as illustrated in a previous discussion [Ref-1]. You have to be careful in case you are using a strain in the z axis since you will not be able to plot the current. It is somehow tricky in this scenario. [Ref-1] https://www.mail-archive.com/kwant-discuss@kwant-project.org/msg00574.html I hop this will help Best wishes Le jeu. 15 juil. 2021 à 14:19, <shrushti.tapar07@gmail.com> a écrit :
I want to create the graphene strained superlattice-like structure, having uniaxial strain defined at the specified region. Please, someone can suggest to me how to define the strained region and interface between unstrained and strained graphene regions. Thanks in Advance Shrushti
Dear Adel Belayadi, I am new to Kwant, as per our previous discussion, here I am attaching my program, I have defined my strain region and function for uniaxial strain in armchair direction. Now I struggling with *how to introduce this function* in the program. Please correct me if I have understood it wrongly, Firstly we have to define graphene, by using the position of lattice points we have to apply pos_transform in a specified region, which will use the values of x and y co-ordinate from lattice placement due to lattice structure. The next doubt is like, as I using uniaxial strain in Y direction, it will squeeze lattice in X direction. So, the unstrained lattice point also has to shift accordingly. Right now I am only focusing on position displacement and not hopping. import numpy as np import scipy.io as spio from numpy import * import scipy.linalg as la import matplotlib as mpl import sympy as sym import kwant #%%###################### # parameters L=20 # Length of device on both sides W=5 # Width of device t=-2.7 pot=0.5 c=0.05 angle=pi/2 # lattice type graphene = kwant.lattice.general([(1, 0), (sin(pi/6), cos(pi/6))], [(0, 0), (0, 1 / sqrt(3))], norbs=1) a, b = graphene.sublattices # scattering region def rectangle(pos): x, y = pos return 0 <= x <= L and 0<= y <= W # strain_pos def pos_transform(pos,c,angle): x,y= pos if 5<x<10: ux=(cos(angle)**2-0.165*sin(angle)**2)*c*x uy=(sin(angle)**2-0.165*cos(angle)**2)*c*y return x+ux,y+uy else: return x,y syst = kwant.Builder() syst[graphene.shape(rectangle, (0, 0))] = 0 syst=pos_transform((5,0),0.05,pi/2) #hoppings = (((0, 0), a, b), ((0, 1), a, b), ((-1, 1), a, b)) #syst[[kwant.builder.HoppingKind(*hopping) for hopping in hoppings]] = t kwant.plot(syst); Thanks in advance. Regards, shrushti On Fri, Jul 16, 2021 at 2:38 AM Adel Belayadi <adelphys@gmail.com> wrote:
Dear Shrushti, it is straightforward to deal with strained graphene. In the region you want to make strain, just set the function which modifies the site position (this mainly depends on the amount and type of the strain). Second, once you have shifted the site position you need to shift the hopping as illustrated in the file provided by Mr. Antonio. here you find a simple script about how to shift the site position in the case of uniaxial strain (adjusted to you case)
def Triaxial_transform(pos, center, I): """ I is a parameter that lets us to control the intensity of the strain""" x, y = pos cx, cy = center[0], center[1] # the center of the strain r=sqrt((x-cx)**2+(y-cy)**2) if r<sigma: ux = 2*I * x*y uy = I * (x**2 - y**2) return x + ux, y + uy else: return x, y
Then use this position to set your hopping as illustrated in a previous discussion [Ref-1].
You have to be careful in case you are using a strain in the z axis since you will not be able to plot the current. It is somehow tricky in this scenario.
