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The electron WF (psi left(vec{r}right)) in a quantum structure is determined by the Schrdinger equation,

$$begin{array}{c}nabla cdot left[-frac{{mathrm{hslash }}^{2}}{2{m}^{*}}nabla psi left(vec{r}right)right]+Uleft(vec{r}right)psi left(vec{r}right)=Epsi left(vec{r}right),end{array}$$

(1)

where (hslash) is the reduced plank constant, (vec{r}) the position, ({m}^{*}) the effective mass, (Uleft(vec{r}right)) the potential energy, and (E) the total energy of the electron. To reduce the DoF for the numerical WF solution, the Schrdinger equation is projected onto a mathematical space constituted by an optimal set of basis functions (or modes) (eta left(vec{r}right)). The POD process13,14 is chosen to generate the optimal set of modes, which maximizes the mean square of the WF projection onto the mode over multiple sets of collected WF solution data,

$$begin{array}{*{20}c} { leftlangle {left[ {frac{psi cdot eta }{{left| eta right|}}} right]^{2} } rightrangle ,} \ end{array}$$

(2)

where the brackets denote the average over the WF data sets accounting for the parametric variations in the electric field/potential and BCs, and the inner product (psi cdot eta) and the L2 norm of (eta) are given below

$$begin{array}{c}psi cdot eta =intlimits_{Omega}psi left(vec{r}right)eta left(vec{r}right)dOmega , and left|eta right|=sqrt{{int }_{Omega }eta {left(vec{r}right)}^{2}dOmega }.end{array}$$

(3)

This data-projection process ensures that the modes (eta (vec{r})) contain the maximum least squares (LS) information of the system embedded in the collected WF data and leads to an eigenvalue problem for the spatial autocorrelation function ({varvec{R}}left(vec{r},{vec{r}}^{prime}right)) of the WF data,

$$begin{array}{*{20}c} {intlimits_{{Omega^{prime}}} {{varvec{R}}left( {vec{r},vec{r}^{prime}} right)eta left( { vec{r}^{prime}} right)dvec{r}^{prime} = lambda eta left( {vec{r}} right)} ,} \ end{array}$$

(4)

where the eigenvalues (lambda) represent the mean squared WF information captured by each mode and ({varvec{R}}left(vec{r},{vec{r} }^{ prime}right)) is given by

$$begin{array}{*{20}c} {varvec{R}left( {vec{r},vec{r}^{prime}} right) = psi leftlangle {left( {vec{r}} right) otimes psi left( {vec{r}^prime } right)} rightrangle } \ end{array}$$

(5)

with (otimes) as the tensor product. Using the POD modes generated from (4), the WF can be expressed via a linear combination of these modes,

$$begin{array}{c}psi left(vec{r}right)={sum }_{j=1}^{M}{a}_{j}{eta }_{j}(vec{r}),end{array}$$

(6)

with (M) as the selected number of modes, where M determines the DoF and the simulation efficiency and accuracy, and ({a}_{j}) are weighting coefficients responding to the parametric variations in the simulation.

In a multi-dimension structure with a fine spatial resolution, the large dimension of R may be difficult to manage. The method of snapshots51,52 is applied to convert the eigenvalue problem in (4) from a discrete space domain with a large dimension of the NrNr to a sampling domain with a dimension of NsNs, where Nr and Ns are the numbers of spatial grid points and data samples/snapshots, respectively, and in general Ns< (M) and ({lambda }_{{N}_{s}}) is many orders smaller than ({lambda }_{1}) to minimize the numerical error resulting from the POD prediction since the theoretical LS error with an M-mode POD model is given as7,

$$begin{array}{c}Er{r}_{M,ls}=sqrt{{sum }_{i=M+1}^{{N}_{s}}{lambda }_{i}/{sum }_{i=1}^{{N}_{s}}{lambda }_{i}},end{array}$$

(7)

where the eigenvalues are arranged in descending order. Theoretically, (7) is valid only if the data quality is sufficient; namely the parametric settings in the POD simulation fall within the bounds of the parametric variations used in the training and the data for the training is accurate enough. Also, differently from POD applications to many other fields, the global POD approach in this study collects WFs from all NQ selected QSs and the modes thus represent the solution for WFs in all NQ selected QSs. As a result, (7) offers a theoretical prediction of the LS error averaged over all selected WFs, as will be discussed later in the demonstrations.

