Chapter 1: Introduction
1.4 Organization of this dissertation
Chapter 2 presents theoretical studies of the “W” type structure of QWs. The nature of the tradeoff between extending of wavelength and optical oscillation strength is analyzed by the results of calculations based on simple one-band effective mass approximation. The E-k band structures of “W” QWs are simulated by using the 8-band k.p theory to look more insight to
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the characteristics like band edge effective masses and momentum matrix elements. The results lead to calculations of optical gain based on basic laser theorem, which can be treated as a design guideline of “W” laser active region.
Chapter 3 presents experimental techniques including how to prepare samples, material analysis methods, and device measurement setups. Our samples are grown by a MBE system.
X-ray diffraction helps to understand the lattice constant of epi layers. Photoluminescence (PL) studies could characterize the type-II nature of grown samples. And the apparatus to measure optically pumped “W” mid-IR lasers will be detailed.
Chapter 4 describes PL experiment results of the designed samples, which will be compared and found consistent with theoretical predictions. The peak of emission wavelength shifting to short wavelength as increasing PL excitation power (Pex) reveals the nature of type-II structure; however which does not follow the Pex 1/3
law as described in published literature. This observation will be discussed more and reasonably explained by a proposed model.
Chapter 5 describes the structure of the designed “W” QWs mid-IR laser along with the waveguide calculation. The lasing results are demonstrated via optical pumping experiments.
Studies of different laser cavity lengths and operation temperatures are implemented to analyze “W” lasers. Laser performance related parameters such characteristic temperature (T0) and internal loss (i) are extracted. It also includes a detailed discussion of Auger process and an estimation of Auger coefficient, which are important for mid-IR lasers.
Chapter 6 gives conclusions and suggestions for future works.
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Chapter 2
Theoretical studies of InGaAs/GaAsSb type-II “W” type quantum wells on InP substrates
The type-II InGaAs/GaAsSb heterostructure has staggered band alignment that offers a good strategy for developing mid-IR light sources on InP substrates. The “W” type structure introduces barrier layers for better electron confinement and carriers wavefunction overlap and hence optical oscillator strength. This chapter gives the theoretical view of the “W”
structure. One band effective mass approximation provides a simple way to study the trade-off nature of “W” type QWs, longer emission wavelength accompanied with reduced wavefunction overlap. More deliberate calculations are carried based on the 8-band k.p theory, which eventually leads to optical gain calculations for needs of design considerations of “W”
mid-IR lasers. The gain spectra and their peak values at various carrier densities were calculated. We have found that a more balanced electron and hole masses in the type-II “W”
QWs can benefit the material gain. In our designed cases, we have seen that the reduced hole effective mass due to a higher Sb content can partially compensate the gain loss caused by the reduced transition matrix element. Based on the optimized design, a material gain above 103 cm-1 is readily achievable for a single properly designed “W” quantum well. In order to be self-contained, all simulation parameters used will be listed as completely as possible.
2.1 Band alignments of “W” type quantum well
Four kinds of ternary semiconductors can be lattice matched to InP substrate with a lattice constant of 5.87Å , which are InAlAs, InGaAs, GaAsSb, and AlAsSb. Table 2.1 lists their
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alloy compositions and band gaps. Their band alignments together with band offset values are depicted in Fig. 2.1, which are based on experiment results in literatures[32-34].
In this material system, the two narrower bandgap materials, InGaAs and GaAsSb, form a type II heterjunction. The transition between the conduction band state in InGaAs and the valence band state can give out light emission with wavelength longer that the direct band-to-band transition in individual materials. Based on the band offset, the energy
In0.52Al0.48As In0.53Ga0.47As GaAs0.49Sb0.51 AlAs0.56Sb0.44
- direct band gap (eV)
0K 1.51 0.81 0.81 2.5
300K 1.42 0.73 0.72 2.4
Table 2.1 Room temperature (300K) and low temperature (0K) direct band gaps of ternary alloys lattice matched to InP substrate.
