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

Takuya Ishikawa, Member, IEEE, and John E. Bowers, Fellow, IEEE
Takuya Ishikawa，IEEE 会员；John E. Bowers，IEEE 会员

Abstract 摘要

We describe the band lineups of InGaAlAs on (001) InP as well as InGaAsP on (001) InP system with strain effects, based on the Harrison model. We show that the compressive strain does not affect the band position so much, and tensile strain raises the band position in the InGaAsP system. It is also shown that both compressive and tensile strains raise the band positions in the InGaAlAs system. The conduction and valence band positions of InGaAs, InGaAsP, and InGaAlAs relative to InP valence band are given in approximate formulas as a function of the strain. We calculate the energy versus in-plane wave vector relationship of the $InGaAsP / InGaAs (P)$ on $InP$ and InGaAlAs/InGa(Al)As on InP strained quantum-well systems. We obtain the in-plane effective mass of the strained quantumwell system by fitting the dispersion relationship to a parabolic curve. The in-plane effective masses of several kinds of strained quantum-well systems are listed.
我们以 Harrison 模型为基础，描述了 InGaAlAs on (001) InP 和 InGaAsP on (001) InP 系统的应变效应带阵列。我们的研究表明，压缩应变对带位置的影响不大，而拉伸应变会提高 InGaAsP 系统中的带位置。研究还表明，在 InGaAlAs 系统中，压缩应变和拉伸应变都会提高能带位置。InGaAs 、InGaAsP 和 InGaAlAs 相对于 InP 价带的导带和价带位置以近似公式给出，与应变成函数关系。我们计算了 $InGaAsP / InGaAs (P)$ on $InP$ 和 InGaAlAs/InGa(Al)As on InP 应变量子阱系统的能量与平面内波矢量的关系。通过将色散关系拟合为抛物线，我们得到了应变量子阱系统的面内有效质量。下面列出了几种应变量子阱系统的面内有效质量。

I. Introduction I.导言

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

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

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

InGaAsP

{In}_{1 - x} {Ga}_{x} {As}_{y} P_{1 - y}

(x, y)

, and the composition of InGaAlAs as

{In}_{1 - x - z} {Ga}_{x} {Al}_{z}

As or (

x, 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

InGaAsP

的组成描述为

{In}_{1 - x} {Ga}_{x} {As}_{y} P_{1 - y}

或

(x, y)

，把 InGaAlAs 的组成描述为

{In}_{1 - x - z} {Ga}_{x} {Al}_{z}

As 或 (

x, z

)。

II. Band Lineup II.乐队阵容

A. Calculation Procedure
A.计算程序

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

[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

[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。

B. Band Lineup with No Strain
B.无压力乐队阵容

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

InP, InAs, GaP

, and GaAs. The lineups are shown in Fig. 1. We set the valence band energy of

InP

as the origin of the energy. Experimentally determined value of

Δ E_{c} : Δ E_{v} =

0.65 : 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

InP, InAs, GaP

和 GaAs 的能带阵列。这些谱线如图 1 所示。我们将

InP

的价带能量设定为能量的原点。实验确定的

Δ E_{c} : Δ E_{v} =

0.65 : 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

{In}_{1 - x} {Ga}_{x} {As}_{y} P_{1 - y}

(or

{In}_{1 - x - z} {Ga}_{x} {Al}_{z} 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

{In}_{1 - x} {Ga}_{x} {As}_{y} P_{1 - y}

（或

{In}_{1 - x - z} {Ga}_{x} {Al}_{z} 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

InP

is set to zero. Harrison model is used for determine the lineups of InP, InAs, GaP, and GaAs. Experimentally determined value of

Δ E_{c} : Δ E_{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。

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

Δ E_{c} : Δ E_{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

A (x_{0}, 1)

and

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

A (x_{0}, 1)

和

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。

We calculated the bandgap energy of quatemary InGaAsP by
我们通过以下方法计算出了四价 InGaAsP 的带隙能

\begin{aligned} E_{g} (x, y) = & x y E_{g} (GaAs) + (1 - x) y E_{g} (InAs) \\ + x (1 - y) E_{g} (GaP) \\ + (1 - x) (1 - y) E_{g} (InP) \\ + x (x - 1) [y C_{In - Ga} (InGaAs) \\ + (1 - y) C_{In - Ga} (InGaP)] \\ + y (y - 1) [x C_{As - P} (GaAsP) \\ + (1 - x) C_{As - P} (InAsP)] \end{aligned}

where

C_{In - Ga} (InGaAs)

is the bandgap nonlinearity of InGaAs, for example.
例如，

C_{In - Ga} (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)

and

B (0, y_{0})

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)

点和

B (0, y_{0})

点。
3) We determine the band lineups of two ternary materials specified above. We assume that the band discontinuity

Δ E_{c} : Δ E_{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

Δ E_{c} : Δ E_{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]

