Current density for given lead currents
Hello Kwant community, I am considering a 6-terminal sample with 0,1,2,3 the voltage leads and 4,5 the current lead. Normally, with the conductance_matrix M found by kwant I can calculate transport properties (the voltages in vector V) as per the linear equation I = M V, given I = [0,0,0,0,1,-1]. Now I want to plot the current density corresponding to this transport state, but not sure how to do it. For instance, with wf = kwant.wave_function(sys, energy), I can have all scattering wavefunctions wf(4) incoming from the current lead 4. But how to (or is it necessary to) reflect the above current configuration vector I we specified? I presume the current density may vary with different I vectors. Or maybe I misunderstand some basics? Thank you!
Dear Zhang, You can calculate the current for each lead separately (sum over the modes in the lead) and then multiply each current by the corresponding V_i that you obtained from the conductance matrix. Total_current=Sum (current_i *V_i). You might claim that the dimension is not correct, but keep in mind that what we usually calculate is the transmission probability and thus we need to multiply by V (and the quantum of conductance) to get the current in the linear response theory. This will give you, *qualitatively*, the current map in your system. (I am interested to see the result) I hope this helps, Adel On Wed, May 1, 2024 at 12:23 PM X.-X. Zhang <xiaoxiao.zhang@riken.jp> wrote:
Hello Kwant community, I am considering a 6-terminal sample with 0,1,2,3 the voltage leads and 4,5 the current lead. Normally, with the conductance_matrix M found by kwant I can calculate transport properties (the voltages in vector V) as per the linear equation I = M V, given I = [0,0,0,0,1,-1].
Now I want to plot the current density corresponding to this transport state, but not sure how to do it.
For instance, with wf = kwant.wave_function(sys, energy), I can have all scattering wavefunctions wf(4) incoming from the current lead 4. But how to (or is it necessary to) reflect the above current configuration vector I we specified? I presume the current density may vary with different I vectors. Or maybe I misunderstand some basics?
Thank you!
-- Abbout Adel
Dear Adel Thank you for your helpful reply. I tried to understand i) why it works this way and ii) why it works 'qualitatively'. i) I presume the current density incoming from lead i (your 'current_i') can be calculated from scattering modes in wf(i), given wf = kwant.wave_function(sys, energy). Is your formula connected to the Landauer-Buttiker formula? I'm still a little confused why it works. ii) My naive guess is that there exist scattering states and (equilibrium) circulating states. Your formula accounts for the former and is justified because only the former directly affects transport. So neglecting the latter makes it 'qualitatively' the relevant current density plot. Look forward to your response. Thank you! Abbout Adel wrote:
Dear Zhang,
You can calculate the current for each lead separately (sum over the modes in the lead) and then multiply each current by the corresponding V_i that you obtained from the conductance matrix. Total_current=Sum (current_i *V_i).
You might claim that the dimension is not correct, but keep in mind that what we usually calculate is the transmission probability and thus we need to multiply by V (and the quantum of conductance) to get the current in the linear response theory.
This will give you, *qualitatively*, the current map in your system. (I am interested to see the result)
I hope this helps, Adel
On Wed, May 1, 2024 at 12:23 PM X.-X. Zhang <xiaoxiao.zhang@riken.jp> wrote:
Hello Kwant community, I am considering a 6-terminal sample with 0,1,2,3 the voltage leads and 4,5 the current lead. Normally, with the conductance_matrix M found by kwant I can calculate transport properties (the voltages in vector V) as per the linear equation I = M V, given I = [0,0,0,0,1,-1]. Now I want to plot the current density corresponding to this transport state, but not sure how to do it. For instance, with wf = kwant.wave_function(sys, energy), I can have all scattering wavefunctions wf(4) incoming from the current lead 4. But how to (or is it necessary to) reflect the above current configuration vector I we specified? I presume the current density may vary with different I vectors. Or maybe I misunderstand some basics? Thank you! -- Abbout Adel
Dear Zhang, The current is related to the conductance via the Landauer-Buttiker formula (kwant uses a different equivalent approach). In the linear regime, the current is proportional to the voltage in that lead. The reason I said that it will work "qualitatively" is because the voltage here is a free parameter (multiplying V and I with the same factor will always give you the same conductance matrix). V needs to be small enough such that the transmission doesn't change in the interval of energies eV (linear response theory.) The reason it works is that for such experiments, that is how someone would calculate the current: calculate the wavefunctions, then the current densities, then summing over the lead/system interface to get the current. I hope this helps, Adel On Mon, May 6, 2024 at 5:16 PM X.-X. Zhang <xiaoxiao.zhang@riken.jp> wrote:
Dear Adel Thank you for your helpful reply. I tried to understand i) why it works this way and ii) why it works 'qualitatively'.
i) I presume the current density incoming from lead i (your 'current_i') can be calculated from scattering modes in wf(i), given wf = kwant.wave_function(sys, energy). Is your formula connected to the Landauer-Buttiker formula? I'm still a little confused why it works.
ii) My naive guess is that there exist scattering states and (equilibrium) circulating states. Your formula accounts for the former and is justified because only the former directly affects transport. So neglecting the latter makes it 'qualitatively' the relevant current density plot.