[Ref-1] https://www.mail-archive.com/kwant-discuss@kwant-project.org/msg00574.html
I hop this will help Best wishes
Le jeu. 15 juil. 2021 à 14:19, <shrushti.tapar07@gmail.com> a écrit :
I want to create the graphene strained superlattice-like structure, having uniaxial strain defined at the specified region. Please, someone can suggest to me how to define the strained region and interface between unstrained and strained graphene regions. Thanks in Advance Shrushti
Dear Shrushti, From a tight binding point of view, once we have defined our lattice and shape, we need to care about the hopping and on-site energy, not the positions. In your case and in order to parameterise the hopping in the strained region you need to know the position transforms. In fact, parameterizing the hopping is the most important task. In strained graphene, parameterising the hopping is somehow similar to adding a magnetic field in the system. *In your script you have written: syst=pos_transform((5,0),0.05,pi/2)* This is completely wrong and Kwant will not be able to recognize the system. Since you are recently discovering Kwant, I recommend you to go through Kwant FAQ: https://kwant-project.org/doc/dev/tutorial/faq Back to your issue, I guess you are caring to see the strained shape. In this case, you need to use pos_transorm like: *kwant.plot(syst, pos_transform=lambda pos: pos_transform(pos,c=c,angle=angle), * Just one more thing, donot forget to uncommenthoppings = (((0, 0), a, b), ((0, 1), a, b), ((-1, 1), a, b))syst[[kwant.builder.HoppingKind(*hopping) for hopping in hoppings]] = totherwise you will not be able to see the strained position. Finally, you need to focus on using the correct hopping BestAdel Le jeu. 22 juil. 2021 à 09:48, Shrushti Tapar <shrushti.tapar07@gmail.com> a écrit :
Dear Adel Belayadi, I am new to Kwant, as per our previous discussion, here I am attaching my program, I have defined my strain region and function for uniaxial strain in armchair direction. Now I struggling with *how to introduce this function* in the program. Please correct me if I have understood it wrongly, Firstly we have to define graphene, by using the position of lattice points we have to apply pos_transform in a specified region, which will use the values of x and y co-ordinate from lattice placement due to lattice structure. The next doubt is like, as I using uniaxial strain in Y direction, it will squeeze lattice in X direction. So, the unstrained lattice point also has to shift accordingly. Right now I am only focusing on position displacement and not hopping. import numpy as np import scipy.io as spio from numpy import * import scipy.linalg as la import matplotlib as mpl import sympy as sym import kwant
#%%###################### # parameters L=20 # Length of device on both sides W=5 # Width of device t=-2.7 pot=0.5 c=0.05 angle=pi/2 # lattice type graphene = kwant.lattice.general([(1, 0), (sin(pi/6), cos(pi/6))], [(0, 0), (0, 1 / sqrt(3))], norbs=1) a, b = graphene.sublattices # scattering region def rectangle(pos): x, y = pos return 0 <= x <= L and 0<= y <= W
# strain_pos def pos_transform(pos,c,angle): x,y= pos if 5<x<10: ux=(cos(angle)**2-0.165*sin(angle)**2)*c*x uy=(sin(angle)**2-0.165*cos(angle)**2)*c*y return x+ux,y+uy else: return x,y
syst = kwant.Builder() syst[graphene.shape(rectangle, (0, 0))] = 0 syst=pos_transform((5,0),0.05,pi/2) #hoppings = (((0, 0), a, b), ((0, 1), a, b), ((-1, 1), a, b)) #syst[[kwant.builder.HoppingKind(*hopping) for hopping in hoppings]] = t kwant.plot(syst);
Thanks in advance. Regards, shrushti
On Fri, Jul 16, 2021 at 2:38 AM Adel Belayadi <adelphys@gmail.com> wrote:
Dear Shrushti, it is straightforward to deal with strained graphene. In the region you want to make strain, just set the function which modifies the site position (this mainly depends on the amount and type of the strain). Second, once you have shifted the site position you need to shift the hopping as illustrated in the file provided by Mr. Antonio. here you find a simple script about how to shift the site position in the case of uniaxial strain (adjusted to you case)
def Triaxial_transform(pos, center, I): """ I is a parameter that lets us to control the intensity of the strain""" x, y = pos cx, cy = center[0], center[1] # the center of the strain r=sqrt((x-cx)**2+(y-cy)**2) if r<sigma: ux = 2*I * x*y uy = I * (x**2 - y**2) return x + ux, y + uy else: return x, y
Then use this position to set your hopping as illustrated in a previous discussion [Ref-1].
