key words

capture time , photons density, quantum spin lasers

Introduction

The laser is a type of excited and energetic light that is not normally seen in nature, but it can be created with special technology and equipment. Laser has differences with normal light, which these properties causes its particular capabilities and applications. Laser light can break down the hardest metals and easily pass from hard objects such as diamond and in it create hole. Low-power and delicate extraordinary strips other lasers can be used to perform for very delicate works, such as human eye surgery. Laser light can brought under control with great precision and used as a continuous strip called a continuous laser or fast explosions called a pulse laser. Unlike normal light, laser light has a perfectly concerted energy, which through this creates a lot of power to do different works. The word laser is derived from the initials of words that describe its properties, which means “light amplification by stimulated emission of radiation”1-5.

Spin-to-semiconductor electric injection

Another possible spintronic application is the injection of a polarized spin current into a semiconductor system. The polarized spin current, is current which in it a population of a spin specie is more than the other species. Since spin coherence time in semiconductors is much longer than that of spin coherence time in metals, the discussion of spin injection into semiconductors is important. When electrons are injected from a ferromagnetic matter to a nonmagnetic matter, they can keep their spin on a determined distance. The necessary conditions for maintaining this spin polarization are successful spin injections, spin transitions in the semiconductor interior with a multi-micron spin, spin Lifetime greater than 100 ns, and ultimately successful detection. In order to select the suitable material for spin injections, the material should include the polarized spin Carriers Unit Orientation at room temperature. There are many suggestions for electron injectors but the most obvious and definite selection of ferromagnetic materials is due to high temperature Curie, we have chosen their low induction and fast magnetization switching. The induction of a ferromagnetic material is defined as the magnetic field intensity applied to reduce the magnetization of that material to zero, after the sample magnetization has reached saturation. However, there are problems with the use of ferromagnetic materials as spin Injectors to semiconductors. The main problem is the mismatch of conductivity that happens at the interface between ferromagnetic and semiconductor. There are three ways to fix this problem: use of 100% polarized spin material as an injector, use of materials with similar conductivity with semiconductor and use of tunnel barrier6, 7. First possible solution is the use of semi-metallic ferromagnetism. The idea for semi-metallic ferromagnetic was first proposed by Devere et al. In 19836. Semi-metallic ferromagnetism are materials that have a band gap in a Fermi level for a spin substrate that they make them 100% spin polarized. Although research on semi-metallic ferromagnetism suggests the absence of a system with a spin polarization of 100% at room temperature.