Look forward to your response. Thank you!
Dear Zhang,
You can calculate the current for each lead separately (sum over the modes in the lead) and then multiply each current by the corresponding V_i that you obtained from the conductance matrix. Total_current=Sum (current_i *V_i).
You might claim that the dimension is not correct, but keep in mind that what we usually calculate is the transmission probability and thus we need to multiply by V (and the quantum of conductance) to get the current in
Abbout Adel wrote: the
linear response theory.
This will give you, *qualitatively*, the current map in your system. (I am interested to see the result)
I hope this helps, Adel
On Wed, May 1, 2024 at 12:23 PM X.-X. Zhang <xiaoxiao.zhang@riken.jp> wrote:
Hello Kwant community, I am considering a 6-terminal sample with 0,1,2,3 the voltage leads and 4,5 the current lead. Normally, with the conductance_matrix M found by kwant I can calculate transport properties (the voltages in vector V) as per the linear equation I = M V, given I = [0,0,0,0,1,-1]. Now I want to plot the current density corresponding to this transport state, but not sure how to do it. For instance, with wf = kwant.wave_function(sys, energy), I can have all scattering wavefunctions wf(4) incoming from the current lead 4. But how to (or is it necessary to) reflect the above current configuration vector I we specified? I presume the current density may vary with different I vectors. Or maybe I misunderstand some basics? Thank you! -- Abbout Adel
-- Abbout Adel
Dear Adel, Thank you for your reply. In the approach/formula you sketched (i: lead index, n: mode index for scattering state from any lead) J = Sum_i V_i Sum_n j_{i n} is the translation invariance of all voltages guaranteed? Since the current density j_{i n} of a particular mode from a lead is fixed, shifting all V_i by the same amount seems to affect the total current density J. Or do I misunderstand your formula? From: Abbout Adel <abbout.adel@gmail.com> Date: Sunday, May 12, 2024 at 0:22 To: X.-X. Zhang <xiaoxiao.zhang@riken.jp> Cc: kwant-discuss@python.org <kwant-discuss@python.org> Subject: Re: [Kwant] Re: Current density for given lead currents Dear Zhang, The current is related to the conductance via the Landauer-Buttiker formula (kwant uses a different equivalent approach). In the linear regime, the current is proportional to the voltage in that lead. The reason I said that it will work "qualitatively" is because the voltage here is a free parameter (multiplying V and I with the same factor will always give you the same conductance matrix). V needs to be small enough such that the transmission doesn't change in the interval of energies eV (linear response theory.) The reason it works is that for such experiments, that is how someone would calculate the current: calculate the wavefunctions, then the current densities, then summing over the lead/system interface to get the current. I hope this helps, Adel On Mon, May 6, 2024 at 5:16 PM X.-X. Zhang <xiaoxiao.zhang@riken.jp<mailto:xiaoxiao.zhang@riken.jp>> wrote: Dear Adel Thank you for your helpful reply. I tried to understand i) why it works this way and ii) why it works 'qualitatively'. i) I presume the current density incoming from lead i (your 'current_i') can be calculated from scattering modes in wf(i), given wf = kwant.wave_function(sys, energy). Is your formula connected to the Landauer-Buttiker formula? I'm still a little confused why it works. ii) My naive guess is that there exist scattering states and (equilibrium) circulating states. Your formula accounts for the former and is justified because only the former directly affects transport. So neglecting the latter makes it 'qualitatively' the relevant current density plot. Look forward to your response. Thank you! Abbout Adel wrote:
Dear Zhang,
You can calculate the current for each lead separately (sum over the modes in the lead) and then multiply each current by the corresponding V_i that you obtained from the conductance matrix. Total_current=Sum (current_i *V_i).
You might claim that the dimension is not correct, but keep in mind that what we usually calculate is the transmission probability and thus we need to multiply by V (and the quantum of conductance) to get the current in the linear response theory.
This will give you, *qualitatively*, the current map in your system. (I am interested to see the result)
I hope this helps, Adel
On Wed, May 1, 2024 at 12:23 PM X.-X. Zhang <xiaoxiao.zhang@riken.jp<mailto:xiaoxiao.zhang@riken.jp>> wrote:
Hello Kwant community, I am considering a 6-terminal sample with 0,1,2,3 the voltage leads and 4,5 the current lead. Normally, with the conductance_matrix M found by kwant I can calculate transport properties (the voltages in vector V) as per the linear equation I = M V, given I = [0,0,0,0,1,-1]. Now I want to plot the current density corresponding to this transport state, but not sure how to do it. For instance, with wf = kwant.wave_function(sys, energy), I can have all scattering wavefunctions wf(4) incoming from the current lead 4. But how to (or is it necessary to) reflect the above current configuration vector I we specified? I presume the current density may vary with different I vectors. Or maybe I misunderstand some basics? Thank you! -- Abbout Adel
-- Abbout Adel
participants (3)
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Abbout Adel -
X.-X. Zhang -
Xiao-Xiao Zhang