You have to be careful in case you are using a strain in the z axis since you will not be able to plot the current. It is somehow tricky in this scenario.
[Ref-1] https://www.mail-archive.com/kwant-discuss@kwant-project.org/msg00574.html
I hop this will help Best wishes
Le jeu. 15 juil. 2021 à 14:19, <shrushti.tapar07@gmail.com> a écrit :
I want to create the graphene strained superlattice-like structure, having uniaxial strain defined at the specified region. Please, someone can suggest to me how to define the strained region and interface between unstrained and strained graphene regions. Thanks in Advance Shrushti
Thanks a lot for clarifying my doubts, just first I wanted to observed deformation and now I will be trying to incorporate hopping parametrization. Regards, Shrushti On Fri, Jul 23, 2021 at 3:32 PM Adel Belayadi <adelphys@gmail.com> wrote:
Dear Shrushti,
From a tight binding point of view, once we have defined our lattice and shape, we need to care about the hopping and on-site energy, not the positions. In your case and in order to parameterise the hopping in the strained region you need to know the position transforms. In fact, parameterizing the hopping is the most important task. In strained graphene, parameterising the hopping is somehow similar to adding a magnetic field in the system.
*In your script you have written: syst=pos_transform((5,0),0.05,pi/2)*
This is completely wrong and Kwant will not be able to recognize the system. Since you are recently discovering Kwant, I recommend you to go through Kwant FAQ: https://kwant-project.org/doc/dev/tutorial/faq
Back to your issue, I guess you are caring to see the strained shape. In this case, you need to use pos_transorm like: *kwant.plot(syst, pos_transform=lambda pos: pos_transform(pos,c=c,angle=angle), *
Just one more thing, donot forget to uncommenthoppings = (((0, 0), a, b), ((0, 1), a, b), ((-1, 1), a, b))syst[[kwant.builder.HoppingKind(*hopping) for hopping in hoppings]] = totherwise you will not be able to see the strained position.
Finally, you need to focus on using the correct hopping BestAdel
Le jeu. 22 juil. 2021 à 09:48, Shrushti Tapar <shrushti.tapar07@gmail.com> a écrit :
Dear Adel Belayadi, I am new to Kwant, as per our previous discussion, here I am attaching my program, I have defined my strain region and function for uniaxial strain in armchair direction. Now I struggling with *how to introduce this function* in the program. Please correct me if I have understood it wrongly, Firstly we have to define graphene, by using the position of lattice points we have to apply pos_transform in a specified region, which will use the values of x and y co-ordinate from lattice placement due to lattice structure. The next doubt is like, as I using uniaxial strain in Y direction, it will squeeze lattice in X direction. So, the unstrained lattice point also has to shift accordingly. Right now I am only focusing on position displacement and not hopping. import numpy as np import scipy.io as spio from numpy import * import scipy.linalg as la import matplotlib as mpl import sympy as sym import kwant
#%%###################### # parameters L=20 # Length of device on both sides W=5 # Width of device t=-2.7 pot=0.5 c=0.05 angle=pi/2 # lattice type graphene = kwant.lattice.general([(1, 0), (sin(pi/6), cos(pi/6))], [(0, 0), (0, 1 / sqrt(3))], norbs=1) a, b = graphene.sublattices # scattering region def rectangle(pos): x, y = pos return 0 <= x <= L and 0<= y <= W
# strain_pos def pos_transform(pos,c,angle): x,y= pos if 5<x<10: ux=(cos(angle)**2-0.165*sin(angle)**2)*c*x uy=(sin(angle)**2-0.165*cos(angle)**2)*c*y return x+ux,y+uy else: return x,y
syst = kwant.Builder() syst[graphene.shape(rectangle, (0, 0))] = 0 syst=pos_transform((5,0),0.05,pi/2) #hoppings = (((0, 0), a, b), ((0, 1), a, b), ((-1, 1), a, b)) #syst[[kwant.builder.HoppingKind(*hopping) for hopping in hoppings]] = t kwant.plot(syst);