Parameter 参数	Symb	olUnit	InP	InAs 砷化镓	GaP	GaAs 砷化镓	AlAs
Lattice constant 晶格常数	d	$Å$	5.8688	6.0684	5.4512	5.6533	5.6611
Bandgap energy 带隙能	$E_{5}$	eV 电子伏特	1.35	0.36	2.74	1.42	2.95
Elastic stiffness constant 弹性刚度常数	$dyn / {cin}^{2}$			8.329	14.120	11.880	12.020
Elastic stiffness constant 弹性刚度常数	$dyn / {cm}^{2}$			4.526	6.253	5.380	5.700
Hydrostatic deformation potential for 的静水变形潜能值	$a^{'}$	eV 电子伏特	4.5	4.1	7.9	6.8	6.3
conduction band Hydrostatic deformation potential for valence band 导带价带的静水变形势能	$a$	eV 电子伏特	2.9	2.5	3.0	2.7	2.6
Shear deformation potential for Yalence band 雅伦斯带的剪切变形潜力	$b$	eV 电子伏特	-2.0	-1.8	-1.5	-1.7	-1.5

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

Bandgap Nonlinearity 带隙非线性	(eV)
$C_{In - Ga} (InGaP)$	0.70
$C_{In - Ga}$ (InGaAs) $C_{In - Ga}$ （InGaAs）	0.51
$C_{In - A 1}$ (InAlAs) $C_{In - A 1}$ （InAlAs）	0.99
$C_{As - P}$ (InAsP) $C_{As - P}$ （InAsP）	0.23
$C_{As - P}$ (GaAsP) $C_{As - P}$ （砷化镓）	0.30
$C_{Al - Ga}$ (AIGaAs) $C_{Al - Ga}$ （AIGaAs）	-0.04

C. Energy Correction Induced by Strain
C.应变引起的能量修正

Suppose that the lattice constant of a substrate is

d_{s}

and the lattice constant of an epitaxial layer with no strain is

d_{e}

, a strain

ε

is defined as
假设基底的晶格常数为

d_{s}

，无应变外延层的晶格常数为

d_{e}

，应变

ε

的定义为

ε = \frac{d_{e} - d_{s}}{d_{s}}

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]

Δ E_{str, c} = 2 a^{'} \frac{C_{11} - C_{12}}{C_{11}} ε

where

a^{'}

is the hydrostatic deformation potential for the conduction band,

C_{11}

and

C_{12}

are the elastic stiffness constants. The energy corrections in the heavy and light hole valence bands are [15]
其中，

a^{'}

是导带的静力学变形势，

C_{11}

和

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] 。

\begin{aligned} Δ E_{str, hh} & = (2 a \frac{C_{11} - C_{12}}{C_{11}} + b \frac{C_{11} + 2 C_{12}}{C_{11}}) ε \\ Δ E_{str, lh} & = (2 a \frac{C_{11} - C_{12}}{C_{11}} - b \frac{C_{11} + 2 C_{12}}{C_{11}}) ε \end{aligned}

where

a

and

b

are the hydrostatic deformation potential and shear deformation potential for the valence band, respectively.
其中

a

和

b

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^{'}

, and

b

of ternary or quaternary materials, and the interpolation weighted by the lattice constant for calculating

C_{11}

and

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^{'}

和

b

时，我们使用了线性插值法；在计算三元或四元材料的

C_{11}

和

C_{12}

时，我们使用了由晶格常数加权的插值法。

D. Band Discontinuity Ratio
D.频带不连续率

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：

\begin{aligned} Q_{e} (for HH) = \frac{Δ E_{c}}{| Δ E_{c} | + | Δ E_{v, HH} |} \\ Q_{e} (for LH) = \frac{Δ E_{c}}{| Δ E_{c} | + | Δ E_{v, LH ∣} |} \end{aligned}

where

Δ E_{c}

is the energy difference of conduction band between two materials,

Δ E_{v, HH}

and

Δ E_{v, LH}

are the energy differences of heavy and light hole valence bands.
其中，

Δ E_{c}

是两种材料导带的能量差，

Δ E_{v, HH}

和

Δ E_{v, 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 μ 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

InGaAlAs / 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

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 μ 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

InGaAlAs / 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

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。

E. Band Lineup Change Purely Induced by Strain
E.纯粹由应变引起的乐队阵容变化

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 - μ m

lattice-matched quaternary barrier / InGaAs well system together with the ratios determined experimentally, and (b)

1.2 - μ m

strain compensated quaternary barrier / InGaAs well system.
图 3.(a)

1.2 - μ 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 - μ 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

In

GaAs well as a function of the strain.
图 4.

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

through

B

C

, in Fig. 5(a) for example. Note that the pass we took for plotting Fig. 4 corresponds to the line from point

D

through

E

F

.
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

点经过

B

到

C

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

经过

E

到

F

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。

F. Approximate Expression of Band Position
F.乐队位置的近似表达式

From Fig. 4 and Figs. 6 and 7, we extracted approximate expressions of band position of InGaAs, InGaAsP and InGaAlAs.