Analytical study of Polarized spin semiconducting lasers

The laser is light amplification by stimulated emission of radiation. The importance of lasers to their widespread use, one of the key characteristics of the laser is the dependence of emitted light to injection. Two polarized and non-polarized spin semiconductor lasers can be separated. Depending on injected the spin carriers are, polarized or unconsolidated. Both lasers have are active region, resonance cavity and carriers injector In polarized spin semiconducting laser, laser operations are directed by charge carriers current in a laser cavity, while the polarization of light emitted by the spins of the carriers is defined. The spin laser heart is a fragment that is called vertical-cavity surface-emitting laser, which is basically a semiconductor laser, the laser beams in the direction z, emitted perpendicular to the sub-layer material on which it grows. Spin electric injections are made using two ferromagnetic connections. Laser resonance cavity consists of two Bragg reflector mirrors parallel to the surface of the sub-layer and laser active region including a well or dot quantum to produce laser light among them. Distributed Bragg Reflector (DBR) mirror consisting of layers with a low and high the refractive coefficient intermittent. Typically, the higher and lower mirrors are p and n types doping materials respectively, which make up a p-n bond. Stimulated emission occurs in the active region the semiconductor Laser of the vertical Cavity Surface Emission. In the electronically active region in the conduction band, the semiconductor is subjected passage to empty state in the valiancy band that produces photon and leaves the active region. This process of recombination is called electron-hole irradiation. To replenish the electron and hole populations, the charge carriers must be injected into the active region with electrical current. Population inversion is achieved by threshold injecting which the number of electrons in the conduction band is greater than the number of electrons remaining near the top of the valence band. When this happens, the laser cavity losses are defeated and the laser action begins. Threshold current is the current that must be injected to allow laser operations to occur.If this current is too high, the laser may be impossible for particular applications. The elective rules for optical transitions in quantum well and quantum dot structures depend on participating carriers spin. A spin-up electron of conduction band can only be recombination with a spin-down hole. In this action, a photon is emission with a leftward circular polarization. Conversely, a spin-down electron recombination with a spin-up hole that a photon is emission with a rightward circular polarization. Accordingly, polarized spin semiconductor lasers host two unequal-intensity laser states with left and right circular polarizations. When the semiconductor laser is powered by a polarized spin current, a decrease in threshold current density is observed. Note that if the laser is powered by up-polarized spin current, then the two laser states have equal intensities. It is important to note that electrons spin can be conversion without light emission, and this spin relaxation can limit many of the properties of polarized spin semiconductor laser.In this paper, we examine the similarities between quantum well (QW) and quantum dot (QD) used as material gain in semiconductor lasers. For active region based on QD, a more complex description is needed. Therefore, mapping between QD and QW lasers has the ability to easily describe for potential returns. To create such a mapping, our focus is on two conventional lasers (spin un-polarized) and spin-lasers that in it the carriers are spin-polarized. And are injected by circular polarized light or electric injection using magnetic contact. Figure 1 shows a diagram of the conduction band using rate equations for quantum well lasers and quantum dots. In quantum wells, a spinal alignment between injectable carriers leads to the recombination of a number of electron-holes, which causes emitted circular light polarization. S represents the emitted photons with positive and negative helicity, respectively. But the quantum dot laser has several additional processes due to the wetting layer compared to the quantum well laser. The wetting layer (WL) acts as the reservoir of carriers7-9. Carriers conquered from WL to QD or, conversely, they can escape from QD to WL.

Figure 1. Conduction band diagram in semiconductor lasers. (a) Quantum well (QW) laser. (b) Quantum dot (QD) laser.

Figure 1. Conduction band diagram in semiconductor lasers. (a) Quantum well (QW) laser. (b) Quantum dot (QD) laser.

Bucket model of lasers

One of the most useful comparisons to show the operation of polarized spin semiconductor laser is by the Bucket model, which was previously intended only for un-polarized spin lasers. To describe the Bucket model for polarized spin semiconductor lasers, must first described this model for un-polarized semiconductor lasers. In Figure 2, first showed the bucket model for conventional lasers (spin-unpolarized) and then for polarized spin lasers using rate equations. For conventional lasers, the water level represents the density of the charge carriers, the water faucet represents the carrier injection, the output water represents the emitted light which is the density of the photon, the small leaks on the surface indicate the processes of spontaneous recombination and the large slit on the bucket indicates the lasing threshold (resonance threshold). In low injection (low pumping), the water level is low and some of this water will drip out of the leaks. In this state, there is only insignificant amount of output light. The operation of such laser is like light emissary, spontaneous recombination is a response for light emissary. At high injection (high pumping), the water will reach a large slit and begin to erupt, reaching the lasing threshold (resonance threshold), and Induction recombination dominates the light emission. Excess water will only lead to a small change in water level, but the output will increase rapidly compared to low injection state. At the injection thresholdJ_T, the spontaneous emission begins and the intensity of the emitted light increases significantly. J>JT Refers to the inductive recombination leasing performance, which indicates how to work the light emission. The description of section (a) in Figure 2 is similar to Figure 1-a mechanism.

Figure 2. Bucket model of lasers. a) Bucket model simulation for conventional lasers. b) Bucket model simulation for spin lasers

Figure 2. Bucket model of lasers. a) Bucket model simulation for conventional lasers. b) Bucket model simulation for spin lasers.

