# The ASA layer Green's function program lmpg

This package implements a modification of the ASA Crystal Green’s function, an implemention of the Atomic Spheres Approximation adapted to layer geometry. In the layer geometry periodic boundary conditions are used in two dimensions, but not the third. Along the third axis, there is an “active” cladded by semi-infinite leads on each end, whose potential is assumed to repeat periodically in a half-plane. Thus **lmpg** is the layer counterpart to the crystal code **lmgf**.

A layer geometry makes transport calculations possible. The Landauer-Buttiker formalism has been implemented, including a capability for nonequilibrium Green’s functions.

There a tutorial showing how to translate a system set up for periodic boundary conditions into one with a geometry of the form **lmpg** uses.

*Note:* This documentation was translated from a latex document, and needs some refinement.

### Table of Contents

- Definition of the layer geometry
- Green’s functions for the end regions
- Self-consistency and charge neutrality
- Electrostatics
- Input for lmpg
- Program operation

### Definition of the layer geometry

**lmpg** runs in much the same way as **lmgf**, except that **lmpg** has a `special direction’ which defines the layer geometry. The Green’s function *G* is generated in real space along this axis, and in *k* space along the other two. Along this axis the system is divided into a “left” semi-infinite region , a “right” semi-infinite region , and an “active” or “scattering” region .

.

The special axis is specified by the third primitive lattice translation vector (token **PLAT**). For the other two axes, Bloch sums are taken in the usual way. Thus for each **k** in a plane defined by the first two reciprocal lattice vectors, the hamiltonian becomes one-dimensional and is thus amenable to solution in order-*N* time in the number of layers, as defined below.

The active region is further divided into “principal layers” 0…*N*−1. The left region is designated principal layer −1; the right layer *N*.

.

Layers −1 and *N* are replicated to the left and right, respectively, extending outside of the region shown. Both are replicated implicitly an infinite number of times to form the semi-infinite leads. In the “left” case a replica of each site belonging to PL −1 is translated to the left by multiples of **PLATL**; similarly, a replica of each site belonging to PL *N* are tranlated to the right by multiples of **PLATR**. PL −1 and *N* are not specified in the input file; their structures are taken from PL 0 and *N*−1, translated respectively by **PLATL** and **PLATR**.

If the left semi-infinite region covered all space (“crystal- or bulk-like”), it would be periodic with a unit cell (**PLAT(1)**, **PLAT(2)**, **PLATL**); similarly the right layer periodic with unit cell (**PLAT(1)**, **PLAT(2)**, **PLATR**). **PLAT** is read in the same way all programs read structural data, while **PLATL**, **PLATR** are read from the PGF category.

The central or scattering region ( or ) should be constructed so that PL −1 and *N* are are already bulk-like. Thus charge densities of the PL 0 and *N*−1 should rather closely resembles densities of the semi-infinite (bulk) leads. This point is discussed later.

While **lmpg** is similar in many respects to **lmgf** there is some additional complexity as a consequence of special treatment of the third dimension. To begin with, the semi-infinite layers must be specially treated in several contexts:

Surface analogs of for the end regions are needed to supply the boundary conditions for the embedding Green’s function. This is discussed later.

The left and right semi-infinite regions must be treated as periodic crystals for the special purpose of constructing the self-consistent densities and Fermi levels in those regions.

**lmpg**requires special treatment for the electrostatics joining the three regions.

### Green’s functions for the end regions

By its construction the PL of each end ( and ) region would, if repeated throughout all space, constitute a periodic solid. **lmpg** has a special branch for the and PL that enable it to generate the self-consistent left- and right- bulk *G* for each corresponding periodic solid.

This is needed to make the potential of each end region. Note that it is assumed to be bulk-like. The *G* should be the same as that generated by 3D Green’s function program **lmgf**, except that this *G* has a mixed-real and k-space representation, and separate Green’s functions and potentials are needed for each end region. Also, owing to the mixed-real and k-space representation, the methodology for constructing *G* is different. We will call the periodic solid of repeating PL the “bulk” crystal of the region; similarly for the PL. Thus, there is a well-defined “bulk” *G* and potential for the PL, and one also for the PL. A self-consistent density generated by **lmpg** for these regions should also be self-consistent when calculated by **lmgf**. In practice this becomes the case in the limit of a fine **k** mesh: **k** convergence is different in the two cases because **lmpg** uses only two dimensions rather than three.