To close the POD simulation methodology, the Schrdinger Eq. in (1) is projected onto the POD modes via the Galerkin projection. The projection along the ith POD mode ({eta }_{i}) is given as

$$begin{array}{c} intlimits_{Omega }nabla {eta }_{i}left(vec{r}right)cdot frac{{mathrm{hslash }}^{2}}{2{m}^{*}}nabla psi left(vec{r}right)dOmega + intlimits_{Omega }{eta }_{i}left(vec{r}right)Uleft(vec{r}right)psi left(vec{r}right)dOmega -intlimits_{s}{eta }_{i}left(vec{r}right)frac{{mathrm{hslash }}^{2}}{2{m}^{*}}nabla psi left(vec{r}right)cdot dvec{S} =E intlimits_{Omega }{eta }_{i}left(vec{r}right)psi left(vec{r}right)dOmega .end{array}$$

(8)

Using (6) in (8), the quantum POD-Galerkin simulation methodology is thus represented by an (Mtimes M) eigenvalue problem for (vec{a}={left[{a}_{1}, {a}_{2}, ldots , {a}_{j},ldots , {a}_{M}right]}^{T}),

$${{varvec{H}}}_{eta }vec{a}=E vec{a},$$

(9)

where ({{varvec{H}}}_{eta }) is the Hamiltonian in the POD space denoted as

$${{varvec{H}}}_{eta }={{varvec{T}}}_{eta }+{{varvec{U}}}_{eta }+{{varvec{B}}}_{eta }$$

(10)

with the interior kinetic energy matrix expressed as

$${T}_{eta i,j }=intlimits_{Omega }nabla {eta }_{i}left(vec{r}right)cdot frac{{hslash }^{2}}{2{m}^{*}}nabla {eta }_{j}left(vec{r}right)dOmega ,$$

(11)

the potential energy matrix expressed as

$${U}_{eta i,j}=intlimits_{Omega }{eta }_{i}left(vec{r}right)Uleft(vec{r}right)psi left(vec{r}right)dOmega ,$$

(12)

and the boundary kinetic energy matrix given as

$${B}_{eta i,j}=-intlimits_{s}{eta }_{i}left(vec{r}right)frac{{hslash }^{2}}{2{m}^{*}}nabla psi left(vec{r}right)cdot dvec{S} .$$

(13)

The ith eigenvector ({vec{a}}_{i}) in (9) corresponds to the ith QS and eigenenergy ({E}_{i}), and the jth element of ({vec{a}}_{i}) in (6) represents the jth modes weight for the ith QS. With homogeneous Neumann and Dirichlet BCs, the boundary kinetic energy matrix elements in (13) vanish. For a structure with periodic BCs, the surface normal vectors at the periodic boundary surfaces in (13) cancel out causing the boundary kinetic energy matrix to vanish as well.

Note that the formulations given in (8)(11) apply to the projection of the Schrdinger equation onto any selected basis, including the Fourier plane-wave basis. The number of DoF needed to reach a desired accuracy in simulation depends on how well and concisely the selected basis functions portray the solution resulting from the potential variation and BCs.

In this work, the QDs are formed via the conduction band offset at the GaAs/InAs heterojunction; the material parameters in the simulations include effective masses ({m}_{GaAs}^{*}=0.067{m}_{o}) and ({m}_{InAs}^{*}=0.023{m}_{o}), and the band offset (Delta E=0.544mathrm{eV}). Training data of WFs needed for POD mode generation are collected from DNSs using the finite difference method53. Two test structures are investigated below. In each structure, the training process for the QD structure is described first followed by the demonstrations.