1.38eV
ΔEc 0.52eV
ΔEv 0.19eV
0.36eV
In0.52Al0.48As
In0.53Ga0.47As
Ga0.49As0.51Sb
AlAs0.56Sb 0.36eV
0.33eV T= 0K
Fig 2.1 Band lineups, with values of conduction and valence band offsets (ΔEc and ΔEv) indicated, of the four ternary alloys lattice matched to InP substrate.
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difference between the conduction band edge of InGaAs and the valence band edge of GaAsSb is ~0.5eV, which can generate a ~2.5 m light emission [35]. The two wider bandgap materials, InAlAs and AlAsSb, form type-I heterojunctions with the two narrower bandgap materials and can be used as barrier layers for carrier confinement. The emission wavelength can be further extended by increasing either the indium content in InGaAs alloy or the antimony fraction in GaAsSb alloy. To better understand this, the band edge evolutions by altering the alloy compositions will be described in next paragraph.
The calculations are based on the “model-solid” theory, which uses deformation potentials to predict behavior of band offsets in either lattice matched or pseudomorphic layers[36, 37].
Here only gives the case that normal directions of ternary alloys are all along [001]. In the condition without strained, the lattice constant (a0) of a ternary alloy AxB1-xC is assumed to follow the Vegard law and estimated from the lattice constants of two binary constituents.
a0_A𝑥B1−xC = x a0_AC + (1 − x) a0_BC , (2.1.1) Band gap (Eg), spin-orbit split-off energy (Δ0), average valence band edge (Evav), i.e. the average of the heavy-hole, light-hole, and spin-orbit split-off bands (Ehh+Elh+Eso)/3, and other parameters of the ternary are assumed fitted to a simple empirical quadratic formula:
PA𝑥B1−xC = x PAC+ (1 − x)PBC− x (1 − x)𝐶𝐴𝐵𝐶 , (2.1.2) where P represents an arbitrary semiconductor parameter, and C is so called the bowing parameter, which accounts for the deviation from a linear interpolation of the two constituent binaries. In general, the band gap bowing parameter of ternary alloy is negative, which means the ternary band gap is always smaller than the linear interpolation result. The reason can be originated to the disorder effect formed by the presence of different cations or anions.
Conduction band (Ec) and valence band (Ev) of the alloy can be described as:
Ev = Ehh = Elh= 𝐸𝑣𝑎𝑣 + ∆0/3, (where Ehh and Elh degenerated) (2.1.3) Ec = Eg+ Ev. (2.1.4)
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A pseudomorphic layer implicates a thin epi layer that can bear the lattice mismatch by subjecting its in-plan lattice constant to the underneath host material. The strain, expressed by biaxial strain ε|| and uniaxial strain ε┴ described below, held within the pseudomorphic layer could change its electronic properties:
ε|| = (as− a0)/a0, (2.1.5) ε┴= −2CC12
11 ε||, (2.1.6) where as and a0 denote substrate lattice constant and unstrained alloy lattice constant. C12 andC11 refer to material elastic constants. ε|| isnegative/positive when the pseudomorphic layer is under compressive/ tensile strain.
The shift of band edge levels caused by strain effects can be divided into two components, one of which is hydrostatic contribution and the other is shear contribution. The former push outward (pull inward) conduction and valence band edges from band gap center as the epi layer under compressive (tensile strain). The later splits energy levels of heavy hole and light hole bands at Г point. Now conduction and valence band edges can be expressed as :
E𝑣 = VBO + ∆𝐸𝑣𝑎𝑣ℎ𝑦 + max(∆𝐸ℎℎ𝑠ℎ, ∆𝐸𝑙ℎ𝑠ℎ), (2.1.7) 𝐸𝑐 = VBO + 𝐸𝑔 + ∆𝐸𝑐ℎ𝑦, (2.1.8)
VBO = 𝐸𝑣𝑎𝑣 + ∆0/3. (2.1.9)
where ∆𝐸𝑣𝑎𝑣ℎ𝑦 and ∆𝐸𝑐ℎ𝑦 account for hydrostatic components,
∆𝐸𝑣𝑎𝑣ℎ𝑦 = av (2ε||+ ε┴), (2.1.10)
∆𝐸𝑐ℎ𝑦 = ac (2ε||+ ε┴), (2.1.11) av/ac refers to hydrostatic deformation potentials for conduction/valence bands.