You must carry out this step prior to calculations of the active region. Select this branch with **PGF_MODE=2** (also see **MODE=2** below).

Once the and bulk potentials are made, the surface Green’s function can be computed, which is needed to supply the proper boundary conditions to generate the GF in the embedding region.

### Self-consistency and charge neutrality

Like any band method, generation of the output density requires integration to the Fermi level, which is determined by charge neutrality. Also in general, the electrostatic potential is determined up to an arbitrary constant. You can specify the Fermi level which determines the constant, or vice-versa. In the band code, we supply (or assume) the constant, and find the corresponding Fermi level. In **lmgf** and **lmpg** we indirectly fix this constant potential by specifying the Fermi level as an input. Either could serve as the independent variable for **lmgf**, but not **lmpg** because the Fermi levels of the end regions are fixed by bulk calculations.

Requirement of charge neutrality fixes the constant potential shift. Note that the GF have a complication not present in a band program, where the entire spectrum is computed: the constant shift must be determined by an iterative procedure. In the crystal GF case, it is iteratively determined by a Pade approximant (see lmgf documentation). Metals and nonmetals are distinguished in that in the latter case, there is no DOS in the gap and therefore the Fermi level (or potential shift, if the Fermi level is specified) cannot be specified precisely. As we will describe, something similar happens in the layer case, but it is a little more complicated. The Fermi level and constant potential are stored in array **vshft**, and permanently on disk in file **vshft. ext** described below and in the documentation found in the source code

**iovshf.f**.

Actually, **lmpg** is more general and can accommodate potential shifts at separate sites. This is useful in non-self-consistent or limited self-consistent calculations.

The layer geometry is complicated by the partitioning into three distinct regions. Self-consistency, and therefore the determination of potential, proceeds differently depending on whether one is computing the potential for the bulk and (**PGF_MODE=2**) or for the layer system (**PGF_MODE=1**).

Computing the bulk potential for the and regions (MODE=2), is quite analogous to the crystal case, albeit for two independent regions and . Periodic boundary conditions are assumed separately; in each case the end layer is assumed to be a periodic solid and the (periodic) Green’s function is computed for it. Thus the potential in each end layer may be shifted independently by a constant shift or . Self-consistency proceeds analogously to the crystal GF, independently for the regions and . Constant potential shifts are determined independently for each layer in the same way as in the crystal GF code; the potential shift is computed and satisfies charge neutrality for the corresponding periodic solids; see lmgf documentation for further details. If the () PL is a metal, the constant potential shift is not adjustable, because the Fermi level is specified at the outset. Note, however, that the Fermi level is only sharply defined for a metal; thus it is important to distinguish metal and nonmetal cases independently in the two layers. You can set them with tokens LMET= and RMET= in the PGF category; these tokens play the role of **BZ_METAL=** for the crystal codes.

Charge neutrality in the layer case is more complicated. It should be satisfied independently in each of the , and regions. In practice, we *assume* that the density in the and end regions is bulk-like and does not change once it has been calculated (MODE=2). Thus changes in the density are confined to the region. The region has to be charge-neutral because the and are already neutral, and the entire system must be neutral. The program proceeds by finding a shift that satisfies charge-neutrality in the region and doesn’t worry about the rest. This is reasonable since we assume at the outset that the region has enough PL near and to allow the density to be bulk-like, no charge should spill into the and regions by construction. Thus, when computing the Green’s function of the system in practice, the layer code selects the shift so as to eliminate the deviation from charge neutrality in the region, following the method of the crystal code **lmgf**. Only sites in the region are shifted by ; the potentials and charges in the and regions are left untouched. Once is found and the corresponding Green’s function is generated, **lmpg** returns to the sphere program where it computes the potential functions for the updated sphere charges and moments, computes the electrostatic potential (see below) for the system, and begins another pass in the self-consistency cycle.