E_{c}

is the conduction band position expressed in eV,

E_{v, HH}

and

E_{V, LH}

are the heavy and light hole valence band positions, and the strain

ε

is expressed in

% . E_{g}

is the bandgap energy of the quaternary material. It should be noted again that the conduction band position of

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_{c}

是导带位置，单位是eV，

E_{v, HH}

和

E_{V, LH}

是重空穴和轻空穴价带位置，应变

ε

用

% . E_{g}

表示，

% . E_{g}

是四元材料的带隙能。需要再次指出的是，

InP

的导带位置为 1.35 eV，而 InP 的价带位置为零。

InGaAs: 砷化镓

\begin{aligned} E_{c} & = 1.040 - 0.0474 ε + 0.003303 ε^{2} \\ E_{v, HH} & = 0.3331 + 0.05503 ε - 0.002212 ε^{2} \\ E_{v, LH} & = 0.3331 - 0.01503 ε - 0.003695 ε^{2} . \end{aligned}

InGaAsP: InGaAsP：
a) $ε < 0$

\begin{aligned} E_{c} & = (0.6958 + 0.4836 E_{g}) - 0.03031 ε \\ E_{Y, LH} & = (0.6958 - 0.5164 E_{g}) - 0.03031 ε \end{aligned}

ε > 0

\begin{aligned} E_{c} & = (0.6958 + 0.4836 E_{g}) + 0.003382 ε \\ E_{v, HH} & = (0.6958 - 0.5164 E_{g}) + 0.003382 ε . \end{aligned}

Fig. 6. Dependence of (a) conduction and (b) valence band lineups on the strain in InGaAsP systern.
图 6.InGaAsP 系统中 (a) 传导带和 (b) 价带排列与应变的关系。
3) InGaAlAs: 3) InGaAlAs：
a)

ε < 0

\begin{aligned} E_{c} & = (0.5766 + 0.6561 E_{g}) - 0.02307 ε \\ E_{v, LH} & = (0.5766 - 0.3439 E_{g}) - 0.02307 ε . \end{aligned}

ε > 0

\begin{aligned} E_{c} & = (0.5766 + 0.6561 E_{g}) + 0.01888 ε \\ E_{v, HH} & = (0.5766 - 0.3439 E_{g}) + 0.01888 ε \end{aligned}

III. Valence Band Structure of Strained Quantum Well.
III.应变量子阱的价带结构。

A. Calculation Method A.计算方法

In this section, we calculate the energy versus wave vector relations of heavy hole and light hole in the strained quantum well system. Basically, we followed the procedure in [17] with slight modification, to determine the energy versus in-plane wave vector relations. We will describe the essential parts of the procedure here. The coupled effective mass equations can be expressed as
在本节中，我们将计算应变量子阱系统中重空穴和轻空穴的能量与波矢量关系。基本上，我们按照文献[17]中的步骤略加修改，确定了能量与平面内波矢量的关系。我们将在此介绍该过程的主要部分。耦合有效质量方程可以表示为

[\begin{array}{cc} H_{h} + V_{h} & W \\ W^{†} & H_{l} + V_{l} \end{array}] [\begin{matrix} F_{h} \\ F_{l} \end{matrix}] = E_{v} [\begin{matrix} F_{h} \\ F_{l} \end{matrix}]

where

F_{h, 1}

are the heavy and light hole envelope functions,

E_{v}

is the energy of the hole, and

W^{†}

is the Hermitian conjugate of

W . V_{h}

and

V_{1}

are the potential energy for heavy and light holes, which can be calculated by the procedure described in the previous section. If we define that the

k_{z}

is directed along
其中，

F_{h, 1}

分别为重空穴和轻空穴包络函数，

E_{v}

为空穴能量，

W^{†}

为

W . V_{h}

的赫米共轭，

V_{1}

分别为重空穴和轻空穴的势能，可以通过上一节描述的程序计算出来。如果我们定义

k_{z}

沿着

Fig. 7. Dependence of (a) conduction and (b) valence band lineups on the strain in the InGaAlAs system.
图 7.InGaAlAs 系统中 (a) 传导带和 (b) 价带排列与应变的关系。
a [100] direction, and

k_{t}

is directed either along a [100] or a [110] direction, the Hamiltonians for heavy and light hole

H_{h, 1}

are expressed by using the Luttinger parameters

γ_{1}

and

γ_{2}

as
100]方向，

k_{t}

沿[100]或[110]方向，重孔和轻孔

H_{h, 1}

的哈密顿参数用鲁丁格参数

γ_{1}

和

γ_{2}

表示为

\begin{aligned} H_{h} & = (γ_{1} - 2 γ_{2}) k_{z}^{2} + (γ_{1} + γ_{2}) k_{t}^{2} \\ H_{1} & = (γ_{1} + 2 γ_{2}) k_{z}^{2} + (γ_{1} - γ_{2}) k_{t}^{2} \end{aligned}

where 其中

γ_{1} - 2 γ_{2} \equiv \frac{ℏ^{2}}{2 m_{hh}}, γ_{1} + 2 γ_{2} \equiv \frac{ℏ^{2}}{2 m_{lh}}

m_{hh}

and

m_{lh}

are the effective masses of the heavy and light holes. The coupling term

W

can be given by

m_{hh}

和

m_{lh}

是重洞和轻洞的有效质量。耦合项

W

可由以下公式给出

\begin{array}{ll} W = \sqrt{3} k_{t} (γ_{2} k_{t} - i 2 γ_{3} k_{z}) & for {100} planes, (3.5a) \\ W = \sqrt{3} k_{t} (γ_{3} k_{t} - i 2 γ_{3} k_{z}) & for {110} planes. (3.5b) \end{array}

The Hermitian conjugate is given by
赫米共轭赋值为

W^{†} = \sqrt{3} k_{t} (γ_{2} k_{t} + i 2 γ_{3} k_{z}) .

