Band Lineup and In-Plane Effective Mass of InGaAsP or InGaAlAs on InP Strained-Layer Quantum Well
InP 应变层量子阱上 InGaAsP 或 InGaAlAs 的带线和平面内有效质量

IN the last several years, strained quantum wells have attracted the interest of many researchers working in the semiconductor laser field. Many interesting possibilities of strained quantum wells have been proposed and demonstrated [1]-[3]. Higher optical gain [4], or a very large modulation bandwidth [5], [6] may be obtained by use of the compressive strained quantum well. Polarization-insensitive optical gain may be obtained by use of tensile strained quantum wells [7], [8].
最近几年，应变量子阱引起了半导体激光领域许多研究人员的兴趣。人们提出并证明了应变量子阱的许多有趣的可能性[1]-[3]。使用压缩应变量子阱可以获得更高的光学增益[4]或非常大的调制带宽[5]、[6]。拉伸应变量子阱可获得偏振不敏感的光增益 [7]，[8]。

In order to determine the material properties in strained quantum well system, we must know the band lineup of the bulk material first. Although several reports have been presented [9]-[13], the band lineups of InGaAsP on InP system with strain effects are still ambiguous. We describe the band lineups of InGaAlAs on InP as well as InGaAsP on $\mathrm{InP}$$\mathrm{InP}$InP\operatorname{InP} with strain effects, based on the Harrison model [14].
为了确定应变量子阱系统的材料特性，我们必须首先知道块体材料的带排。虽然已有一些报告[9]-[13]，但具有应变效应的 InP 上 InGaAsP 体系的能带谱系仍然模糊不清。我们根据 Harrison 模型 [14]，描述了 InP 上 InGaAlAs 以及 $\mathrm{InP}$$\mathrm{InP}$InP\operatorname{InP} 上 InGaAsP 在应变效应下的能带阵列。

Precise calculation of the optical gain [4], [18], [19] as well as the analytical approximation of the laser properties [22] have been demonstrated. However, it is still useful to know the effective mass of the strain quantum well system.
光增益的精确计算 [4]、[18]、[19] 以及激光特性的分析近似 [22] 已经得到证实。然而，了解应变量子阱系统的有效质量仍然很有用。

We describe the in-plane effective mass of an InGaAsP on InP and InGaAlAs on InP strained quantum well systems.
我们描述了 InP 上 InGaAsP 和 InP 上 InGaAlAs 应变量子阱系统的面内有效质量。

We assume that the quantum well layers are on a (001) InP substrate, which is commonly used for long-wavelength semiconductor lasers. We will describe the composition of $\mathrm{InGaAsP}$$\mathrm{InGaAsP}$InGaAsP\operatorname{InGaAsP} as ${\mathrm{In}}_{1-x}{\mathrm{Ga}}_x{\mathrm{As}}_y{\mathrm{P}}_{1-y}$${\mathrm{In}}_{1-x}{\mathrm{Ga}}_x{\mathrm{As}}_y{\mathrm{P}}_{1-y}$In_(1-x)Ga_(x)As_(y)P_(1-y)\mathrm{In}_{1-x} \mathrm{Ga}_{x} \mathrm{As}_{y} \mathrm{P}_{1-y} or $(x,y)$$(x,y)$(x,y)(x, y), and the composition of InGaAlAs as ${\mathrm{In}}_{1-x-z}{\mathrm{Ga}}_x{\mathrm{Al}}_z$${\mathrm{In}}_{1-x-z}{\mathrm{Ga}}_x{\mathrm{Al}}_z$In_(1-x-z)Ga_(x)Al_(z)\operatorname{In}_{1-x-z} \mathrm{Ga}_{x} \mathrm{Al}_{z} As or ( $x,z$$x,z$x,zx, z ) in this paper.
我们假设量子阱层位于 (001) InP 衬底上，这种衬底通常用于长波长半导体激光器。我们将在本文中把 $\mathrm{InGaAsP}$$\mathrm{InGaAsP}$InGaAsP\operatorname{InGaAsP} 的组成描述为 ${\mathrm{In}}_{1-x}{\mathrm{Ga}}_x{\mathrm{As}}_y{\mathrm{P}}_{1-y}$${\mathrm{In}}_{1-x}{\mathrm{Ga}}_x{\mathrm{As}}_y{\mathrm{P}}_{1-y}$In_(1-x)Ga_(x)As_(y)P_(1-y)\mathrm{In}_{1-x} \mathrm{Ga}_{x} \mathrm{As}_{y} \mathrm{P}_{1-y} 或 $(x,y)$$(x,y)$(x,y)(x, y) ，把 InGaAlAs 的组成描述为 ${\mathrm{In}}_{1-x-z}{\mathrm{Ga}}_x{\mathrm{Al}}_z$${\mathrm{In}}_{1-x-z}{\mathrm{Ga}}_x{\mathrm{Al}}_z$In_(1-x-z)Ga_(x)Al_(z)\operatorname{In}_{1-x-z} \mathrm{Ga}_{x} \mathrm{Al}_{z} As 或 ( $x,z$$x,z$x,zx, z )。

