Elements of a scattering matrix
Dear Kwant community, I am a physics student and I am approaching to Kwant to solve simple problems of scattering. I am new in Kwant and, for this reason, I'm trying to use Kwnat for basic problems of scattering. The most simple problem in my mind is the scattering of a plane-wave (1 dimension) from the left of a potential barrier. If we now suppose the potential barrier is 'zero' everywhere, the scattering matrix, according to the scattering theory, should be 'S={{0,1},{1,0}}'. If I try to compute the reflection and transmission coefficient, I obtained the correct values (1 for reflection and 0 for transmission). But it occurs only for some energies. For example in the code I attached, if I put the value of '10' in 'S=kwant.solvers.default.smatrix(syst,10)' I obtaine for reflection and transmission the value of '0'. Where is my errors? And, is there a way to print the entire elements of the scattering matrix ? Yours sincerely Lorenzo Bagnasacco
Hi Lorenzo, You need to compute the transmission at an energy where the leads have modes. At E = 10 in your system there aren't any. Plotting the band structure of the leads helps. Check out also the Kwant tutorial at https://kwant-project.org/doc/1/tutorial/ Best, Anton On Tue, 18 Jan 2022 at 15:37, Lorenzo BAGNASACCO <lorenzo.bagnasacco@sns.it> wrote:
Dear Kwant community,
I am a physics student and I am approaching to Kwant to solve simple problems of scattering. I am new in Kwant and, for this reason, I'm trying to use Kwnat for basic problems of scattering. The most simple problem in my mind is the scattering of a plane-wave (1 dimension) from the left of a potential barrier. If we now suppose the potential barrier is 'zero' everywhere, the scattering matrix, according to the scattering theory, should be 'S={{0,1},{1,0}}'. If I try to compute the reflection and transmission coefficient, I obtained the correct values (1 for reflection and 0 for transmission). But it occurs only for some energies. For example in the code I attached, if I put the value of '10' in 'S=kwant.solvers.default.smatrix(syst,10)' I obtaine for reflection and transmission the value of '0'. Where is my errors? And, is there a way to print the entire elements of the scattering matrix ?
Yours sincerely
Lorenzo Bagnasacco
Dear Kwant community, Thanks for the quick reply. I plot the band structure and I obtaine the figure I attached. For a free-particle I expect a band structure like quadratic dispersion (E \sim k^2). The plot of structure band is nearly quadraic. Is it due to discretization of the system ? So it means that the wave-function sent to the left lead (in my example) is not a free-plane-wave ? Is there a way to control the wave-function injected in the lead ? Yours sincerely Lorenzo Bagnsacco Il giorno mar 18 gen 2022 alle ore 15:40 Anton Akhmerov < anton.akhmerov+kd@gmail.com> ha scritto:
Hi Lorenzo,
You need to compute the transmission at an energy where the leads have modes. At E = 10 in your system there aren't any.
Plotting the band structure of the leads helps. Check out also the Kwant tutorial at https://kwant-project.org/doc/1/tutorial/
Best, Anton
On Tue, 18 Jan 2022 at 15:37, Lorenzo BAGNASACCO < lorenzo.bagnasacco@sns.it> wrote:
Dear Kwant community,
I am a physics student and I am approaching to Kwant to solve simple problems of scattering. I am new in Kwant and, for this reason, I'm trying to use Kwnat for basic problems of scattering. The most simple problem in my mind is the scattering of a plane-wave (1 dimension) from the left of a potential barrier. If we now suppose the potential barrier is 'zero' everywhere, the scattering matrix, according to the scattering theory, should be 'S={{0,1},{1,0}}'. If I try to compute the reflection and transmission coefficient, I obtained the correct values (1 for reflection and 0 for transmission). But it occurs only for some energies. For example in the code I attached, if I put the value of '10' in 'S=kwant.solvers.default.smatrix(syst,10)' I obtaine for reflection and transmission the value of '0'. Where is my errors? And, is there a way to print the entire elements of the scattering matrix ?
Yours sincerely
Lorenzo Bagnasacco
Hi Lorenzo, Indeed, you are right; please see for example this tutorial: https://kwant-project.org/doc/1/tutorial/discretize#limitations-of-discretiz... Best, Anton On Tue, 18 Jan 2022 at 16:31, Lorenzo BAGNASACCO <lorenzo.bagnasacco@sns.it> wrote:
Dear Kwant community,
Thanks for the quick reply.
I plot the band structure and I obtaine the figure I attached. For a free-particle I expect a band structure like quadratic dispersion (E \sim k^2). The plot of structure band is nearly quadraic. Is it due to discretization of the system ? So it means that the wave-function sent to the left lead (in my example) is not a free-plane-wave ? Is there a way to control the wave-function injected in the lead ?