We solved the relationship between

k_{t}

and

E_{y}

by using the condition that the four following quantities match across the well-barrier interface.
我们利用以下四个量在井-壁垒界面上相匹配的条件，解决了

k_{t}

和

E_{y}

之间的关系。

\begin{aligned} F_{h} and (γ_{1} - 2 γ_{2}) \frac{d F_{h}}{d z} + \sqrt{3} γ_{3} k_{t} F_{1} \\ F_{l} and (γ_{1} + 2 γ_{2}) \frac{d F_{1}}{d z} + \sqrt{3} γ_{3} k_{t} F_{h} \end{aligned}

The Luttinger parameters of the binary materials are listed in Table III [23] together with the electron effective masses. We used the linear interpolation of the binary materials for a compound material.
表 III [23] 列出了二元材料的卢廷格参数和电子有效质量。我们使用二元材料的线性插值来计算复合材料。

TABLE III 表 III
Luttinger Parameters and Electron Effective Mass of the Binary Materials [23]
二元材料的鲁丁格参数和电子有效质量 [23]

	$InP$	$lnAs$	GaP	GaAs 砷化镓	AlAs
$γ_{1}$	5.15	20.40	4.05	6.95	3.45
$γ_{12}$	0.94	8.30	0.49	2.25	0.68
$γ_{3}$	1.62	9.10	1.25	2.86	1.29
$m_{e}$	0.0765	0.0239	0.254	0.0670	0.150

B. Energy Dispersion of InGaAsP/InGaAs on InP System
B.InGaAsP/InGaAs on InP 系统的能量散布

Figure 8 shows the example of the valence band energy dispersion calculated for an InGaAs ternary quantum well with an InGaAsP lattice-matched

1.2 μ m

barrier. The quantum well width is set to 8 nm . The origin of the energy is set to the valence band position of

InP

. In the lattice-matched well, the dispersion is complicated due to the heavy and light hole mixing effect. The mixing effect can be greatly reduced with the

1.0 %

compressive strained well, and the dispersion is close to a parabolic curve. In the tensile-well case, the lighthole subband energy is larger than the heavy-hole subband energy. The dispersion of the light-hole subband (LH1) is also close to a parabolic shape, however, the curvature is very large compared with the curvature of the heavy-hole subband (HH1) of the compressive strained case.
图 8 显示了对带有 InGaAsP 晶格匹配

1.2 μ m

势垒的 InGaAs 三元量子阱计算的价带能量色散示例。量子阱宽度设置为 8 nm。能量原点设置为

InP

的价带位置。在晶格匹配阱中，由于重空穴和轻空穴的混合效应，色散非常复杂。在

1.0 %

压缩应变井中，混合效应可以大大降低，色散接近抛物线曲线。在拉伸井情况下，轻洞子带能量大于重洞子带能量。轻洞子带（LH1）的色散也接近抛物线形状，但与压缩应变情况下的重洞子带（HH1）的曲率相比，曲率非常大。

C. Energy Dispersion of InGaAlAs/InGaAs on InP System
C.InGaAlAs/InGaAs on InP 系统的能量散布

Figure 9 shows the example of the valence band energy dispersion calculated for InGaAs ternary quantum well with InGaAlAs lattice-matched

1.2 - μ m

barrier. The quantum well width is set to 8 nm . In this material system, the band discontinuity energy in the valence band is much smaller than that of the InGaAsP/InGaAs system. However, the tendencies of the dispersion are quite similar to those of

InGaAsP / InGaAsP

system, except for the number of the subbands.
图 9 显示了 InGaAs 三元量子阱与 InGaAlAs 晶格匹配

1.2 - μ m

势垒计算得出的价带能量色散示例。量子阱宽度设置为 8 nm。在该材料体系中，价带的能带不连续度远小于 InGaAsP/InGaAs 体系。然而，除了子带的数量之外，色散的趋势与

InGaAsP / InGaAsP

系统十分相似。

IV. In-Plane Effective Mass of Strained Quantum Well
IV.应变量子阱的平面内有效质量

A. Parabolic Approximation of Energy Dispersion
A.能量扩散的抛物线近似法

Once the energy dispersion is obtained, we can calculate the material parameters for designing a quantum well laser, such as optical gain [18], [19]. However, it is not necessary to solve the energy dispersion every time a quantum well laser is designed. The optical gain is proportional to a reduced density of states. When the energy dispersion is parabolic, the reduced density of states can be described as
获得能量色散后，我们就可以计算设计量子阱激光器的材料参数，如光学增益 [18], [19]。然而，并不是每次设计量子阱激光器时都需要求解能量色散。光增益与降低的态密度成正比。当能量色散为抛物线时，还原态密度可描述为

ρ_{red} = \frac{1}{1 / m_{e} + 1 / m_{h}} \cdot \frac{1}{π ℏ^{2}} .