Thanks in advance. Regards, shrushti
On Fri, Jul 16, 2021 at 2:38 AM Adel Belayadi <adelphys@gmail.com> wrote:
Dear Shrushti, it is straightforward to deal with strained graphene. In the region you want to make strain, just set the function which modifies the site position (this mainly depends on the amount and type of the strain). Second, once you have shifted the site position you need to shift the hopping as illustrated in the file provided by Mr. Antonio. here you find a simple script about how to shift the site position in the case of uniaxial strain (adjusted to you case)
def Triaxial_transform(pos, center, I): """ I is a parameter that lets us to control the intensity of the strain""" x, y = pos cx, cy = center[0], center[1] # the center of the strain r=sqrt((x-cx)**2+(y-cy)**2) if r<sigma: ux = 2*I * x*y uy = I * (x**2 - y**2) return x + ux, y + uy else: return x, y
Then use this position to set your hopping as illustrated in a previous discussion [Ref-1].
You have to be careful in case you are using a strain in the z axis since you will not be able to plot the current. It is somehow tricky in this scenario.
[Ref-1] https://www.mail-archive.com/kwant-discuss@kwant-project.org/msg00574.html
I hop this will help Best wishes
Le jeu. 15 juil. 2021 à 14:19, <shrushti.tapar07@gmail.com> a écrit :
I want to create the graphene strained superlattice-like structure, having uniaxial strain defined at the specified region. Please, someone can suggest to me how to define the strained region and interface between unstrained and strained graphene regions. Thanks in Advance Shrushti
Dear Shrushti, The error you see is due to the edge_disorder function you have defined. Surprisingly the function outputs that error while using *and/or argument* in the function (x<X1 and 3<y<W-3). It works only if we *remove the boolean *conditions (and/or). At this point, *kwant developer might state why!.* However, your problem might be solved by averting the boolean condition in your onsite function. I see that you have used so many conditions in the onsite function (which might be written with fewer scripts) and your code is also hardly organized. The following propositions (two steps) have solved your issue with fewer codes. 1-Fist you need to define the* pn-junction* and *disorder* by separate functions def *onsite_pn*(site): return -pot if site.pos[0]<X1 else +pot def *onsite_disorder*(site,salt=8, w0=3): return 0 if -(W/2-w0)<site.pos[1]< (W/2-w0) else U0*(kwant.digest.uniform(repr(site), repr(salt))-0.5)+2 2-merge them in one onsite function called *onsite_PN_and_disorder* and given as def *onsite_PN_and_disorder*(site): return onsite_pn(site)+onsite_disorder(site,salt=8). Final remark.* You said* *I can include it, but I am not able to calculate the conductance/( in my case valley transmission). * I am surprised that the transmittance is given by kwant function *smatrix.transmission(0, 1)* gives you the valley transmittance (as you said). What I do know is that the valley transmittance is somehow more tricky than you think. Best, Adel *Here is the corrected code:* W=8 w0=2 t = 2.7 # hopping parameters pot=0.25 # Potential value in eV # angle of wings with x axis X1=L/2 a_uc=0.246 U0=6 salt=8 #%%Functions def shape(pos): x,y=pos rectangle= 0 <= x <= L and -W/2<= y <= W/2 return rectangle # def onsite(site): # x, y = site.pos # if x<X1: # return -pot # if x>X1: # return pot # def edge_disorder(site,salt=8): # x,y= site.pos # if x<X1 and 3<y<W-3: # return - pot # if