In section (b) from Figure 2 illustrates spin laser structural with polarized spin carriers. If you pay attention, the bucket is divided into two halves by the connecting wall. Which indicates the injection of two spin populations, which are filled particularly with hot and cold water, respectively. The openings on connecting wall in Segmentation allow the hot and cold water to mix, the openings on connecting wall in segmentation allow the hot and cold water to mix, which by doing so, it intends to show the spin relaxing that cause mixes the spin-up and spin-down population5. If connecting wall has a small diameter that can be ignorable. The two populations will not be mixed, which corresponds to the time when is unlimited spin relaxation. If the connecting wall is too wide, the immediately unequal populations will be in equilibrium. This leads to carrier polarization of ignorable. With an unequal injection of hot and cold water, the spin injection polarization is defined as follows11:

PJ=((J+-J )⁄J (1)

In it, J+ injections are of two spin images that together form the total injection (electric spin injection) J=J++J ∆The difference in levels is cold and hot water (see Figure 2, section b), this leads to three functional region of the carrier and two different leasing thresholds JT1,212 Small

leaks shows carrier loss through spontaneous recombination, and large slit near the top part indicate the lasing threshold. Total electron density is written as n=n++ntotal hole density is written as P=P++Pand total photon density is written as S=S++S-In the range of time when spin relaxation is zero, we expect polarized and un-polarized spin semiconductor lasers to behave the same way. Based on the Bucket model, can provide an intuitive understanding of polarized spin semiconductor laser operations. Based on the Bucket model for polarized spin semiconductor lasers, which confirms the existence of two threshold current densities Jt1and Jt2 for carriers with majority and minority spins, for this type of laser, as we said, three functional region are considered. Region I, is region where there is no inductive emission and the laser is off. Region II, is the region where in it carriers have the majority spin lead to lasing (resonance), known as the normalized spin filtering interval. Region III, is the region where carriers both types of spin lead to resonance (lasing).In low pumping (when both hot and cold water levels are below the large slit), both carriers spin-up and a spin-down are in off region of the light emitting diode (LED), thus, the emitting is insignificant. In higher pumping, the hot water reaches a large slit and erupts as shown in Figure (2, section b). While the amount of cold water that flows out is static and insignificant. This indicates that a region of spin majority is in the leasing state. While the spin minority is still in the LED region, therefore the inductive emission is due to the recombination of the spin majority carriers. By describing section b in Figures 1 and 2, we described the operation of quantum dot lasers with polarized spin carriers as a schematic and a bucket model.

Investigating rate equations in conventional and spin lasers QD and QW:

Rate equations in conventional lasers:

In this paper, we first consider rate equations for conventional lasers that used quantum wells as their material gain. Structurally, this laser is similar to Section (a) in Figure 1, also in the previous section structure simulation of this laser with Bucket model was presented (see Figure 2 of Section (a)). Therefore, according to the rate equations, direct relationship between material properties and laser device parameters can be provided. For QW conventional laser, the rate equations are as follows13

dn/dt=J-g(n,S)S-Rsp “(2)”

ds/dt=Γg(n,S)S+ΓβRsp-S⁄τph “(3)”

n Demonstrative the carriers density and S is photons density. The optical gain is as follows14:

g(n,S)=(g0 (n-ntran ))⁄((1 +ϵS) ) “(4)”

In the paper, g0 is the gain coefficient reagent14, ntran is the transparency density, and here ϵ is the gain compression factor13. Rsp Spontaneous recombination can be dependent on different densities8. Here we focus on the quadratic equations, Bn2 , which B is temperature dependent constant. Γ Represents the optical confinement factor, β represents the spontaneous emission factor12, 15. τPh Indicates the photon lifetime.

According to scrutiny rate equations of (4) and (5) and the optical gain of (6), we find that rate equations are exponentially increased on the variations of t from zero to 50 for n (carrier density), as well as for S (photon density), as shown in Fig. 3, is negligible in relation to t, This means that n is increased more than S versus t.