Therefore, it is very informative that we obtain the “in-plane effective mass” around the states where the optical transition for the lasing occurs. In this section, we try to apply the parabolic approximation to the energy dispersion calculated in the previous section.
因此，我们在发生激光跃迁的态周围获得 "面内有效质量 "是非常有意义的。在本节中，我们尝试将抛物线近似应用于上一节计算的能量色散。

The threshold current density of a quantum well laser has been reduced to less than

1 kA / {cm}^{2}

, for example

500 A / {cm}^{2}

量子阱激光器的阈值电流密度已降低到

1 kA / {cm}^{2}

以下，例如

500 A / {cm}^{2}

Fig. 8. Valence band energy dispersion calculated for InGaAs ternary quantum well with

InGaAsP

lattice-matched

1.2 - μ m

barrier. Quantum-well width is set to 8 nm with (a) lattice-matched, (b)

1.0 %

compressive strain, and ©

1.0 %

tensile strain. The origin of the energy is set to the valence band position of

l n P

.
图 8.计算的具有

InGaAsP

晶格匹配

1.2 - μ m

势垒的 InGaAs 三元量子阱的价带能量色散。量子阱宽度设为 8 nm，(a) 晶格匹配，(b)

1.0 %

压缩应变和 ©

1.0 %

拉伸应变。能量原点设置为

l n P

的价带位置。
[24]. This threshold current density corresponds to a carrier density of

3 \cdot 10^{12} {cm}^{- 2}

. Suppose that the carrier lifetime is about 1 ns . The quasi-Fermi energy for this carrier density is about 15 meV from the subband edge, when the effective mass is 0.2 (this effective mass is consistent with the results below). It can be seen from Figs. 8 and 9 that we can describe the energy dispersion of 15 meV range if we take the in-plane wave vector

k

up to 0.2 . Therefore, we picked up the

| k | < 0.2

parts of the dispersion curves and least square fit to parabolic curves.
[24].该阈值电流密度对应于

3 \cdot 10^{12} {cm}^{- 2}

的载流子密度。假设载流子寿命约为 1 ns。当有效质量为 0.2 时，该载流子密度的准费米能距离子带边缘约为 15 meV（该有效质量与下面的结果一致）。从图 8 和图 9 可以看出，如果我们将面内波矢量

k

取到 0.2，就可以描述 15 meV 范围内的能量弥散。因此，我们拾取了频散曲线的

| k | < 0.2

部分，并对抛物线曲线进行了最小二乘法拟合。

B. Effective Mass of InGaAsP/InGaAs(P) and InGaAlAs/InGa(Al)As on InP Systemfor I.55- $μ m$ Operation
B. InGaAsP/InGaAs(P) 和 InGaAlAs/InGa(Al)As 在 I.55- $μ m$ 操作的 InP 系统上的有效质量

Figure 10 shows the strain dependence of the in-plane effective mass for InGaAs ternary well with InGaAsP

1.2 - μ m

图 10 显示了具有 InGaAsP

1.2 - μ m

的 InGaAs 三元阱的面内有效质量的应变依赖性。

Fig. 9. Valence band energy dispersion calculated for InGaAs ternary quantum well with InGaAlAs lattice-matched

1.2 - μ m

barrier. Quantum well width is set to 8 nm with (a) lattice-matched, (b)

1.0 %

1.0 %

tensile strain. The origin of the energy is set to the valence band position of InP .
图 9.计算的具有 InGaAlAs 晶格匹配

1.2 - μ m

势垒的 InGaAs 三元量子阱的价带能量色散。量子阱宽度设为 8 nm，(a) 晶格匹配，(b)

1.0 %

压缩应变和 ©

1.0 %

拉伸应变。能量原点设置为 InP 的价带位置。
lattice-matched barrier. The well width is determined so that the energy difference between the first subband of conduction band and the first subband of the valence band equals to

1.55 μ m

. We fit the in-plane effective mass to relations as follows:
晶格匹配势垒。井宽度的确定是为了使导带第一子带和价带第一子带之间的能量差等于

1.55 μ m

。我们根据以下关系拟合面内有效质量：

\begin{aligned} \frac{1}{m_{lh}} = A + B ε + C ε^{2} \\ \frac{1}{m_{hh}} = D + E ε + F ε^{2} \end{aligned}

where

A, B, C, D, E

, and

F

are the fitting parameters, and the strain

ε

is expressed in %. We also calculate the inplane effective mass for InGaAsP

1.6 μ m

quaternary well with 1.2-

μ m

lattice-matched barrier, and for InGaAs ternary well with strain-compensated

1.2 - μ m

quaternary barrier. The values obtained by fitting are listed in Table IV. It is not appropriate
其中

A, B, C, D, E

和

F

是拟合参数，应变

ε

以 % 表示。我们还计算了具有 1.2-

μ m

晶格匹配势垒的 InGaAsP

1.6 μ m

四元阱和具有应变补偿

1.2 - μ m

四元势垒的 InGaAs 三元阱的平面内有效质量。通过拟合得到的数值列于表 IV。不适合

TABLE IV 表 IV
In-Plane Effective Mass of InGaAsP Barrier (

1.2 μ m

)/nGaAs§ Well and InGaAlAs Barrier

(1.2 μ m) / InGa (Al)

As Well System for

1.55 μ m

Operation
InGaAsP 势垒 (

1.2 μ m

)/nGaAs§ 井和 InGaAlAs 势垒

(1.2 μ m) / InGa (Al)