Because of deviations from self-consistency, and also because of finite-size effects discussed above, there can be some deviation from charge neutrality in the end regions. **lmpg** will generate the GF in each end PL, so that the deviation from neutrality may be computed. (You must have the ‘bigemb’ option set for **lmpg** to generate this information.) The sphere charges are shown in the table headed by the lines:

```
PGFASA: integrated quantities from G
PL D(Ef) N(Ef) E_band 2nd mom Q-Z
```

and deviations of the end regions from charge neutrality are summarized at the end of this table in lines similar to the following:

```
Deviations from charge neutrality:
Left end layer 0.003965
Right end layer 0.003965
```

However, **lmpg** does not use this information; instead it keeps the densities as computed for the bulk and crystals. If these charges are not small, your active region should be enlarged.

### Electrostatics

At self-consistency there is a unique potential defined by the electrostatic potential from the charges at each site and a global constant potential shift that shifts the entire system. This shift makes each region charge neutral and possesses within the region a dipole that will exactly align the Fermi levels in the and regions, as we now describe.

For now let’s restrict ourselves to the case when both and regions are metals. We can compute electrostatic potentials in the and regions by two different constructions:

Electrostatic potentials in and may be computed as in MODE=2, that is by assuming and are bulk solids with periodic boundary conditions in the respective and regions. In each case the potential is completely fixed by charge neutrality. Electrostatic potentials in and may also be computed from solution of the potential of the entire system. This potential is adjustable up to some constant shift. In practice these two constructions produce different potential in the and regions. (Indeed, one might choose the constant shift that 'best reconciles' the mismatch in these two constructions. Versions of lmpg earlier than 6.14 did something like this. The present version chooses the constant shift so as to satisfy charge neutrality in the region, as was discussed in the previous section.)

These potentials might be mismatched for two distinct reasons. One error can arise because the density is not self-consistent. More exactly the density doesn’t possess the requisite dipole, so that a global constant shift that 'best reconciles' the potential mismatch as constructed by the two methods above is different from the one that ‘‘best reconciles’’ the mismatch on the right. Finite-size effect (a region with insufficient PL near the end regions) is another source of error.

Recall that **lmpg** freezes the potentials in the and regions. Thus, only the central region is affected by the shift because the end PL shouldn’t be shifted at all. **lmpg** does print out the electrostatic potentials computed by the two different constructions, and summarizes the deviations in a table similar to the following:

```
Deviations in end potentials:
region met <ves>Bulk <ves>layer Diff RMS diff
L T 0.055210 0.058192 0.002982 0.011265
R T 0.055210 0.058192 0.002982 0.011265
RMS pot difference in L PL = 0.000127 in R PL = 0.000127 total = 0.000127
vconst that minimizes potential mismatch to end layers = 0.058835
vconst is now (estimate to satisfy charge neutrality) = 0.061817
difference = -0.002982
```

The top table compares the average electrostatic potential in an end region computed from the bulk geometry, and computed in the (… | | …) geometry. The later numbers compare the potential that ‘‘best reconciles’’ the the methods of computation to the potential shift the program will actually use. Note that the potential shift you actually should use is the one that meets charge-neutrality in the region; indeed **lmpg** will find this shift on its own if you invoke it in self-consistent mode (sec XX).

When getting started, this table gives you a pretty reasonable guess at the proper choice of the constant shift . That is, you might set as the one that minimizes potential mismatch to end layers. You can adjust if you start out with the choice by invoking **lmpg** in an interactive mode, or by setting file **vshft** appropriately. If you do so, you will alleviate some of the burden on lmpg in determining this shift.

If the potential in either the or the differs significantly (see case and are both metals, below), or if there is significant deviation from charge neutrality in the end PL, the user should enlarge the embedding region, as the end PL are not sufficiently bulk-like. This difference should vanish at self-consistency if you construct the embedding region with PL near the end regions similar to those of the last region. Typically having 2 PL (say , and does a pretty good job keeping the discrepancies between the bulk- and layer potentials small.

Now we must distinguish between metal and nonmetal cases. If either end layer is a nonmetal, its potential can shift by a constant without affecting charge neutrality. Therefore, if either or regions is a nonmetal, there can be a constant shift in that region (change in band offset). Consider the following cases:

and are both metals. No potential shifts are allowed in the end layers. In the self-consistency cycle (MODE=1) the program checks for deviations from charge neutrality and adjusts the potential in the region until neutrality is achieved. However, the density resulting from this Green’s function will generate charges and electrostatic potentials at sites in the and layers. As mentioned above, potentials computed from the (… | | …) system will reveal some differences relative to the electrostatic potentials when computed using just the or charges and a geometry for infinitely repeating and layers. The potentials calculated these two ways is printed out, as described above.