Figure 3. Calculation time variations of rate equations for carrier and photon density, n and S in conventional lasers.

Figure 3. Calculation time variations of rate equations for carrier and photon density, n and S in conventional lasers.

Rate equations in spin lasers:

Now, according to Figures 1 and 2, we describe the rate equations for spin lasers with polarized spin carriers that use quantum dots as material gain. As we have said, since QD spin lasers structure has several processes addition to QW conventional lasers, the rate equations of QD spin laser are more complex than QW conventional laser. The rate equations for QD spin lasers are described as follows13:

dn±/dt=J±-g±(n±,S± ) S±-((n±-n ))⁄(τsn -Rsp± “(5)”

ds±/dt=Γg (n,S± ) S±-S±⁄τph -βΓRsp “(6)”

In this paper, – / + subscript (superscript) represent the electron spin (photon helicity). Here, τsn represents the electron spin relaxation time, which for PJ=0 in conventional lasers, accordance with spin relaxation F=(n±-n )/τS. Spontaneous recombination rewritten as follows: Rsp±=2 Bn± p±, which τsp→0 approves hole spin instantaneous relaxation. The hole density is thus eliminated p+=p=p⁄2 =(n++n )/2 , While the results in Rsp±=B n± (n++n) are assumed to charge negative.

In the review of the rate equation for spin lasers according to equation (5) and (6), we find that electrons spin number in positive and negative states increased exponentially, also, Calculation time variations of electrons spin number from electrons helicity number in positive and negative states are higher. Electrons helicity in both states, there is a negligibly increased versus t.

In this section, we compare the rate equations for conventional and spin lasers according to Figures 3,4and Table 1. We find that time variations of electrons spin number (n) in the conventional lasers are higher than spin lasers. But, time variations of electrons helicity (S) in the conventional lasers in relation to spin lasers is decreased by approximately 0.03. The carrier’s density in conventional lasers are initially zero and increase with lapse time, and after 25 seconds, the laser emits a constant light. The photons density is such that only increasing it is very small compared to increasing the carriers density. The carrier’s density in spin lasers are initially zero and increase with lapse time, and after 5 seconds, the laser emits a constant light. The photons density is such that only increasing it is very small compared to increasing the carriers density similar to conventional lasers.

Figure 4. Calculation time variations of rate equations for carrier and photon density, n and S in spin lasers

Figure 4. Calculation time variations of rate equations for carrier and photon density, n and S in spin lasers.

Table 1. Investigation numerical values of time variations of Rate equation for convention and spin lasers.

50 40 30 20 10 0 t
16.191 16.190 16.176 15.981 13.761 0 n
1.007 1.007 1.007 1.007 1.007 0 n+
1.007 1.007 1.007 1.007 1.007 0 n
0.043 0.043 0.043 0.042 0.031 0 S
0.061 0.061 0.061 0.061 0.060 0 S+
0.061 0.061 0.061 0.061 0.060 0 S

The net rate of levels occupation probability by electrons and holes in rate equations for QW conventional lasers

Here is we writes briefly time dependence of levels occupying probability by electrons, for QW conventional lasers (according to reference 16, 17):

(dfw)/dt=I-C+2/K E-Rw “(7)”

(dfq)/dt=K/2 C-E-Rq-G “(8)”

(dfS)/dt=ΓQD G+ΓQD βRq-fS⁄τPh “(9)”

Here, w and q subscripts represent the WL and QD regions (according to Figures 1 and 2), and the S subscript is related to photon (place the laser light is emitted). Here fw,q are indicative the probability of electrons occupying in QD and WL regions, that are written as fw=n ̅w⁄Nw and fq=n ̅q⁄(2Nq ) (0<fw,q<1) . In these, n ̅w,q are indicative the electrons corresponding number, and Nw the states number in WL and Nq are quantum dots number, that each dot has a twofold spin degenerate level. The relationship between the states number in WL and quantum dot number is shown by K=N_w⁄Nq . fS Is indicative the probability of photon occupation (fS=S ̅⁄(2 Nq )), which S ̅ is the number of cavity photons, that doesn’t have a high limit.