井系统在

1.55 μ m

操作中的平面内有效质量

	-2.0< Strain $(%) < - 0.5$ -2.0< 应变 $(%) < - 0.5$			$0 < Strain (%) < 2.0$
	A	$B$	$C$	D	E	$F$
Lattice-matched InGaAsP barrier/InGaAs well 晶格匹配的 InGaAsP 势垒/InGaAs 井	-4.238	-6.913	-1.687	4.260	2.253	-0.584
InGaAsP ( $1.6 - μ m$ ) well InGaAsP ( $1.6 - μ m$ ) 井	-0.936	-4.825	-1.754	2.986	5.505	-1.736
Strain-compensated InGaAsP barrier/InGaAs well 应变补偿 InGaAsP 势垒/InGaAs 井	-7.761	-13.601	-5.100	4.166	1.933	-0.558
Lattice-matched InGaAlAs barrier/InGaAs well 晶格匹配的 InGaAlAs 势垒/InGaAs 井	-3.469	-6.133	-1.481	3.9134	2.336	-0.633
InGaAlAs ( $1.6 - μ m$ ) well InGaAlAs ( $1.6 - μ m$ ) 井	-1.297	$- 5.598$	-2.104	2.725	6.317	-1.766
Strain compensated InGaAlAs barrier/InGaAs well 应变补偿 InGaAlAs 势垒/InGaAs 井	-5.889	-9.467	-2.730	4.193	1.075	-0.155

TABLE V 表 V
In-Plane Effective Mass of InGaAsP Barrier (

1.1 μ m

)/InGaAs§ Well and InGaAlAs Barrier (

1.1 μ m

)/InGa(Al)As Well System for

1.30 - μ m

Operation
InGaAsP 势垒（

1.1 μ m

）/InGaAs§ 井和 InGaAlAs 势垒（

1.1 μ m

）/InGa(Al)As 井系统在

1.30 - μ m

操作中的平面内有效质量

	$- 2.0 < Strain (%) < - 1.0$			$- 0.5 < Strain (%) < 2.0$
	A	$B$	C	D	E	$F$
Lattice-matched InGaAsP barrier/InGaAs well 晶格匹配的 InGaAsP 势垒/InGaAs 井	-9.558	-8.634	$- 1.847$	4.631	1.249	-0.313
InGaAsP ( $1, 4 - μ m)$ well InGaAsP ( $1, 4 - μ m)$ 井	-2.453	-4.432	-1.222	3.327	3.425	-1.542
Strain-compensated InGaAsP barrier/InGaAs well 应变补偿 InGaAsP 势垒/InGaAs 井				4.720	0.421	-0.014
Lattice-matched InGaAlAs barrier/InGaAs well 晶格匹配的 InGaAlAs 势垒/InGaAs 井	-8.332	-8.582	-2.031	3.931	1.760	-0.543
InGaAlAs ( $1.4 - μ m$ ) well InGaAlAs ( $1.4 - μ m$ ) 井	-3.269	-5.559	-1.594	3.164	4.065	-1.321
Strain-compensated InGaAlAs barrier/InGaAs well 应变补偿 InGaAlAs 势垒/InGaAs 井				4.078	0.516	-0.0621

Fig. 10. Strain dependence of the (a) well width and quantized energy position and (b) in-plane effective mass for InGaAs ternary well with lattice-matched InGaAsP

1.2 - μ m

barrier. The well width is set such that the energy difference between the first subband of conduction band and the first subband of the valence band is

1.55 μ m

.
图 10.具有晶格匹配 InGaAsP

1.2 - μ m

势垒的 InGaAs 三元阱的 (a) 阱宽和量化能量位置以及 (b) 面内有效质量的应变依赖性。阱宽设定为导带第一子带与价带第一子带之间的能量差为

1.55 μ m

。
to define the approximate effective mass at the strain around

- 0.2 %

, because the energy dispersion is far beyond a parabolic curve due to the fact that the subband energies of heavy and light holes are very close to each other.
来定义

- 0.2 %

附近应变的近似有效质量，因为重洞和轻洞的子带能量非常接近，能量弥散远远超出了抛物线曲线。

Figure 11 shows the strain dependence of the in-plane effective mass for InGaAs ternary well with InGaAlAs 1.2

μ m

lattice-matched barrier. The values obtained similarly are listed in the lower rows in Table IV.
图 11 显示了具有 InGaAlAs 1.2

μ m

晶格匹配势垒的 InGaAs 三元阱的面内有效质量的应变依赖性。类似得到的数值列于表 IV 的下一行。
C. Effective Mass of

InGaAsP / InGaAs (P) and

InGaAlAs/InGa(Al)As on InP System for 1.30-

μ m

Operation
C. InP 系统上的

InGaAsP / InGaAs (P) and

InGaAlAs/InGa（Al）As 在 1.30-

μ m

操作条件下的有效质量
We also calculated the in-plane effective mass of the strained quantum well system for

1.30 - μ m

operation. The barrier bandgap wavelength was

1.1 μ m

, and the bandgap wavelength of quaternary well was

1.4 μ m

. The results are listed in Table V. In this case, the mixing between heavy and light hole valence bands is strong at the strain around

- 1.0 %

.
我们还计算了

1.30 - μ m

工作时应变量子阱系统的面内有效质量。势垒带隙波长为

1.1 μ m

，四元阱带隙波长为

1.4 μ m

。在这种情况下，在

- 1.0 %

附近的应变处，重空穴价带和轻空穴价带之间的混合很强。

D. Discussion D.讨论情况

Generally, symmetric band structure between conduction and valence bands are suitable for a good gain material. In this sense, the in-plane effective mass of the hole should be close to the electron effective mass. The inverse electron effective mass