This test case first performs the training of POD modes via a set of single-component electric fields from two orthogonal directions in x and y. The domain of the underlining nanostructure shown in Fig.1 consists of a 44 grid of QDs, where each QD with a size of 4nm4nm is separated by 1nm from its adjacent QDs with a GaAs boundary spacer of 2.5nm on each side of the domain. DNSs of the Schrdinger equation for this QD structure are carried out using a fine resolution of 241241 (a grid size of 0.1nm in either direction) with homogeneous Dirichlet and Neumann BCs to collect the WF data for the first 6 QSs. A fine mesh is needed to offer accurate eigenenergies and WFs of several nearly degenerate QSs in this QD structure. To account for field variations, in addition to a zero-bias simulation, DNSs of the QD structure are performed at 8 electric fields in each of the x and y directions with linearly spaced magnitudes between -35kV/cm and+35kV/cm. The sampled WF data in the QD structure collected from the 2 sets of orthogonal electric fields, together with the unbiased WF data, are combined to generate one set of POD modes that represent all 6 trained QSs. With 6 selected QSs for each field, a total of 102 samples/snapshots of the WF data is implemented in the method of snapshots51,52 to solve the first 102 eigenvalues and POD modes from (4). This quantum POD-Galerkin model for (vec{a}) given in (9) is therefore constructed with the coefficients evaluated from (11) to (13) via its POD modes. After POD-Galerkin simulation using (9) to determine (vec{a}), post processing is needed to calculate the WF from (6).

(a) Underlining GaAs/InAs QD structure for the demonstration of the POD-Galerkin simulation methodology. (b) Eigenvalue for the WF data in descending order.

It is always informative to first observe the eigenvalues that offer information on the effectiveness of the trained POD modes. Eigenvalues of the first 6 modes shown in Fig.1b vary very little, which reveals that the first 6 POD modes tend to maximize the information on all the 6 selected trained QSs. This is because the global quantum POD approach50 implemented in this study generates POD modes that represent WFs in all NQ selected QSs, where NQ=6 in this case. Apparently, the most essential LS information is captured by the first NQ POD modes, and thus the eigenvalue decreases sharply beyond the NQ-th mode. By Mode 11, the eigenvalue drops more than 3 orders of magnitude from the first mode and continues decreasing drastically beyond the 11th mode. As shown in Fig.2a, the theoretical LS error estimated by (7) for the first 6 QSs becomes below 1% with 10 more modes. Moreover, one can see in Fig.1b that after the 52nd mode the eigenvalue drops nearly 16 orders of magnitude from the first value and starts decreasing very slowly due to the limited computer precision.

(a) POD LS errors for WFs in QSs 18. The average LS error (blue dashed line) is averaged over the 6 trained QSs, and the theoretical LS error (Er{r}_{M,ls}) is also for the first 6 QSs. (b) Absolute error of the estimated POD eigenenergy, where two insets include the eigenenergy obtained from the DNS, one in a table and the other in a band diagram along the diagonal direction of the QS structure from (0,0) to (24nm,24nm).

To test the validity of the POD-Galerkin methodology, a test electric field of (vec{E}=(25widehat{x}-10widehat{y}) mathrm{kV}/mathrm{cm}) is applied to the QD structure. The LS error of the POD WF in each state is first illustrated in Fig.2a, compared to the theoretical LS error and the average LS error. Figure2a shows that the LS errors of WFs in all trained QSs predicted by the POD simulation are near or below 1% with 11 or more modes and their errors are all below 0.37% when incorporating 13 modes. The POD model is in general more effective in the lower QSs. As can been seen, the LS errors in QSs 14 are all near or below 1% when 8 modes are included, and POD WFs in QSs 13 reach an error near or below 0.6% and 0.42% with 9 and 10 modes, respectively. Even the POD WFs in the untrained 7th and 8th states reach an LS error near 1.6% and 1.1% with 13 modes and stay near 1.25% and 0.95% with more modes, respectively. The theoretical error (Er{r}_{M,ls}) in (7) predicts the average POD LS error for the first 6 QSs quite well until 13 modes where the average POD LS error is as low as 0.3%; this indicates a good data quality for this test case. Due to the numerical errors and computer precision, the average POD LS error starts deviating from (Er{r}_{M,ls}) beyond 13 modes and eventually converges to a range between 0.28 and 0.32%.