The max(∆𝐸ℎℎ𝑠ℎ, ∆𝐸𝑙ℎ𝑠ℎ) in equation (2.1.7) means only select the maximum term between the
∆𝐸ℎℎ𝑠ℎ and ∆𝐸𝑙ℎ𝑠ℎ. The shear contribution couples to spin-orbit interaction and can be described below:
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∆𝐸ℎℎ𝑠ℎ = −𝑏(ε┴− ε|| ), (2.1.12)
∆𝐸𝑙ℎ𝑠ℎ = 1/2(−∆0+ 𝑏(ε┴− ε|| ) + √∆02+ 2∆0𝑏(ε┴− ε||) + 9𝑏2(ε┴− ε||)2 [37], (2.1.13) where b is the tetragonal deformation potential.
Fig. 2.2 is depicted below for better understanding of strain effects induced energy shift of energy levels. In general, if pseudomorphic layer is biaxial compressive-strained, ∆𝐸ℎℎ𝑠ℎ will be larger than ∆𝐸𝑙ℎ𝑠ℎ and make heavy hole band shifted above light hole band. In the tensile-strained case ∆𝐸𝑙ℎ𝑠ℎ>∆𝐸ℎℎ𝑠ℎ and light hole is above heavy hole band.
Band structure parameters for binary semiconductors of InAs, GaAs, and GaSb are listed in Table 2.2(a), including and which are used to empirically fit binary band gap as a function of temperature,
Fig 2.2 Strain effects induced band edge shifts for the three cases: tensile-strained, unstrained, and compressive-strained.
∆𝑬𝒍𝒉𝒔𝒉+ ∆𝑬𝒗𝒂𝒗𝒉𝒚
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Table 2.2(a) Band structure parameters for binary semiconductors of InAs, GaAs, and GaSb.
E𝑔 ( ) = Eg− 2
+ (2.1.14)
Bowing parameters of InGaAs and GaAsSb ternary semiconductors are listed in Table 2.2(b).
These values of parameters are from the literature [4]. Other material parameters also can be found there, while only most related InGaAs/GaAsSb ternaries are presented here. It is known the material parameter values are varied among references and should be treated within a varying indium fraction and antimonite fraction respectively. In order to see explicitly the energy shifts of conduction band (CB) edge of InGaAs and valence band (VB) edge of GaAsSb, their energy values have been reset to zero when they are lattice matched to InP substrate. Notice that, CB edge of InGaAs drops as increasing indium fraction, while VB edge of GaAsSb goes up as increasing antimony fraction. Either of them leads to decrease energy difference between CB edge of InGaAs and VB edge of GaAsSb for InGaAs/GaAsSb heterostructure, and hence helps to extend emission wavelength.
InAs GaAs GaSb
17 Table 2.2(b) Bowing parameters for InGaAs and GaAsSb.
Fig. 2.3(a) Energy shifts of band edges, including conduction (Ec), heavy hole (Ehh), and light hole (Elh) bands, vs. indium fraction for InGaAs pseudomorphic layer, and Fig. 2.3(b) those for GaAsSb pseudomorphic layer with varied antimony fraction on InP substrate.
VBOeV
GaAs1-xSbx pseudomorphic layer
(b)
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A typical band alignment of the “W” type QW along with the electron and hole fundamental wavefunctions are shown in Fig. 2.4. The “W” structure mainly consists of symmetric InGaAs/GaAsSb/InGaAs layers sandwiched by InAlAs barrier layers. The structure gains the name from the “W”-like shape of conduction band profile. Because of the type-II band alignment, holes are confined inside the valence band (VB) of GaAsSb QW, forming heavy hole (HH) and light hole (LH) sub-bands, and electrons are confined in the two coupled InGaAs QWs, which have the split symmetric (E1) and the anti-symmetric (E2) states. The fundamental E1-HH1 optical transition has a smaller effective band gap energy not limited by the bandgaps of the constituent layers, and therefore gives a longer emission wavelength. The barrier layers provide quantum confinement for the electrons and increase the electron-hole wavefunction overlap, which leads to enhanced momentum matrix elements for optical device operation. Two dimensional densities of states for both carries are preserved in the “W” structure, and the type II configuration suppresses the Auger recombination.