We need to consider two things in order to determine the band lineup of a certain strained quantum-well system. First, we must know the unstrained band lineup of a bulk material, when the material has its own lattice constant. It has been reported that the band discontinuity ratio of the conduction band of lattice-matched InGaAsP on InP system is about $0.35-0.4$$0.35-0.4$0.35-0.40.35-0.4 [9]-[13]. We might use the band discontinuity ratio to determine the unstrained band lineup, and this could make the procedure very simple. However, we cannot apply this method to an arbitrary composition of InGaAsP. Our goal in this section is to determine the band lineup for any composition of InGaAsP (or InGaAlAs), therefore we develop a method which enables us to do so. The method will be described in the next part. The second procedure we pursue is to consider the change in energy induced by a strain. The procedure is straightforward, and will be given in the later part.
要确定某个应变量子阱系统的能带阵容，我们需要考虑两点。首先，我们必须知道块体材料在具有自身晶格常数时的非应变带阵容。据报道，晶格匹配的 InGaAsP on InP 系统导带的带不连续比约为 $0.35-0.4$$0.35-0.4$0.35-0.40.35-0.4 [9]-[13]。我们可以利用带不连续比来确定非约束带的排列，这样可以使过程变得非常简单。但是，我们无法将这种方法应用于任意组成的 InGaAsP。本节的目标是确定任何成分的 InGaAsP（或 InGaAlAs）的带阵，因此我们开发了一种方法，使我们能够做到这一点。该方法将在下一部分中介绍。我们采用的第二种方法是考虑应变引起的能量变化。该过程简单明了，将在下一部分介绍。

It should be noted that “unstrained” has a different meaning from “lattice-matched.” We start with the band lineups of the binary materials, which compose either InGaAsP or InGaAlAs. We use the Harrison model [14] for describing the band lineups of $\mathrm{InP},\mathrm{InAs},\mathrm{GaP}$$\mathrm{InP},\mathrm{InAs},\mathrm{GaP}$InP,InAs,GaP\operatorname{InP}, \operatorname{InAs}, \mathrm{GaP}, and GaAs. The lineups are shown in Fig. 1. We set the valence band energy of $\mathrm{InP}$$\mathrm{InP}$InP\operatorname{InP} as the origin of the energy. Experimentally determined value of $\mathrm{\Delta }{E}_{\mathrm{c}}:\mathrm{\Delta }{E}_{\mathrm{v}}=$$\mathrm{\Delta }{E}_{\mathrm{c}}:\mathrm{\Delta }{E}_{\mathrm{v}}=$DeltaE_(c):DeltaE_(v)=\Delta E_{\mathrm{c}}: \Delta E_{\mathrm{v}}= $0.65:0.35$$0.65:0.35$0.65:0.350.65: 0.35 used for AlAs/GaAs to determine the AlAs position [15], because Harrison model cannot give the good value only for AlAs.
需要注意的是，"未受约束 "与 "晶格匹配 "的含义不同。我们从组成 InGaAsP 或 InGaAlAs 的二元材料的能带排列开始。我们使用 Harrison 模型 [14] 来描述 $\mathrm{InP},\mathrm{InAs},\mathrm{GaP}$$\mathrm{InP},\mathrm{InAs},\mathrm{GaP}$InP,InAs,GaP\operatorname{InP}, \operatorname{InAs}, \mathrm{GaP} 和 GaAs 的能带阵列。这些谱线如图 1 所示。我们将 $\mathrm{InP}$$\mathrm{InP}$InP\operatorname{InP} 的价带能量设定为能量的原点。实验确定的 $\mathrm{\Delta }{E}_{\mathrm{c}}:\mathrm{\Delta }{E}_{\mathrm{v}}=$$\mathrm{\Delta }{E}_{\mathrm{c}}:\mathrm{\Delta }{E}_{\mathrm{v}}=$DeltaE_(c):DeltaE_(v)=\Delta E_{\mathrm{c}}: \Delta E_{\mathrm{v}}= $0.65:0.35$$0.65:0.35$0.65:0.350.65: 0.35 值用于确定 AlAs/GaAs 的 AlAs 位置 [15]，因为 Harrison 模型不能只给出 AlAs 的良好值。