In addition to the accuracy of the predicted WFs, Fig.2b illustrates the excellent agreement between the eigenenergies predicted by the POD-Galerkin methodology and by DNS of the Schrdinger equation, where the deviation for QSs 18 for the POD-Galerkin prediction ranges from 0.217% to 0.485%. It is interesting to observe the consistent increase in the deviation from the lower to higher QSs except for the nearly degenerate states, QSs 23, 56 and 78, where the eigenenergy differences in these degenerate states estimated from DNS are 2.733meV, 2.523meV and 0.233meV, respectively. The eigenenergy differences in these 3 sets of nearly degenerate states (2.728meV, 2.52meV, 0.231meV, respectively) predicted by the POD-Galerkin methodology are well preserved, including the untrained 7th and 8th QSs.

Contours and cross-sectional profiles of ({left|psi right|}^{2}) in each QS obtained from the POD and DNS are illustrated in Figs. 3 and 4, respective, where the maximum number of POD modes in each state is selected for its LS error near or below 1.5%. With a deviation near 1.5%, contours and profiles between these two approaches are nearly indistinguishable. The first mode of the POD-Galerkin solution always provides the mean of the training data. As seen in Fig.4, the one-mode POD model offers the unbiased (symmetric) ({left|psi right|}^{2}) in each QS since the training electric fields are symmetric about the zero field. Similar to Fig.2, it is shown in Figs. 3 and 4 that 68 modes are needed in QSs 14 for the POD methodology to achieve a very good agreement (near or below a 1.5% LS error) with the DNS, while 9 or 10 modes are needed in the higher states (QSs 5 and 6). For the untrained 7th and 8th states POD WFs to reach a similar accuracy, 13 modes are needed. Figures2, 3, 4 clearly illustrate that the POD-Galerkin methodology is capable of extrapolating the WFs and the eigenenergies with a high accuracy even in the untrained 7th and 8th states if a few more modes are included.

Contour of ({left|psi right|}^{2}) predicted by POD-Galerkin and DNS in upper and lower rows, respectively, in QSs 18. The horizontal and vertical red dashed lines reference the cross-sectional profiles of ({left|psi right|}^{2}) visualized in Fig.4.

Profiles of ({left|psi right|}^{2}) in each state along (a) x and (b) y via the red dashed lines indicated in Fig.3.

To further illustrate the extrapolation capability, an electric field of (vec{E}=50 widehat{x}-10 widehat{y}) (kV/cm) is applied to the QD structure. With the x-component field beyond the maximum training field (35k/cm), as expected in Fig.5a, the error in each state declines more slowly and more modes are needed to reach a similar accuracy to the previous test case presented in Fig.2. For example, 6, 7, 6, 8, 10, 9, 13 and 13 modes are needed from QSs 1 to 8 in sequence to reach an error below 2% in Fig.2, while in Fig.5 for this case beyond the training field, 6, 8, 9, 12, 11, 12, 14 and 15 modes are needed. Due to the applied field greater than the training fields, the data quality in this case is not as good as that in the previous case. As a result, the average LS error of the POD approach in Fig.5 is slightly greater than that in Fig.2 beyond 5 modes and does not follow the theoretical LS error (estimated based on the training data for the first 6 QSs) as close as the previous case. Nevertheless, compared to the interpolation case presented in Figs. 2, 3, 4, the POD-Galerkin model still offers a very good prediction in the extrapolation case if a few more modes are included. Even in the untrained 7th and 8th QSs beyond the training fields, the POD LS error in QS 7 is as small as 2.2% and 1.67% with 13 and 15 modes, respectively, and 2.67% and 1.6% with 13 and 15 modes in QS 8, respectively. Excellent accuracy of the POD eigenenergy is also observed at this higher field, as displayed in Fig.5b, where the deviation for QSs 18 for the POD prediction ranges from 0.213% to 0.463%. Unlike the lower field case, only QSs 7 and 8 are nearly degenerate in this case, whose eigenenergy difference estimated in DNS is as small as 0.422meV and is 0.419meV predicted by the POD-Galerkin model.