Fig. 2.5 shows the relative position of the band edges of the materials used in the “W” QW,
E2
LH1 HH2 HH1 E1
EV GaAsSb
InGaAs InAlAs
EC
Fig. 2.4 A Scheme of a typical band alignment for the “W” type QW along with fundamental electron and hole wavefunctions based calculations of one band effective mass model.
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Fig. 2.5 Band edge energy levels (conduction band edge Ec denoted by solid line and valence band edge Ev by dashed line) of In0.53Ga0.47As and In0.52Al0.48As which are lattice matched to InP substrate, and GaAs1-xSbx pseudomorphic layer with Sb fraction varied from 0 to1. The biaxial strain is indicated on the top horizon axis. The band gap values for each material are indicated by the arrows.
The Δ denotes the energy separation between the Ec of In0.53Ga0.47As and the Ev of In0.53Ga0.47As
including In0.53Ga0.47As and In0.52Al0.48As which are lattice matched to InP substrate, and The Sb mole fraction in GaAs1-xSbx is varied from 0 to 1. The top horizontal axis indicates the biaxial strain in the GaAsSb layer, which is assumed to be pseudomorphically grown with the same in-plane lattice constant as InP. The plot of band edge energy shifts of GaAsSb are calculated based on “model solid” theory same as Fig. 2.3(b) but takes an offset to meet consistency with the band lineup for lattice matched case (GaAs0.51Sb0.49) depicted in Fig. 2.1.
The energy difference (Δ) between CB edge of In0.53Ga0.47As and VB edge of GaAsSb shrinks as the Sb fraction, x, increases. It is only around 0.25eV when x is 0.8. This energy corresponds to a very respectable 4.96 m mid-IR light emission (without adding the electron and hole quantized energies). In next section, emission wavelength and wavefunction overlap of “W” QWs with varied structure parameters, including thickness of InGaA and GaAsSb and Sb fraction of GaAsSb, will be simulated by simple one band effective mass approximation.
0.0 0.2 0.4 0.6 0.8 1.0
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2.2 The tradeoff feature for the “W” type quantum well explored by one band effective mass approximation
One band effective mass approximation provides basic way to calculate quantized energy levels and wavefunction profile for quantum structure, which is time saving for computation so that it can be used to simulate cases with various structure parameters to reveal overall behaviors. Once knowing band alignment values of “W” structure in early section, electron and hole confinement energies are computed by solving one band Schrödinger wave equation with conduction band effective mass and valence band effective mass respectively. It also needs to fit proper boundary conditions for wavefunction across hetero interfaces described below.