We can obtain the conduction and valence band positions of ${\mathrm{In}}_{1-x}{\mathrm{Ga}}_x{\mathrm{As}}_y{\mathrm{P}}_{1-y}$${\mathrm{In}}_{1-x}{\mathrm{Ga}}_x{\mathrm{As}}_y{\mathrm{P}}_{1-y}$In_(1-x)Ga_(x)As_(y)P_(1-y)\mathrm{In}_{1-x} \mathrm{Ga}_{x} \mathrm{As}_{y} \mathrm{P}_{1-y} (or ${\mathrm{In}}_{1-x-z}{\mathrm{Ga}}_x{\mathrm{Al}}_z\mathrm{As}$${\mathrm{In}}_{1-x-z}{\mathrm{Ga}}_x{\mathrm{Al}}_z\mathrm{As}$In_(1-x-z)Ga_(x)Al_(z)As\mathrm{In}_{1-x-z} \mathrm{Ga}_{x} \mathrm{Al}_{z} \mathrm{As} ) relative to the InP valence band as the following steps. The procedure for InGaAsP will be explained, and the procedure for InGaAlAs is similar.
我们可以通过以下步骤获得 ${\mathrm{In}}_{1-x}{\mathrm{Ga}}_x{\mathrm{As}}_y{\mathrm{P}}_{1-y}$${\mathrm{In}}_{1-x}{\mathrm{Ga}}_x{\mathrm{As}}_y{\mathrm{P}}_{1-y}$In_(1-x)Ga_(x)As_(y)P_(1-y)\mathrm{In}_{1-x} \mathrm{Ga}_{x} \mathrm{As}_{y} \mathrm{P}_{1-y} （或 ${\mathrm{In}}_{1-x-z}{\mathrm{Ga}}_x{\mathrm{Al}}_z\mathrm{As}$${\mathrm{In}}_{1-x-z}{\mathrm{Ga}}_x{\mathrm{Al}}_z\mathrm{As}$In_(1-x-z)Ga_(x)Al_(z)As\mathrm{In}_{1-x-z} \mathrm{Ga}_{x} \mathrm{Al}_{z} \mathrm{As} ）相对于 InP 价带的导带和价带位置。我们将解释 InGaAsP 的步骤，InGaAlAs 的步骤与此类似。