(a) POD LS errors for WFs in QSs 18. The average POD LS error is averaged over the 6 trained QSs, and the theoretical LS error (Er{r}_{M,ls}) is identical to that in Fig.2. (b) Absolute error of the estimated POD eigenenergy with two insets of the eigenenergy obtained from the DNS, one in a table and the other in a band diagram along the diagonal direction of the QS structure from (0,0) to (24nm,24nm).

Contours and profiles of ({left|psi right|}^{2}) along x and y directions in several QSs predicted by the POD-Galerkin simulation and DNS are compared in Fig.6a and b, where the maximum number of modes is chosen for the POD LS error near or below 2% in QSs 1, 2 and 6 and 2.5% in the untrained states 7 and 8. Accurate results for both WFs and eigenenergies presented in Figs. 5 and 6 highlight the capability of the POD-Galerkin methodology even in the extrapolation situations beyond both training fields and trained QSs, which is difficult to achieve in typical machine learning methods. Unlike most machine learning methods, the POD-Galerkin simulation methodology incorporates the first principles, as described in (8), by projecting the Schrdinger equation onto the POD modes. This projection offers a clear guideline to accurately predict the solution in response to the field variations even beyond the training fields in the untrained QSs. It is also worth noting that, even though the WF data were collected for electric fields in x and y directions separately, the POD modes generated from the combined data sets are able to accurately predict the WFs in each state subjected to an electric field in any direction.

(a) Contours and (b) cross sectional profiles of ({left|psi right|}^{2}) in QSs 1, 2, 6, 7 and 8. In (a), the upper and lower rows show ({left|psi right|}^{2}) obtained from POD-Galerkin and DNS, respectively. The dashed horizontal and vertical lines in (a) reference the profiles visualized in (b) along x and y in upper and lower rows, respectively.

The next case extends the quantum POD-Galerkin methodology to a nanostructure with internal potential variation and periodic BCs. The underlining nanostructure of a 33 grid of GaAs/InAs QDs is shown in Fig.7a. Each QS is 4nm4nm in size adjacently separated by 1.5nm with a 1.25nm GaAs boundary spacer on each side. In addition, a potential energy profile of 5 pyramids each with a 4.8nm4.8nm base shown in Fig.7b is superposed on the band energy of the QD structure. The potential energy profile with variation of each pyramid height is chosen in this investigation to demonstrate the proposed physics-informed learning algorithm derived from the first principles to predict the WFs in response to the variation of the internal potential profile with periodic BCs, which appear in many applications in quantum nanostructures and materials, including DFT simulations36,45,47,48,49.

(a) GaAs/InAs QD structure with periodic BCs. (b) Potential energy with a profile of 5 pyramids applied to the QD structure in (a). (c) Eigenvalue of WF data in descending order.

To account for variation of the internal potential in the WF data collection, DNS of the QD structure described in Fig.7a and b is first performed with only one of the five pyramids at a time. Five different heights for each pyramid potential energy varying equally from 0.07 to 0.35eV are included in DNSs. When using only these 25 pyramid potential samples to train the POD modes, it was found that the LS error of the POD-Galerkin model is relatively large, as expected due to the poor data quality. More specifically, the influences among the five pyramids are not included in the collected data. To thoroughly account for the influences among different heights of the pyramid potentials, if five different heights are selected for each pyramid, a combination of all different heights for the five pyramids should be included in the DNSs to collect the WF data. This would lead to an enormous number of additional samples, i.e., 55=3125, which requires an immense computational effort to collect the WF data from DNSs and evaluate the POD Hamiltonian elements in (11)(13). To minimize the training effort, five pyramids varying together with the same (five) heights from 0.07 to 0.35eV in DNS are used to collect just one additional sample of WF data with the hope that the physical principles enforced by the Galerkin projection would intelligently predict the influences among different heights of pyramids. As will be seen later, this setting actually works reasonably well.

In each simulation, only the WFs in the first 6 (NQ) QSs are collected. Using data collected from the QD structure with variations of five pyramid potentials, together with the structure without any pyramid, there are in total 186 samples/snapshots (i.e., (65+1)6) of WF data collected to solve POD modes and eigenvalues from (4) via the method of snapshots51,52. Similar to Fig.1b, the eigenvalues displayed in Fig.7c remain nearly unchanged for the first NQ modes, and decrease sharply beyond the NQ-th mode. The eigenvalue also reveals a more than 3-order drop from the first to the 11 modes and decreases considerably more slowly after dropping 16 orders of magnitude from the first mode due to the computer precision.