1 / m_{e}

is around 20 in most cases. Therefore, the inverse hole effective mass should be close to 20. From Figs. 10 and 11 and the other calculation results, the compressive strained well is suitable for a good gain material.
一般来说，导带和价带之间的对称能带结构适用于良好的增益材料。从这个意义上说，空穴的面内有效质量应接近电子有效质量。在大多数情况下，反电子有效质量

1 / m_{e}

约为 20。因此，反向空穴有效质量也应接近 20。从图 10 和图 11 以及其他计算结果来看，压缩应变井适合作为良好的增益材料。

However, we have to consider the other aspects to obtain a high performance laser. Important parameters are the Auger recombination rate [20] and gain saturation coefficient [21], [22], which are not investigated here. Another parameter is the bandgap discontinuity ratio which determines the carrier confinement. If we only want to obtain a strong electron confinement structure by InGaAsP system, then tensile-strained barrier and lattice-matched well is the best pair, for example. Therefore, we cannot conclude that the best material for the laser is compressive strained well. It depends on what we want
然而，要获得高性能激光器，我们还必须考虑其他方面。重要参数包括奥杰尔重组率 [20] 和增益饱和系数 [21], [22]，在此不做研究。另一个参数是决定载流子约束的带隙不连续率。如果我们只想通过 InGaAsP 系统获得强电子约束结构，那么例如拉伸应变势垒和晶格匹配阱就是最佳配对。因此，我们不能得出结论说激光器的最佳材料是压缩应变阱。这取决于我们想要

Fig. 11. Strain dependence of the (a) well width and quantized energy position and (b) in-plane effective mass for InGaAs ternary well with latice-matched InGaAlAs

1.2 - μ m

barrier. The well width is set such that the energy difference between the first subband of conduction band and the first subband of the valence band is

1.55 μ m

.
图 11.具有 Latice 匹配 InGaAlAs

1.2 - μ m

势垒的 InGaAs 三元阱的 (a) 阱宽和量化能量位置以及 (b) 面内有效质量的应变依赖性。阱宽设定为导带第一子带与价带第一子带之间的能量差为

1.55 μ m

。
the laser to be. We believe that the set of listed data here are quite convenient for choosing the ideal material.
的激光器。我们相信，这里列出的一组数据对于选择理想材料非常方便。

V. SUMMARY V.总结

We described the band lineups of InGaA1As on (001) InP as well as InGaAsP on (001) InP system with strain effects, based on the Harrison model. We showed that the compressive strain does not affect the band lineup much, and tensile strain raises the band lineups in the InGaAsP system. It was also shown that the both compressive and tensile strains raise the band lineups in the InGaA1As system. The conduction and valence band positions of InGaAs, InGaAsP, and InGaAlAs relative to InP valence band are given in approximate formulas as a function of the strain.
我们根据 Harrison 模型描述了 InGaA1As on (001) InP 和 InGaAsP on (001) InP 系统的应变效应带阵列。我们的研究表明，压缩应变对带阵列影响不大，而拉伸应变会提高 InGaAsP 系统的带阵列。研究还表明，在 InGaA1As 系统中，压缩应变和拉伸应变都会提高能带阵列。InGaAs 、InGaAsP 和 InGaAlAs 的导带和价带位置相对于 InP 价带的位置与应变的函数关系用近似公式给出。

We calculated the energy versus in-plane wave vector relation of the

InGaAsP / InGaAs (P)

on InP and InGaAlAs/InGa(Al)As on InP strained quantum well systems. We obtained the in-plane effective mass of the strained quantum well system by fitting the dispersion to a parabolic curve. We listed the in-plane effective mass of several kinds of strained quantum well system. We believe that these values are very convenient for designing strained quantum well based devices.
我们计算了InP和InGaAlAs/InGa(Al)As on InP应变量子阱系统的

InGaAsP / InGaAs (P)

能量与面内波矢量关系。通过将色散拟合为抛物线，我们得到了应变量子阱系统的面内有效质量。我们列出了几种应变量子阱系统的面内有效质量。我们相信，这些数值对于设计基于应变量子阱的器件非常方便。

ACKNOWLEDGMENT 致谢

The authors acknowledge useful discussions with Prof. H. Kroemer, D. Babic, and M. Ishikawa.
作者感谢与 H. Kroemer、D. Babic 和 M. Ishikawa 教授进行的有益讨论。

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Takuya Ishikawa (S’87-M’89) was born in Ehime, Japan, in 1961. He received the B.E.M.E. and Ph.D. degrees in electronic engineering from the University of Tokyo, Japan, in 1984, 1986, and 1990, respectively.
Takuya Ishikawa（1987-1989 中学毕业）1961 年出生于日本爱媛县。他分别于 1984 年、1986 年和 1990 年获得日本东京大学电子工程学士和博士学位。
In 1990, he joined the optotechnology laboratory of Furukawa Electric Co., Ltd., where he is currently working for the research and development of semiconductor-based waveguide devices. He spent two years from 1991 to 1993 as a visiting researcher at the University of California, Santa Barbara.
1990 年，他加入古河电工株式会社的光学技术实验室，目前从事基于半导体的波导器件的研究和开发工作。1991 年至 1993 年，他在美国加州大学圣巴巴拉分校做了两年访问研究员。