Fig. 1. Band lineups of the binary materials which compose InGaAsP or InGaAlAs. The valence band energy of $\mathrm{InP}$$\mathrm{InP}$InP\operatorname{InP} is set to zero. Harrison model is used for determine the lineups of InP, InAs, GaP, and GaAs. Experimentally determined value of $\mathrm{\Delta }{E}_{\mathrm{c}}:\mathrm{\Delta }{E}_{\mathrm{v}}=0.65:0.35$$\mathrm{\Delta }{E}_{\mathrm{c}}:\mathrm{\Delta }{E}_{\mathrm{v}}=0.65:0.35$DeltaE_(c):DeltaE_(v)=0.65:0.35\Delta E_{\mathrm{c}}: \Delta E_{\mathrm{v}}=0.65: 0.35 is used for AlAs/GaAs tc determine the AlAs position.
图 1.构成 InGaAsP 或 InGaAlAs 的二元材料的能带排列。 $\mathrm{InP}$$\mathrm{InP}$InP\operatorname{InP} 的价带能被设为零。哈里森模型用于确定 InP、InAs、GaP 和 GaAs 的谱线。实验确定的 $\mathrm{\Delta }{E}_{\mathrm{c}}:\mathrm{\Delta }{E}_{\mathrm{v}}=0.65:0.35$$\mathrm{\Delta }{E}_{\mathrm{c}}:\mathrm{\Delta }{E}_{\mathrm{v}}=0.65:0.35$DeltaE_(c):DeltaE_(v)=0.65:0.35\Delta E_{\mathrm{c}}: \Delta E_{\mathrm{v}}=0.65: 0.35 值用于确定 AlAs/GaAs 的 AlAs 位置。

Fig. 2. Working plane to determine the band lineup of a certain composition of quaternary InGaAsP. Ternary materials corresponding the point $\mathrm{A}(x_0,1)$$\mathrm{A}\left(x_0,1\right)$A(x_(0),1)\mathrm{A}\left(x_{0}, 1\right) and $B(0,yo)$$B(0,yo)$B(0,yo)B(0, y o) have the same bandgap energy as the quatemary material.
图 2.确定某种成分的四元 InGaAsP 带阵的工作平面。与 $\mathrm{A}(x_0,1)$$\mathrm{A}\left(x_0,1\right)$A(x_(0),1)\mathrm{A}\left(x_{0}, 1\right) 和 $B(0,yo)$$B(0,yo)$B(0,yo)B(0, y o) 点相对应的三元材料具有与四元材料相同的带隙能。

where $C_{\mathrm{In}-\mathrm{Ga}}(\mathrm{InGaAs})$$C_{\mathrm{In}-\mathrm{Ga}}(\mathrm{InGaAs})$C_(In-Ga)(InGaAs)C_{\mathrm{In}-\mathrm{Ga}}(\mathrm{InGaAs}) is the bandgap nonlinearity of InGaAs, for example.
例如， $C_{\mathrm{In}-\mathrm{Ga}}(\mathrm{InGaAs})$$C_{\mathrm{In}-\mathrm{Ga}}(\mathrm{InGaAs})$C_(In-Ga)(InGaAs)C_{\mathrm{In}-\mathrm{Ga}}(\mathrm{InGaAs}) 是 InGaAs 的带隙非线性。
2) We calculate the ternary composition whose bandgap energy is the same as the quaternary material calculated by (2.1). Usually, there exist two ternary compositions to satisfy the situation. These two ternary compositions correspond to points $A(x_0,1)$$A\left(x_0,1\right)$A(x_(0),1)A\left(x_{0}, 1\right) and $B(0,y_0)$$B\left(0,y_0\right)$B(0,y_(0))B\left(0, y_{0}\right) in Fig. 2.
2) 我们计算带隙能与 (2.1) 所计算的四元材料相同的三元成分。通常，有两种三元成分可以满足这种情况。这两种三元成分分别对应图 2 中的 $A(x_0,1)$$A\left(x_0,1\right)$A(x_(0),1)A\left(x_{0}, 1\right) 点和 $B(0,y_0)$$B\left(0,y_0\right)$B(0,y_(0))B\left(0, y_{0}\right) 点。
3) We determine the band lineups of two ternary materials specified above. We assume that the band discontinuity $\mathrm{\Delta }{E}_{\mathrm{c}}:\mathrm{\Delta }{E}_{\mathrm{v}}$$\mathrm{\Delta }{E}_{\mathrm{c}}:\mathrm{\Delta }{E}_{\mathrm{v}}$DeltaE_(c):DeltaE_(v)\Delta E_{\mathrm{c}}: \Delta E_{\mathrm{v}} is constant if we go along on a boundary of the square in Fig. 2 from a certain binary material to another.
3) 我们确定了上述两种三元材料的能带排列。我们假定，如果我们沿着图 2 中的正方形边界从某种二元材料走到另一种二元材料，那么带不连续性 $\mathrm{\Delta }{E}_{\mathrm{c}}:\mathrm{\Delta }{E}_{\mathrm{v}}$$\mathrm{\Delta }{E}_{\mathrm{c}}:\mathrm{\Delta }{E}_{\mathrm{v}}$DeltaE_(c):DeltaE_(v)\Delta E_{\mathrm{c}}: \Delta E_{\mathrm{v}} 是恒定的。
4) We can obtain the band lineup of the quaternary material by interpolating the band lineups of two temary materials.
4) 我们可以通过对两种三元材料的带状排列进行内插，得到四元材料的带状排列。