In this demonstration, the heights of Pyramids 1 to 5, as labeled in Fig.7b, are randomly selected as 0.23eV, 0.3eV, 0.14eV, 0.12eV and 0.25eV in sequence. The average LS error of the WF predicted by the POD-Galerkin model shown in Fig.8a for the first 6 QSs agrees quite well with the theoretical LS error estimated in (7) until 11 modes although not as well as that in Fig.2a. This indicates that the data quality associated with Fig.2a is better than that for this test case. However, the simple training in this case to account for influences among the pyramids with unequal heights still offers very accurate prediction of WFs (near 1% LS error) in the trained (first 6) stateswith just 12 to 13 modes. Because of a better data quality, the POD-Galerkin model for the external field case in Fig.2b is more effective than the model for the periodic BC case. For example, to reach an LS error near 1% for all trained states, 10 or 11 modes are needed in Fig.2a while 13 modes in Fig.8a. When using 13 or more modes, the average LS error near 0.3% is observed in Fig.2a while 0.78% in Fig.8a. Also note that the LS errors of the untrained 7th and 8th states in this periodic BC case are as small as 1.5% and 2.25% with 13 modes, respectively, reduce to 1.2% and 1.4% with 14 modes, and below 1% beyond 16 modes. The eigenenergy predicted by the POD-Galerkin simulation in Fig.8b is as accurate as that in Fig.2b, and QSs 2 and 3 are nearly degenerate with an eigenenergy difference of 1.249meV calculated in DNS, as shown in the insets. The POD-Galerkin model however predicts a difference of 1.269meV.

(a) POD LS errors for WFs in QSs 18. The average LS error is averaged over the 6 trained QSs, and the theoretical LS error (Er{r}_{M,ls}) is also for the first 6 QSs. (b) Absolute error of the estimated POD eigenenergy with two insets including the eigenenergy obtained from the DNS, one in a table and the other in a band diagram along the diagonal of the QS structure in Fig.7 from (0,0) to (17.5nm, 17.5nm).

Contours and profiles of ({left|psi right|}^{2}) are illustrated in Fig.9a and b, respectively, for QSs 1, 2, 4, 6 7 and 8, where the maximum number of POD modes in each state is selected for the LS error below 1.5%. Similar to results presented in Figs. 2, 3, 4, to reach a high accuracy, more modes are needed when the eigenenergy is closer to the QD barrier energy. Again, the result of the first mode represents the mean of the training WF data which are symmetric about the center of the QD structure, as revealed in the one mode POD WF solution for all states. In this demonstration, the WF in QS 4 appears to be nearly symmetric and thus can be well represented by the one-mode POD solution (see the QS 4 profiles in Fig.9b along both x and y directions) with just a 1% LS error, as shown in Fig.8a. The QS-4 LS error decreases to 0.55% between 4 and 15 modes and drops to 0.24% beyond 15 modes. This indicates that the QS-4 WF is not perfectly symmetric, and 4 or more modes are needed if higher accuracy is desired. For the untrained 7th and 8th states POD WFs, high accuracy can still be achieved when using 13 or 14 modes.

(a) Contours and (b) cross sectional profiles of ({left|psi right|}^{2}) in QSs 1, 2, 4, 6, 7 and 8. In (a), the upper and lower rows show ({left|psi right|}^{2}) obtained from POD-Galerkin and DNS, respectively. The dashed horizontal and vertical lines in (a) reference the profiles visualized in (b) along x and y in the upper and lower rows, respectively.