Conduction band: 2𝑚−ℏ2
0 ∇𝑚1
𝑐∗∇ϕ𝑒+ 𝐸𝑐 = 𝐸𝑛𝑐ϕ𝑒, (2.2.1) Valence band: −ℏ2
2𝑚0 ∇ 1
𝑚𝑣∗∇ϕℎ+ 𝐸𝑣 = 𝐸𝑚𝑣ϕℎ, (2.2.2) Boundary conditions:
ϕ𝑒,ℎ|ℎ𝑒𝑡𝑒𝑟𝑜+ = ϕ𝑒,ℎ|ℎ𝑒𝑡𝑒𝑟𝑜− (2.2.3)
1
𝑚𝑐,𝑣∗ ∇ϕ𝑒,ℎ|ℎ𝑒𝑡𝑒𝑟𝑜+ = 𝑚1
𝑐,𝑣∗ ∇ϕ𝑒,ℎ|ℎ𝑒𝑡𝑒𝑟𝑜− (2.2.4) Notations: 𝑚0 : Free electron mass
𝑚𝑐∗/𝑚𝑣∗ : Conduction band/valence band effective mass divided by 𝑚0 ϕ𝑒/ϕℎ : Electron / hole wavefunction
𝐸𝑐/𝐸𝑣 : Conduction/ valence band edge potential profile 𝐸𝑛𝑐/𝐸𝑚𝑣 : Conduction/ valence band quantum confined levels ℏ : Reduced plank constant
The effective mass values used for simulations are listed in Table 2.3. For simplicity, the effective mass for GaAsSb remains the same value without considering changes contributed
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by strain effect and different composition. Calculations were performed with InGaAs and GaAsSb thickness varied from 2nm to 7 nm, and with Sb fraction of GaAsSb varied from 0.5 to 0.9. We used In0.52Al0.48As as the barrier material in our “W” structure, which has a higher conduction band offset as compared to the (Ga)InP layer used in ref. 28 and the GaAsSb layer used in ref. 29. Our structure has a better electron confinement and a higher energy level when the same InGaAs width is used in the structure. In this parameter range, the ground state of valence band is always from heavy hole band because of the compressive strain of GaAsSb layer, which splits heavy hole band above light hole band, and the heavier effective mass of heavy hole as compared to light hole band makes the first heavy hole confined level more deeper. One can deduce the emission wavelengths of ground states optical transitions (E1 to HH1) and their corresponding electron-hole wavefunction overlaps. The square of wavefunction overlap is known to proportional to optical transition rate. Figure 2.6(a) shows the 3D contour plot for the three chosen wavelengths of 2m, 2.5m, and 3m, where the x, y, and z axes are, in turn, values of the InGaAs layer width, GaAsSb layer width, and Sb mole fraction in GaAsSb. Figure 2.6(b) shows their corresponding wavefunction overlap square displayed in in grayscale, where the brightness increases with the magnitude. The maximum and minimum values in the plot are 0.261 and 0.015 respectively.
In0.53Ga0.47As In0.52Al0.48As GaAs1-xSbx
𝑚𝑐∗ 0.043 0.075 0.047
𝑚𝑣∗ -0.38 -0.34 -0.3
As shown in Fig. 2.6(a), wavelength can be tuned in the range from 2 to 3m via modifying by varying the three parameters mentioned above. However, as the wavelength is increased
Table 2.3 The conduction and valence band effective mass values of ternaries used for simulations
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the wave function overlap decreases resulting in a smaller matrix element. A gradual increase in darkness in the plot is clearly seen as the wavelength changed from 2m to 3m in Fig.
2.6(b). This is a trade-off between the long wavelength emission and the optical matrix element and it is an intrinsic feature for the “W” type QW. The electron and hole are confined separately in different layers, the electron in the two coupled InGaAs layers and the hole in the GaAsSb layer, as shown in Fig. 2.4. Since the confinement potentials are large (~0.4eV) for both carriers, the quantization energies are decided by their own layer thickness; a thicker InGaAs or GaAsSb layer leads to a smaller electron or hole quantization energy and hence a longer emission wavelength. Although the increase of the InGaAs or GaAsSb layer thickness extends the emission wavelength, it also makes the electron and hole wave functions more concentrated in individual layers, and causes the reduction of electron-hole wavefunctions overlap. A similar trade-off exists when the Sb content in the GaAsSb layer is varied. The wavelength can be extended longer with a higher Sb content due to the reduced energy separation, Δ indicated in Fig 2.5, between the electron and the hole states. The raised conduction band edge, however, blocks the electron wave function penetration into the GaAsSb layer causing a reduction of the electron-hole wave function overlap. In order to a better elucidation, Fig. 2.7 (a) gives the wavefunctions at a chosen “W” structure indicated with values of ground state wavelength emission and overlap, and for comparisons, Fig.
2.7(b), Fig. 2.7 (c) ,and Fig. 2.7 (d) give the cases with each varying thickness of InGaAs, GaAsSb, and Sb fraction respectively.
As shown in Fig. 2.6(a), there are many possibilities in choosing the layer parameters to achieve the same emission wavelength. However, a different combination of the three parameters has a different strength in the transition matrix element. Looking carefully at Fig.