TABLE I  表 I
Parameters Used for the Calculations (2) [15]
用于计算的参数 (2) [15]

TABLE II  表 II
Parameters. Used for the Calculations (1) [15]
参数。用于计算 (1) [15]

Suppose that the lattice constant of a substrate is $d_{\mathrm{s}}$$d_{\mathrm{s}}$d_(s)d_{\mathrm{s}} and the lattice constant of an epitaxial layer with no strain is $d_e$$d_e$d_(e)d_{e}, a strain $\epsilon$$\epsilon$epsi\varepsilon is defined as
假设基底的晶格常数为 $d_{\mathrm{s}}$$d_{\mathrm{s}}$d_(s)d_{\mathrm{s}} ，无应变外延层的晶格常数为 $d_e$$d_e$d_(e)d_{e} ，应变 $\epsilon$$\epsilon$epsi\varepsilon 的定义为

The energy correction in the conduction band induced by the strain can be described as [15]
应变引起的导带能量修正可描述为 [15]

where $a^{\prime }$$a^{\prime }$a^(')a^{\prime} is the hydrostatic deformation potential for the conduction band, $C_{11}$$C_{11}$C_(11)C_{11} and $C_{12}$$C_{12}$C_(12)C_{12} are the elastic stiffness constants. The energy corrections in the heavy and light hole valence bands are [15]
其中， $a^{\prime }$$a^{\prime }$a^(')a^{\prime} 是导带的静力学变形势， $C_{11}$$C_{11}$C_(11)C_{11} 和 $C_{12}$$C_{12}$C_(12)C_{12} 是弹性刚度常数。重空穴价带和轻空穴价带的能量修正为 [15] 。

where $a$$a$aa and $b$$b$bb are the hydrostatic deformation potential and shear deformation potential for the valence band, respectively.
其中 $a$$a$aa 和 $b$$b$bb 分别是价带的静力变形势和剪切变形势。
The parameters used in this paper are listed in Tables I and II [15]. We used the linear interpolation for calculating $d,a,a^{\prime }$$d,a,a^{\prime }$d,a,a^(')d, a, a^{\prime}, and $b$$b$bb of ternary or quaternary materials, and the interpolation weighted by the lattice constant for calculating $C_{11}$$C_{11}$C_(11)C_{11} and $C_{12}$$C_{12}$C_(12)C_{12} of ternary or quaternary materials.
本文使用的参数列于表 I 和表 II [15]。在计算三元或四元材料的 $d,a,a^{\prime }$$d,a,a^{\prime }$d,a,a^(')d, a, a^{\prime} 和 $b$$b$bb 时，我们使用了线性插值法；在计算三元或四元材料的 $C_{11}$$C_{11}$C_(11)C_{11} 和 $C_{12}$$C_{12}$C_(12)C_{12} 时，我们使用了由晶格常数加权的插值法。