Yours sincerely
Lorenzo Bagnsacco
Il giorno mar 18 gen 2022 alle ore 15:40 Anton Akhmerov < anton.akhmerov+kd@gmail.com> ha scritto:
Hi Lorenzo,
You need to compute the transmission at an energy where the leads have modes. At E = 10 in your system there aren't any.
Plotting the band structure of the leads helps. Check out also the Kwant tutorial at https://kwant-project.org/doc/1/tutorial/
Best, Anton
On Tue, 18 Jan 2022 at 15:37, Lorenzo BAGNASACCO < lorenzo.bagnasacco@sns.it> wrote:
Dear Kwant community,
I am a physics student and I am approaching to Kwant to solve simple problems of scattering. I am new in Kwant and, for this reason, I'm trying to use Kwnat for basic problems of scattering. The most simple problem in my mind is the scattering of a plane-wave (1 dimension) from the left of a potential barrier. If we now suppose the potential barrier is 'zero' everywhere, the scattering matrix, according to the scattering theory, should be 'S={{0,1},{1,0}}'. If I try to compute the reflection and transmission coefficient, I obtained the correct values (1 for reflection and 0 for transmission). But it occurs only for some energies. For example in the code I attached, if I put the value of '10' in 'S=kwant.solvers.default.smatrix(syst,10)' I obtaine for reflection and transmission the value of '0'. Where is my errors? And, is there a way to print the entire elements of the scattering matrix ?
Yours sincerely
Lorenzo Bagnasacco
Dear Kwant community, I created a 2D system for a 'free particle like' (Hamiltonian with only kinetic energy in the direction of 'x' and 'y'). Then I created two leads (left and right) with three bands (cosine-like bands caused by discretization of the previous Hamiltonian). If I try to calculate the scattering matrix at the energy of '300', I obtain a square scattering matrix with dimensions 4*4. I expected a square scattering matrix with dimensions 6*6, because I have 3 bands for the left lead. Where am I wrong? I attached the notebook of the previous system I created for a clear visualization. Yours sincerely Lorenzo Bagnasacco Il giorno mar 18 gen 2022 alle ore 17:44 Anton Akhmerov < anton.akhmerov+kd@gmail.com> ha scritto:
Hi Lorenzo,
Indeed, you are right; please see for example this tutorial: https://kwant-project.org/doc/1/tutorial/discretize#limitations-of-discretiz...
Best, Anton
On Tue, 18 Jan 2022 at 16:31, Lorenzo BAGNASACCO < lorenzo.bagnasacco@sns.it> wrote:
Dear Kwant community,
Thanks for the quick reply.
I plot the band structure and I obtaine the figure I attached. For a free-particle I expect a band structure like quadratic dispersion (E \sim k^2). The plot of structure band is nearly quadraic. Is it due to discretization of the system ? So it means that the wave-function sent to the left lead (in my example) is not a free-plane-wave ? Is there a way to control the wave-function injected in the lead ?
Yours sincerely
Lorenzo Bagnsacco
Il giorno mar 18 gen 2022 alle ore 15:40 Anton Akhmerov < anton.akhmerov+kd@gmail.com> ha scritto:
Hi Lorenzo,
You need to compute the transmission at an energy where the leads have modes. At E = 10 in your system there aren't any.
Plotting the band structure of the leads helps. Check out also the Kwant tutorial at https://kwant-project.org/doc/1/tutorial/
Best, Anton
On Tue, 18 Jan 2022 at 15:37, Lorenzo BAGNASACCO < lorenzo.bagnasacco@sns.it> wrote:
Dear Kwant community,
I am a physics student and I am approaching to Kwant to solve simple problems of scattering. I am new in Kwant and, for this reason, I'm trying to use Kwnat for basic problems of scattering. The most simple problem in my mind is the scattering of a plane-wave (1 dimension) from the left of a potential barrier. If we now suppose the potential barrier is 'zero' everywhere, the scattering matrix, according to the scattering theory, should be 'S={{0,1},{1,0}}'. If I try to compute the reflection and transmission coefficient, I obtained the correct values (1 for reflection and 0 for transmission). But it occurs only for some energies. For example in the code I attached, if I put the value of '10' in 'S=kwant.solvers.default.smatrix(syst,10)' I obtaine for reflection and transmission the value of '0'. Where is my errors? And, is there a way to print the entire elements of the scattering matrix ?
Yours sincerely
Lorenzo Bagnasacco
Lorenzo BAGNASACCO wrote:
If I try to calculate the scattering matrix at the energy of '300', I obtain a square scattering matrix with dimensions 4*4. I expected a square scattering matrix with dimensions 6*6, because I have 3 bands for the left lead. Where am I wrong? I attached the notebook of the previous system I created for a clear visualization.
At energy 300 there are two incoming and two outgoing modes in each lead. Hence, the scattering matrix is 4x4. Christoph
participants (3)
-
Anton Akhmerov -
Christoph Groth -
Lorenzo BAGNASACCO