For periodic functions, the Fourier basis is usually assumed to approximate the solution of the problem. An additional demonstration is therefore illustrated using FPWs as the basis functions in (8) and (9) to derive an FPW model and to perform simulation of the periodic structure given in Fig.7a with the pyramid potential variation in Fig.7b. The LS errors against the DNS are shown in Fig.10, where even with periodic BCs a large number of modes of FPWs is still needed to reach a reasonably small error due to complicated internal pyramid potential variation. For example, when using 40 modes, the minimum error is near 5.4% in QS 1 and the maximum is near 11% in QS 7. Using 225 modes, the minimum error reduces to 2.78% in QS 1 and the maximum is near 5.51% in QS 3. Beyond 225 modes, the LS error induced by the FPW approach continues decreasing very slowly. Nevertheless, this demonstration further validates the effectiveness of the POD-Galerkin methodology in which use of 1314 modes for this periodic structure offers an LS error of WFs 5 to 11 times smaller in each of all the trained QSs than the FPW approach using 225 modes. For the untrained 7th and 8th states with 1314 modes, POD-Galerkin leads to an LS errors 2 to 3.5 times smaller than the FPW approachusing 225 modes. The error for the eigenstate energy predicted by the FPW approach given in Fig.10 in each state is approximately 4.5 to 7 times as large as that in the POD-Galerkin model given in Fig.8b.

LS errors of WFs in QSs 18 for the first 225 modes, resulting from the FPW approach. The insets include a closer look at the LS errors for the first 40 modes and a table of the absolute error of the eigenenergy estimated from the plane-wave approach. The eigenenergy obtained from DNS is given in an inset of Fig.8b.

To further validate the extrapolation ability of the POD-Galerkin methodology for the QD structure with pyramid potential variation, Pyramids 15 are selected as 0.23eV, 0.3eV, 0.5eV, 0.12eV and 0.6eV in sequence, where heights of Pyramids 3 and 5 are evidently higher than the maximum training height (0.35eV). For this demonstration, the average LS error shown in Fig.11a becomes slightly larger than that in Fig.8a and moves away from the theoretical LS error. Instead of 9 or 10 modes to reach an average error near 2% in Fig.8a, 13 modes are needed for this extrapolation case. Nevertheless, the POD-Galerkin methodology still offers an accurate prediction of the WFs in this case with a small number of DoF, especially in the lower QSs. For example, the LS error is all below 2% with 9 modes in QSs 13, 1.2% in QSs 14 with 15 modes and 0.6% in QSs 14 when using 17 modes. It is interesting to observe that somehow the untrained 7th QS reaches an error smaller than that in the 5th and 6th states with 13 or more modes. The LS error in the higher states (QSs 58) is all lower than 2% beyond 13 modes. The error of the POD eigenenergy in this case with higher pyramid potential heights shown in Fig.11b is very similar to what was observed in Fig.8b. Due to the larger variation of internal pyramid potential in this periodic structure, accuracy of the FPW approach further deteriorates and becomes considerably worse than the POD-Galerkin model. The FPW approach is therefore not presented for this case.

(a) POD LS errors for WFs in QSs 18. The average LS error is averaged over the 6 trained QSs, and the theoretical LS error (Er{r}_{M,ls}) is also for the first 6 QSs. (b) Absolute error of the estimated POD eigenenergy with two insets including the eigenenergy obtained from the DNS, one in a table and the other in a band diagram along the diagonal of the QS structure in Fig.7 from (0,0) to (17.5nm, 17.5nm).

Figures12a and 11b illustrated the contours and profiles of ({left|psi right|}^{2}), respectively, for QSs 1, 2, 3, 6 7 and 8, where the maximum number of POD modes in each state is selected for the LS error below 2%. The contours and profiles of ({left|psi right|}^{2}) derived from the POD-Gakerlin prediction and DNS are nearly identical. Instead of QS 4 in Fig.9, the QS-3 WF in this case is nearly symmetric, as shown in Fig.12; as a result, the one-mode POD WF in QS 3 leads to a 2% LS error. More modes are needed to compensate for the non-symmetric portion of the WF, and the LS error reduces to 1.16% with 4 modes, 0.91% with 6 modes and 0.43% with 16 mods.

(a) Contours and (b) cross sectional profiles of ({left|psi right|}^{2}) in QSs 1, 2, 3, 6, 7 and 8. In (a), the upper and lower rows show ({left|psi right|}^{2}) obtained from POD-Galerkin and DNS, respectively. The dashed horizontal and vertical lines in (a) reference the profiles visualized in (b) along x and y in the upper and lower rows, respectively.

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