Now, we can plot the strain dependence of the band discontinuity ratio between two materials. Since the band positions of the heavy and light holes split with strain, we should define two band discontinuity ratio as follows:
现在，我们可以绘制出两种材料的带不连续比与应变的关系图。由于重孔和轻孔的带位置随应变而分裂，我们应该定义两个带不连续比如下：

where $\mathrm{\Delta }{E}_{\mathrm{c}}$$\mathrm{\Delta }{E}_{\mathrm{c}}$DeltaE_(c)\Delta E_{\mathrm{c}} is the energy difference of conduction band between two materials, $\mathrm{\Delta }{E}_{v,\mathrm{HH}}$$\mathrm{\Delta }{E}_{v,\mathrm{HH}}$DeltaE_(v,HH)\Delta E_{v, \mathrm{HH}} and $\mathrm{\Delta }{E}_{\mathrm{v},\mathrm{LH}}$$\mathrm{\Delta }{E}_{\mathrm{v},\mathrm{LH}}$DeltaE_(v,LH)\Delta E_{\mathrm{v}, \mathrm{LH}} are the energy differences of heavy and light hole valence bands.
其中， $\mathrm{\Delta }{E}_{\mathrm{c}}$$\mathrm{\Delta }{E}_{\mathrm{c}}$DeltaE_(c)\Delta E_{\mathrm{c}} 是两种材料导带的能量差， $\mathrm{\Delta }{E}_{v,\mathrm{HH}}$$\mathrm{\Delta }{E}_{v,\mathrm{HH}}$DeltaE_(v,HH)\Delta E_{v, \mathrm{HH}} 和 $\mathrm{\Delta }{E}_{\mathrm{v},\mathrm{LH}}$$\mathrm{\Delta }{E}_{\mathrm{v},\mathrm{LH}}$DeltaE_(v,LH)\Delta E_{\mathrm{v}, \mathrm{LH}} 是重空穴价带和轻空穴价带的能量差。
InGaAsP quaternary barrier and InGaAs strained ternary well is the most popular pair of the strained quantum-well system for long wavelength lasers. Figure 3(a) shows the strain dependence of the bandgap discontinuity ratio for $1.2\mu \mathrm{m}$$1.2\mu \mathrm{m}$1.2 mum1.2 \mu \mathrm{~m} lattice-matched quaternary barrier/InGaAs well system. The heavy-hole valence band determines the carrier confinement in the compressive strain region and the light-hole valence band determines it in the tensile region. Therefore, the band discontinuity ratio for heavy hole is important for the compressive strain region and that for light hole is important for the tensile region, respectively. When we use the InGaAsP/InGaAs system, the band discontinuity ratio is almost constant around 0.4 in compressive strain region, and the ratio is decreased with tensile strain. This means that electron confinement is poorer if we use the tensile-strained well. When we use $\mathrm{InGaAlAs}/\mathrm{InGaAs}$$\mathrm{InGaAlAs}/\mathrm{InGaAs}$InGaAlAs//InGaAs\mathrm{InGaAlAs} / \mathrm{InGaAs} system, the band discontinuity ratio decreases with both signs of strain, which means that the electron confinement is the best at the lattice-matched system. The bandgap discontinuity ratios determined experimentally are also shown in Fig. 3(a). Our theory agrees very well with the experimental values with no fitting parameters. Figure 3(b) shows the similar plot of the strain compensated system [16], in which the strain of barrier material is the opposite sign of the strain of well material. In the InGaAsP/InGaAs system, the electron confinement is improved from 0.4 to 0.45 in the compressive strain region. No remarkable change from the lattice-matched barrier system is obtained in strain compensated $\mathrm{InGaAlA}s/InGaAssystem.$$\mathrm{InGaAlA}s/InGaAssystem.$InGaAlA s//InGaAssystem.\operatorname{InGaAlA} s / I n G a A s ~ s y s t e m . ~
InGaAsP 四元势垒和 InGaAs 应变三元阱是长波长激光器中最常用的一对应变量子阱系统。图 3(a) 显示了 $1.2\mu \mathrm{m}$$1.2\mu \mathrm{m}$1.2 mum1.2 \mu \mathrm{~m} 晶格匹配四元势垒/InGaAs 井系统带隙不连续比的应变依赖性。重空穴价带决定了压缩应变区的载流子约束，而轻空穴价带则决定了拉伸区的载流子约束。因此，重空穴价带的不连续比对压缩应变区和轻空穴价带的不连续比对拉伸区分别非常重要。当我们使用 InGaAsP/InGaAs 系统时，在压缩应变区，带不连续比几乎恒定在 0.4 左右，而随着拉伸应变的增加，带不连续比减小。这意味着，如果我们使用拉伸应变井，电子约束性会变差。当我们使用 $\mathrm{InGaAlAs}/\mathrm{InGaAs}$$\mathrm{InGaAlAs}/\mathrm{InGaAs}$InGaAlAs//InGaAs\mathrm{InGaAlAs} / \mathrm{InGaAs} 体系时，带隙不连续比随着两种应变符号的变化而减小，这说明在晶格匹配体系中电子约束是最好的。实验测定的带隙不连续比也如图 3（a）所示。在没有拟合参数的情况下，我们的理论与实验值非常吻合。图 3(b) 显示了应变补偿系统的类似曲线[16]，其中势垒材料的应变与阱材料的应变符号相反。在 InGaAsP/InGaAs 系统中，压缩应变区域的电子约束从 0.4 提高到 0.45。在应变补偿的 $\mathrm{InGaAlA}s/InGaAssystem.$$\mathrm{InGaAlA}s/InGaAssystem.$InGaAlA s//InGaAssystem.\operatorname{InGaAlA} s / I n G a A s ~ s y s t e m . ~ 系统中，与晶格匹配的势垒系统相比没有明显变化。