x>X1 and 3<y<W-3: # return +pot # if 0<y<3 or W-3<y<W: # return U0 * (kwant.digest.uniform(repr(site), repr(salt))-0.5)+2 # else : # return 0 def onsite_pn(site): return -pot if site.pos[0]<X1 else +pot def onsite_disorder(site,salt=8, w0=w0): return 0 if -(W/2-w0)<site.pos[1]< (W/2-w0) else U0*(kwant.digest.uniform(repr(site), repr(salt))-0.5)+2 def onsite_PN_and_disorder(site): return onsite_pn(site)+onsite_disorder(site,salt=8) def site_color(site): return onsite_scattering(site) def plot_conductance(syst, energies): # Compute transmission as a function of energy data = [] for energy in energies: smatrix = kwant.smatrix(syst, energy) data.append(smatrix.transmission(0, 1)) plt.figure() plt.plot(energies, data) plt.xlabel("energy [t]") plt.ylabel("conductance [e^2/h]") plt.show() #------------------------------------------------------------------------------------------------------------------------------ # defining complete lattice structure and plotting #------------------------------------------------------------------------------------------------------------------------------ graphene = kwant.lattice.honeycomb(a=a_uc,norbs=1) a, b = graphene.sublattices syst = kwant.Builder() syst[graphene.shape(shape, (0,0))] = onsite_PN_and_disorder hoppings = (((0, 0), a, b), ((0, 1), a, b),((-1, 1), a, b)) syst[[kwant.builder.HoppingKind(*hopping) for hopping in hoppings]] = -t syst.eradicate_dangling() # kwant.plot(syst, site_color=site_color) # Left Lead sym0 = kwant.TranslationalSymmetry(graphene.vec((-1, 0))) def lead0_shape(pos): x,y=pos return (-W/2<y<W/2) lead = kwant.Builder(sym0) sym0.add_site_family(graphene.sublattices[0], other_vectors=[(-1, 2)]) sym0.add_site_family(graphene.sublattices[1], other_vectors=[(-1, 2)]) lead[graphene.shape(lead0_shape, (0, -0))] = 0.001 lead[[kwant.builder.HoppingKind(*hopping) for hopping in hoppings]] = -t lead.eradicate_dangling() syst.attach_lead(lead,add_cells=0) syst.attach_lead(lead.reversed(),add_cells=0) syst=syst.finalized() params=dict(pot=pot,salt=8,U0=U0) kwant.plot(syst) plt.show() #%% n=10 energies= [0+ (0.091* i) for i in range(n)] plot_conductance(syst, energies) #%%------------------------------------------------------------- Le ven. 23 juil. 2021 à 11:55, Shrushti Tapar <shrushti.tapar07@gmail.com> a écrit :
Thanks a lot for clarifying my doubts, just first I wanted to observed deformation and now I will be trying to incorporate hopping parametrization. Regards, Shrushti
On Fri, Jul 23, 2021 at 3:32 PM Adel Belayadi <adelphys@gmail.com> wrote:
Dear Shrushti,
From a tight binding point of view, once we have defined our lattice and shape, we need to care about the hopping and on-site energy, not the positions. In your case and in order to parameterise the hopping in the strained region you need to know the position transforms. In fact, parameterizing the hopping is the most important task. In strained graphene, parameterising the hopping is somehow similar to adding a magnetic field in the system.
*In your script you have written: syst=pos_transform((5,0),0.05,pi/2)*
This is completely wrong and Kwant will not be able to recognize the system. Since you are recently discovering Kwant, I recommend you to go through Kwant FAQ: https://kwant-project.org/doc/dev/tutorial/faq
Back to your issue, I guess you are caring to see the strained shape. In this case, you need to use pos_transorm like: *kwant.plot(syst, pos_transform=lambda pos: pos_transform(pos,c=c,angle=angle), *
Just one more thing, donot forget to uncommenthoppings = (((0, 0), a, b), ((0, 1), a, b), ((-1, 1), a, b))syst[[kwant.builder.HoppingKind(*hopping) for hopping in hoppings]] = totherwise you will not be able to see the strained position.