John E. Bowers (S’78-M’81-SM’85-F’93) received the M.S. and Ph.D. degrees in applied physics from Stanford University, with a dissertation on Sezawa wave signal processing devices.
约翰-鲍尔斯（John E. Bowers，S'78-M'81-SM'85-F'93）获得斯坦福大学应用物理学硕士和博士学位，论文题目为塞泽波信号处理装置。
He is a professor in the Department of Electrical Engineering at the University of Califomia, Santa Barbara. He has also worked for AT&T Bell Laboratories and Honeywell and is currently an optoelectronics consultant. He is a member of the Optoelectronics Technology Center and the NSF Science and Technology Center on Quantized Electronic Structures. His research interests are primarily concemed with high-frequency optoelectronic devices and physics.
他是加州大学圣巴巴拉分校电子工程系教授。他还曾供职于 AT&T 贝尔实验室和霍尼韦尔公司，目前是一名光电顾问。他是光电子技术中心和国家科学基金会量子化电子结构科技中心的成员。他的研究兴趣主要涉及高频光电设备和物理学。
Dr. Bowers was elected as a fellow of the IEEE “for contributions to the understanding and demonstration of novel ultrafast semiconductor lasers, photodetectors, and transmission systems.” He is a recipient of the Thomas F. Andrew prize and the NSF Presidential Young Investigator Award. He has published over 160 papers and has received 12 patents.
鲍尔斯博士因 "对新型超快半导体激光器、光电探测器和传输系统的理解和演示所做的贡献 "而当选为电气和电子工程师协会会员。他还获得了托马斯-安德鲁奖（Thomas F. Andrew prize）和美国国家科学基金会总统青年研究员奖（NSF Presidential Young Investigator Award）。他发表了 160 多篇论文，获得了 12 项专利。

Manuscript received June 7, 1993. This work was supported by the Defense Advanced Research Projects Agency through the ULTRA Program.
手稿于 1993 年 6 月 7 日收到。这项工作得到了美国国防部高级研究计划局（Defense Advanced Research Projects Agency）ULTRA 计划的支持。
T. Ishikawa is with the Department of Electrical and Computer Engineering, University of California, Santa Barbara, CA 93106 on leave from Furukawa Electric Co., Chiba, Japan.
T. Ishikawa现供职于加州大学圣巴巴拉分校电气与计算机工程系，任期为93106年。
J. E. Bowers is with the Department of Electrical and Computer Engineering, University of California, Santa Barbara, CA 93106.
J. E. Bowers 是加州大学圣巴巴拉分校电气与计算机工程系的一员，地址：加州 93106。

IEEE Log Number 9215744.
IEEE Log Number 9215744。

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

Abstract 摘要

I. Introduction I.导言

II. Band Lineup II.乐队阵容

A. Calculation ProcedureA.计算程序

B. Band Lineup with No StrainB.无压力乐队阵容

C. Energy Correction Induced by StrainC.应变引起的能量修正

D. Band Discontinuity RatioD.频带不连续率

E. Band Lineup Change Purely Induced by StrainE.纯粹由应变引起的乐队阵容变化

F. Approximate Expression of Band PositionF.乐队位置的近似表达式

III. Valence Band Structure of Strained Quantum Well.III.应变量子阱的价带结构。

A. Calculation Method A.计算方法

B. Energy Dispersion of InGaAsP/InGaAs on InP SystemB.InGaAsP/InGaAs on InP 系统的能量散布

C. Energy Dispersion of InGaAlAs/InGaAs on InP SystemC.InGaAlAs/InGaAs on InP 系统的能量散布

IV. In-Plane Effective Mass of Strained Quantum WellIV.应变量子阱的平面内有效质量

A. Parabolic Approximation of Energy DispersionA.能量扩散的抛物线近似法

B. Effective Mass of InGaAsP/InGaAs(P) and InGaAlAs/InGa(Al)As on InP Systemfor I.55- μ m μ m mum\mu \mathrm{m} OperationB. InGaAsP/InGaAs(P) 和 InGaAlAs/InGa(Al)As 在 I.55- μ m μ m mum\mu \mathrm{m} 操作的 InP 系统上的有效质量

D. Discussion D.讨论情况

V. SUMMARY V.总结

ACKNOWLEDGMENT 致谢

References 参考资料

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

A. Calculation Procedure
A.计算程序

B. Band Lineup with No Strain
B.无压力乐队阵容

C. Energy Correction Induced by Strain
C.应变引起的能量修正

D. Band Discontinuity Ratio
D.频带不连续率

E. Band Lineup Change Purely Induced by Strain
E.纯粹由应变引起的乐队阵容变化

F. Approximate Expression of Band Position
F.乐队位置的近似表达式

III. Valence Band Structure of Strained Quantum Well.
III.应变量子阱的价带结构。

B. Energy Dispersion of InGaAsP/InGaAs on InP System
B.InGaAsP/InGaAs on InP 系统的能量散布

C. Energy Dispersion of InGaAlAs/InGaAs on InP System
C.InGaAlAs/InGaAs on InP 系统的能量散布

IV. In-Plane Effective Mass of Strained Quantum Well
IV.应变量子阱的平面内有效质量

A. Parabolic Approximation of Energy Dispersion
A.能量扩散的抛物线近似法

B. Effective Mass of InGaAsP/InGaAs(P) and InGaAlAs/InGa(Al)As on InP Systemfor I.55- $μ m$ Operation
B. InGaAsP/InGaAs(P) 和 InGaAlAs/InGa(Al)As 在 I.55- $μ m$ 操作的 InP 系统上的有效质量