Figure 4 shows the band lineup of InGaAs as a function of the strain. Usually, we vary the composition of the InGaAs to introduce the strain. The composition variation often confuses us when we think what the strain will do. In fact, at a first glance, it may look that the compressive strain lowers the position of the conduction band, while the tensile strain raises it. However, it is not true. It is the composition variation that lowers the conduction band positions in compressive strain region and raises it in tensile strain region. The strain induced effect itself is contrary to the above effect. Therefore, we must be very careful if we want to look at the pure strain effect. If we want to investigate the pure strain effect on the band lineup, we should look at the strain dependence of the band position of a material at constant bandgap energy.
图 4 显示了 InGaAs 的带状排列与应变的函数关系。通常，我们会改变 InGaAs 的成分来引入应变。当我们考虑应变的作用时，成分的变化往往会让我们感到困惑。事实上，乍一看，压缩应变会降低导带的位置，而拉伸应变则会提高导带的位置。然而，事实并非如此。是成分的变化降低了压缩应变区域的导带位置，而提高了拉伸应变区域的导带位置。应变诱导效应本身与上述效应相反。因此，如果我们想研究纯应变效应，就必须非常小心。如果我们想研究纯应变对带阵列的影响，我们应该研究材料在带隙能量不变时带位置的应变依赖性。

Fig. 3. Strain dependence of the bandgap discontinuity ratio for (a) $1.2-\mu \mathrm{m}$$1.2-\mu \mathrm{m}$1.2-mum1.2-\mu \mathrm{m} lattice-matched quaternary barrier / InGaAs well system together with the ratios determined experimentally, and (b) $1.2-\mu \mathrm{m}$$1.2-\mu \mathrm{m}$1.2-mum1.2-\mu \mathrm{m} strain compensated quaternary barrier / InGaAs well system.
图 3.(a) $1.2-\mu \mathrm{m}$$1.2-\mu \mathrm{m}$1.2-mum1.2-\mu \mathrm{m} 晶格匹配的四元势垒/InGaAs 井系统的带隙不连续比与实验测定的比率，以及 (b) $1.2-\mu \mathrm{m}$$1.2-\mu \mathrm{m}$1.2-mum1.2-\mu \mathrm{m} 应变补偿的四元势垒/InGaAs 井系统的带隙不连续比的应变依赖性。

Fig. 4. Band lineup of $\mathrm{In}$$\mathrm{In}$In\operatorname{In} GaAs well as a function of the strain.
图 4. $\mathrm{In}$$\mathrm{In}$In\operatorname{In} 砷化镓的带状排列与应变的函数关系。