Only one of or is a metal (LMET=f and RMET=t or vise-versa). Now the nonmetallic end region can shift its potential by a constant to best align to the potential computed from the (… | | …) system. We choose the constant in the nonmetallic PL that best aligns and ; that is, that minimizes the RMS difference and .

Neither the or is a metal (LMET=f and RMET=f). If the region is a metal (specified by BZ METAL=) the global potential shift should conform to the shift in the region. Both endpoints must be allowed to float (there are two distinct band offsets). If no region is a metal, i.e. if there is no DOS at the Fermi level, the system should already be charge neutral and no shifts should be required.

#### Electrostatics in layer geometry

The correct procedure to construct electrostatics in a layer geometry is to carry out 2D Ewald summations for each PL, and add up the contributions from all PL. This is described in the paper by Parry Surf. Sci. 49, 433. Since we haven’t yet made 2D Ewald sums for this program, we use a trick.

We compute the electrostatics via a 3D Ewald summation of the following supercell:

.

It assumes periodic boundary conditions of period in the layer direction `PLAT(3) + 2×(PLATL + PLATR)`. There is an artificial periodicity in the boundary condition on this axis that has to be removed to make it similar to the 2D Ewald summation. There is also a small error in that the electrostatics from layers `3×PLATL` and `3×PLATR`, …, are omitted.

### Input for lmpg

This section describes input specific to **lmpg**. Much of the Green’s function specific part of **lmpg**’s input is the same as input to the crystal Green’s function] **lmgf**, e.g. the **BZ_EMESH** tag.

Each site must be assigned a 'principal layer index’ to tell **lmpg** which principal layer a site belongs to. Each member of category **SITE**, should have a token **PL**:

Sites with the same PL index are grouped together into a single principal layer.

*Note:* the principal layers should be large enough such that the range of the hamiltonian connects only adjacent PL. The range of the hamiltonian is fixed by the range of the structure; you can set it manually with **STR_RMAX**.

There is a **lmpg**-specific category, which includes the following:

```
PGF MODE=#
PLATL=#1 #2 #3
PLATR=#1 #2 #3
GFOPTS=options
SPARSE=#
```

**MODE**, **PLATL**, and **PLATR** are required inputs.

**MODE** tells **lmpg** what to do:

**#=0:**do nothing after the atomic part**#=1:**calculate the diagonal part of*G*, layer-by-layer. This is the appropriate mode for self-consistent calculations.**#=2:**left- and right layer “bulk” Green’s functions, equivalent to crystal Green’s functions for the left and right leads. This mode is designed to generate a self-consistent densities for left- and right- leads as though they were crystalline, with periodicity given by**PLATL**or**PLATR**.**lmpg**must be run in this mode first.**#=3**find k(E) for left bulk, meaning the left end layer as if it were repeated an infinite number of times in both directions. This uses a special trick (see PRB 39, 923 (1989)) to find the wave numbers of the left bulk*G*corresponding to a given energy. This mode is illustrated for a free-electron test case in the the step potential tutorial.**#=4**find k(E) for right bulk.**#=5**Calculate conductance from the Landauer-Buttiker theory. /tutorial/lmpg/steplmpg/#transmission This mode is illustrated in a tutorial for a simple flat potential with a barrier of finite width [the step potential tutorial].**#=7**Calculate reflectance from the Landauer-Buttiker theory**#=8**Calculate spin torques

**PLATL** and **PLATR** specify and specific parts of the lattice vectors for the (… | | …) geometry

**SPARSE**=T tells **lmpg** to compute *G* using LU decomposition. The default is to compute *G* layer-by-layer via a Dyson equation (**SPARSE**=F)

#### Potential shifts

File **vshft. ext** holds information about potential shifts as described in previous sections. The global shifts are contained in the first line and keep information about Fermi level, the global constant shift, and the shifts at the and end regions needed to match the Fermi level. The syntax for the first line is

```
ef=# vconst=# vconst(L)=# vconst(R)=#
```

You can optionally add site-dependent potential shifts. After the first line, add a line **site shifts** followed by as many lines as desired, one line per site shift, e.g.:

```
ef=.03 vconst=-.01 vconst(L)=.02 vconst(R)=.03
site shifts
3 .1
4 .2
```

### Program operation

**lmpg** starts by creating the left surface, and then proceeds ‘left-to-right’ layer-by-layer to generate the surface GF for PL 0,1,2,3,… until the rightmost layer is reached. At that point the right surface GF is generated, and the crystal GF is generated by embedding between the left- and right- surface GF.