Finally, you need to focus on using the correct hopping BestAdel
Le jeu. 22 juil. 2021 à 09:48, Shrushti Tapar <shrushti.tapar07@gmail.com> a écrit :
Dear Adel Belayadi, I am new to Kwant, as per our previous discussion, here I am attaching my program, I have defined my strain region and function for uniaxial strain in armchair direction. Now I struggling with *how to introduce this function* in the program. Please correct me if I have understood it wrongly, Firstly we have to define graphene, by using the position of lattice points we have to apply pos_transform in a specified region, which will use the values of x and y co-ordinate from lattice placement due to lattice structure. The next doubt is like, as I using uniaxial strain in Y direction, it will squeeze lattice in X direction. So, the unstrained lattice point also has to shift accordingly. Right now I am only focusing on position displacement and not hopping. import numpy as np import scipy.io as spio from numpy import * import scipy.linalg as la import matplotlib as mpl import sympy as sym import kwant
#%%###################### # parameters L=20 # Length of device on both sides W=5 # Width of device t=-2.7 pot=0.5 c=0.05 angle=pi/2 # lattice type graphene = kwant.lattice.general([(1, 0), (sin(pi/6), cos(pi/6))], [(0, 0), (0, 1 / sqrt(3))], norbs=1) a, b = graphene.sublattices # scattering region def rectangle(pos): x, y = pos return 0 <= x <= L and 0<= y <= W
# strain_pos def pos_transform(pos,c,angle): x,y= pos if 5<x<10: ux=(cos(angle)**2-0.165*sin(angle)**2)*c*x uy=(sin(angle)**2-0.165*cos(angle)**2)*c*y return x+ux,y+uy else: return x,y
syst = kwant.Builder() syst[graphene.shape(rectangle, (0, 0))] = 0 syst=pos_transform((5,0),0.05,pi/2) #hoppings = (((0, 0), a, b), ((0, 1), a, b), ((-1, 1), a, b)) #syst[[kwant.builder.HoppingKind(*hopping) for hopping in hoppings]] = t kwant.plot(syst);
Thanks in advance. Regards, shrushti
On Fri, Jul 16, 2021 at 2:38 AM Adel Belayadi <adelphys@gmail.com> wrote:
Dear Shrushti, it is straightforward to deal with strained graphene. In the region you want to make strain, just set the function which modifies the site position (this mainly depends on the amount and type of the strain). Second, once you have shifted the site position you need to shift the hopping as illustrated in the file provided by Mr. Antonio. here you find a simple script about how to shift the site position in the case of uniaxial strain (adjusted to you case)
def Triaxial_transform(pos, center, I): """ I is a parameter that lets us to control the intensity of the strain""" x, y = pos cx, cy = center[0], center[1] # the center of the strain r=sqrt((x-cx)**2+(y-cy)**2) if r<sigma: ux = 2*I * x*y uy = I * (x**2 - y**2) return x + ux, y + uy else: return x, y
Then use this position to set your hopping as illustrated in a previous discussion [Ref-1].
You have to be careful in case you are using a strain in the z axis since you will not be able to plot the current. It is somehow tricky in this scenario.
[Ref-1] https://www.mail-archive.com/kwant-discuss@kwant-project.org/msg00574.html
I hop this will help Best wishes
Le jeu. 15 juil. 2021 à 14:19, <shrushti.tapar07@gmail.com> a écrit :
I want to create the graphene strained superlattice-like structure, having uniaxial strain defined at the specified region. Please, someone can suggest to me how to define the strained region and interface between unstrained and strained graphene regions. Thanks in Advance Shrushti
participants (4)
-
Adel Belayadi -
Antonio Manesco -
Shrushti Tapar -
shrushti.tapar07＠gmail.com