Figure 5 shows the contour of the strain and bandgap energy. Bandgap energy includes the correction induced by the strain. When we want to investigate the pure strain effect, we should go along the contour line from point $A$$A$AA through $B$$B$BB to $C$$C$CC, in Fig. 5(a) for example. Note that the pass we took for plotting Fig. 4 corresponds to the line from point $D$$D$DD through $E$$E$EE to $F$$F$FF.
图 5 显示了应变和带隙能的等值线。带隙能包括应变引起的修正。例如，当我们要研究纯应变效应时，我们应该沿着从 $A$$A$AA 点经过 $B$$B$BB 到 $C$$C$CC 的等值线，如图 5（a）所示。请注意，我们在绘制图 4 时所经过的位置对应于从点 $D$$D$DD 经过 $E$$E$EE 到 $F$$F$FF 的直线。

Dependence of conduction and valence band lineups purely on the strain are shown in Fig. 6 for InGaAsP system and in Fig. 7 for InGaAlAs system. From Fig. 6, we can say that the compressive strain does not affect the band lineup so much, and tensile strain raises the band lineups in the InGaAsP system. From Fig. 7, it can be said that the both compressive and tensile strains raise the band lineups in the InGaAlAs system.
图 6 和图 7 分别显示了 InGaAsP 系统和 InGaAlAs 系统的导带和价带排列对应变的纯粹依赖性。从图 6 中可以看出，压缩应变对带阵列的影响不大，而拉伸应变会提高 InGaAsP 系统的带阵列。从图 7 可以看出，在 InGaAlAs 系统中，压缩应变和拉伸应变都会提高带列。

Fig. 5. Contours of the strain and bandgap energy of (a) InGaAsP systern and (b) InGaAlAs system. Bandgap energy includes the correction induced by the strain.
图 5.(a) InGaAsP 系统和 (b) InGaAlAs 系统的应变和带隙能的等值线。带隙能包括应变引起的修正。

From Fig. 4 and Figs. 6 and 7, we extracted approximate expressions of band position of InGaAs, InGaAsP and InGaAlAs. $E_{\mathrm{c}}$$E_{\mathrm{c}}$E_(c)E_{\mathrm{c}} is the conduction band position expressed in eV, $E_{\mathrm{v},\mathrm{HH}}$$E_{\mathrm{v},\mathrm{HH}}$E_(v,HH)E_{\mathrm{v}, \mathrm{HH}} and $E_{\mathrm{V},\mathrm{LH}}$$E_{\mathrm{V},\mathrm{LH}}$E_(V,LH)E_{\mathrm{V}, \mathrm{LH}} are the heavy and light hole valence band positions, and the strain $\epsilon$$\epsilon$epsi\varepsilon is expressed in $\mathrm{\%}.E_g$$\mathrm{\%}.E_g$%.E_(g)\% . E_{g} is the bandgap energy of the quaternary material. It should be noted again that the conduction band position of $\mathrm{InP}$$\mathrm{InP}$InP\operatorname{InP} is 1.35 eV and the valence band position of InP is zero.
从图 4 以及图 6 和图 7 中，我们提取了 InGaAs、InGaAsP 和 InGaAlAs 的带位置近似表达式。 $E_{\mathrm{c}}$$E_{\mathrm{c}}$E_(c)E_{\mathrm{c}} 是导带位置，单位是eV， $E_{\mathrm{v},\mathrm{HH}}$$E_{\mathrm{v},\mathrm{HH}}$E_(v,HH)E_{\mathrm{v}, \mathrm{HH}} 和 $E_{\mathrm{V},\mathrm{LH}}$$E_{\mathrm{V},\mathrm{LH}}$E_(V,LH)E_{\mathrm{V}, \mathrm{LH}} 是重空穴和轻空穴价带位置，应变 $\epsilon$$\epsilon$epsi\varepsilon 用 $\mathrm{\%}.E_g$$\mathrm{\%}.E_g$%.E_(g)\% . E_{g} 表示， $\mathrm{\%}.E_g$$\mathrm{\%}.E_g$%.E_(g)\% . E_{g} 是四元材料的带隙能。需要再次指出的是， $\mathrm{InP}$$\mathrm{InP}$InP\operatorname{InP} 的导带位置为 1.35 eV，而 InP 的价带位置为零。