Then using Dyson’s equation, **lmpg** proceeds layer-by-layer ‘right-to-left’ to generate the crystal GF from the surface GF on the left and the crystal GF on the right. This is done for each energy and k-point in the two-dimensional Brillouin zone.

#### Use in conjunction with other programs

You can use plane-analysis tool **lmplan** to analyze charge distributions by plane, and also use it to create a **site file** to facilitate making of lattices with periodic boundary conditions so that you can run **lm** and **lmgf** for comparison.

*Note:* This documentation is outdated and needs revision.

At present **lmpg** cannot make the static response function, as **lmgf** can; this can vastly improve effiency for self-consistency. However, if you make the response function using **lmgf**, you can use it with the **lmpg** program. Here is an example taken from the directory of **pgf/tests/copt**.

Invoke this command

```
lmplan copt -vpgf=1 -cstrx=file -vlmf=f -vnk1=6 --pr31 -vnit=10 -vgamma=f -vsparse=0
```

At the prompt, type

```
wsite -pad abc
```

**lmplan** creates a site file named ‘abc.copt’ suitable for use with programs lm and **lmpg**. It uses the padding layers as buffer layers. Conveniently, it satisfies periodic boundary conditions. To verify this, try

```
cp abc.copt site.copt
lmchk copt -vpgf=0 -cstrx=file -vlmf=f -vnk1=6 -pr31 -vnit=10 -vgamma=f -vsparse=0
```

Now you can make a suitable ASA static response function suitable for the layer code with

```
lmgf copt -vpgf=0 -cstrx=file -vlmf=f -vnk1=6 -pr31 -vnit=10 -vgamma=f -vsparse=0 -iactiv -time=5 -vscr=1
```

The *q*=0 response function is written to file **psta. ext**. (This file has already been created and sits in pgf/test/copt.) Once the psta is created, it can greatly facilitate convergence to self-consistency. Try running, for example, the test script

```
pgf/test/test.pgf -usepsta copt 5
```

will use the psta file to assist convergence when you use the switch `-usepsta`

. Look in particular at the RMS DQ, viz:

```
grep RMS out.lmpg.self-consistent
```

```
PQMIX: read 0 iter from file mixm. RMS DQ=1.84e-3
PQMIX: read 1 iter from file mixm. RMS DQ=1.59e-4 last it=1.84e-3
PQMIX: read 2 iter from file mixm. RMS DQ=3.30e-4 last it=1.59e-4
PQMIX: read 3 iter from file mixm. RMS DQ=4.66e-5 last it=3.30e-4
```

If you invoke it in the usual way, viz without `-usepsta`

:

```
PQMIX: read 0 iter from file mixm. RMS DQ=1.44e-3
PQMIX: read 1 iter from file mixm. RMS DQ=6.41e-3 last it=1.44e-3
PQMIX: read 2 iter from file mixm. RMS DQ=2.22e-1 last it=6.41e-3
PQMIX: read 3 iter from file mixm. RMS DQ=3.05e-2 last it=2.22e-1
PQMIX: read 4 iter from file mixm. RMS DQ=3.01e-2 last it=3.05e-2
PQMIX: read 5 iter from file mixm. RMS DQ=2.66e-2 last it=3.01e-2
PQMIX: read 6 iter from file mixm. RMS DQ=4.15e-2 last it=2.66e-2
PQMIX: read 7 iter from file mixm. RMS DQ=4.92e-2 last it=4.15e-2
PQMIX: read 8 iter from file mixm. RMS DQ=4.40e-2 last it=4.92e-2
PQMIX: read 8 iter from file mixm. RMS DQ=2.24e-2 last it=4.40e-2
```