2.6(b), one can see that a better optical gain is generally obtained with a higher Sb mole fraction and thinner InGaAs/GaAsSb layers even for the same wavelength. Given such design
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flexibility, one has to be careful, however, when using a high Sb content in the GaAsSb layer because the material quality may degrade.
(a)
Wavelength 3D contour plots(b)
| Wavefucntion overlap |2Fig. 2.6(a) 3D contour plots and (b) corresponding wavefunction overlap square displayed in correlated grayscale at three wavelengths based on calculations using the one-band effective mass approximation.
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structure, indicated with values of structure parameters, emission wavelength, and electron-hole wavefunction overlap. For comparison, (b), (c), and (d) are results with increased thickness of InGaAs, GaAsSb, and Sb fraction of GaAsSb respectively. By comparing to (a), each shows a trade-off behavior, Extending wavelength accompanying with the reduction of wavefunction overlap.
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2.3 Energy band dispersion relations and material gain simulations of “W” quantum wells based on the eight band k.p theory
The k.p theory has been widely used to simulation semiconductors energy dispersions and momentum matrix elements [38], which cannot be provided by the previous one band calculations. It includes minimum number of parameters deduced from the group theoretical treatment as in a high symmetric point at Brillouin zone (BZ). The parameter set can be experimentally determined to truly reflect the band structure. The theory works well at vicinity of band extreme, where the most cared place for the optoelectronics consideration.
The Schrödinger equation incorporated spin-orbit interaction for one electron in a bulk semiconductor material can be described as
Ĥψ(𝐫) = [2𝑚−ℏ2
0∇2+ V(𝐫) +4𝑚ℏ
02𝑐2𝛔 ∙ (∇V × 𝐩)] ψ(𝐫) = 𝐸(𝒌)ψ(𝐫), (2.3.1) where the is the Pauli spin matrix. Based on Bloch’s theorem, the electronic wavefunctions in a periodic potential follows the form:
ψ(𝐫) = 𝑒𝑖𝒌∙𝒓∗ 𝑈𝑛𝒌(𝒓), (2.3.2) which is a slowly varying plane wave, with a wave vector k, times a fast changing cell-periodic Bloch’s function, with a n referring to different bands. In a zinc blende crystal, by taking the Bloch bases in turns of S↑, X↑,Y↑, Z↑, S↓, X↓, Y↓, Z↓, which are spin up and down of anti-bonding S-like atomic orbital and three degenerate bonding P-like symmetry states at k point respectively, the k.p Hamiltonian can be written down as
Ĥ = (𝐻̂4 0
0 𝐻̂4) + Ĥ𝑠𝑜+ Ĥ𝜀 [39, 40]. (2.3.3)
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Here Löwdin perturbation theorem has been used to set conduction band, heavy hole, light hole, and split-off band as Class A group, and other remote bands as Class B group. Then the 𝐻̂4 is 4 x 4 block k related Hamiltonian expressed as
𝐻̂4 = (𝐻̂𝑐𝑐 𝐻̂𝑐𝑣
𝐻̂𝑣𝑐 𝐻̂𝑣𝑣) (2.3.4)
𝐻̂𝑐𝑐 is the scalar
𝐻̂𝑐𝑐 = 𝐸𝑐 + ∑ 𝑘̂𝑖𝐴𝑐𝑘̂𝑖
𝑖=𝑥,𝑦,𝑧
, (2.3.5)
where Ec is the conduction band edge, Ac include the influence of remote bands on the electron effective mass and can be described as
𝐴𝑐 = ℏ2
2𝑚0𝑚𝑐∗ −2P2
3𝐸𝑔− P2
3(𝐸𝑔+ ∆) (2.3.6)
Here P is related to the interband momentum matrix element : P = ℏ
𝑖𝑚0 < 𝑆|𝑝̂𝑥|𝑥 >, (2.3.7) which is often related to the parameter Ep with a dimension of energy,
𝑖𝑚0 < 𝑆|𝑝̂𝑥|𝑥 >, (2.3.7) which is often related to the parameter Ep with a dimension of energy,