5

Figure 5. Calculation time variations of rate equations for carrier and photon density, and in spin lasers.

The process of electrons injecting from witting layer region to quantum dot is I=j(1 -fw ), where j is the number of electrons injected into the laser in witting layer state and unit of time. Also, the process of electrons injecting from witting layer region to quantum dot is C=fw ((1-fq ))⁄τc , and the carrier escape process that is the opposite with capture process, is represented by E=fq ((1-fw ))⁄τe . In these, τc and τe are the capture and escape times, respectively.

According to Figure 2-b, there are two processes of spontaneous recombination and inductive emission, which show the spontaneous recombination process asR_η=bη fη2, and (η=w,q). Thus, the inductive emission process is indicated by G=g(2 fq-1 ) fS, where g is not related to gain compression factor and photons occupied, and used in quantum dot lasers. Here the light confinement factor in quantum dot lasers is equal to one (ΓQD=1).

In here according to table.2, we reviewed first carrier capture and the escape for QW lasers, then carrier injection and at the end responsible for stimulated emission. We find out that in the states fw constant and fq variation between(0-1), carrier escape increase and carrier capture decrease, therefore E and C act on the inverse.

Then according to table.2 and Fig.8, we review variation carriers’ injection in the states that electrons occupation in the wetting layer f_w constant and j variation. In here we conclude with increase fw and, carriers’ injection J decreases. Next, according to table.2 and Fig.9, we find that responsible for stimulated emission in the state f_q=0, G decreases and fq=1/2, G is equal to zero, but in the states fq=1 unlike fq=0, G increased.

Table 2. Investigation numerical values of injection, capture, escape, and the stimulated emission carriers.

G fq f_S I fw J E C fw fq
-0.008 1/2 1 1 1/2 00 1/2
-0.0016 1 2 2 1 00 1
00 1/2 00 0 1/2 00 00 0.25 1/2 00
00 1/2 1/2 1 1/4 0.12 1/2
00 1 1 2 1/2 00 1
00 1 00 00 1 00 00 0. 5 1 00
0.008 1/2 00 1 00 0.25 1/2
0.0016 1 00 2 00 00 1

Figure 6. Variation carriers escape versus electron occupation in the wetting layer (WL) and quantum dot regions (QD)

Figure 6. Variation carriers escape versus electron occupation in the wetting layer (WL) and quantum dot regions (QD).

7

Figure 7. Variation carriers captuer versus electron occupation in the wetting layer (WL) and quantum dot regions (QD).

Figure 8. Variation carriers injection versus electron occupation in the wetting layer (WL) and number of carriers (electrons) injected into the laser per WL state and unit time (j).

Figure 8. Variation carriers injection versus electron occupation in the wetting layer (WL) and number of carriers (electrons) injected into the laser per WL state and unit time (j).

Figure 9. Variation responsible for stimulated emission versus electron occupation in the quantum dot regions (QD) and photon occupancy

Figure 9. Variation responsible for stimulated emission versus electron occupation in the quantum dot regions (QD) and photon occupancy.

The net rate of levels occupation probability by electrons and holes in rate equations for QW spin lasers

In this part of the paper, the time developments of the occupation probability in rate equations of spin laser are analyzed. Then, by analyzing the difference between the rate equations of spin laser with material gain QW and QD, It is found that the rate equations for quantum dot spin lasers have obvious terms for holes occupation. However, in spin lasers with material gain of quantum well, holes densities can be easily filled or replaced by electron densities. Therefore, for spin lasers with quantum dot material gain, unlike quantum wells, for holes, the spin relaxation time is much faster(τspwspq→0 ). This cause more difficult to analyze spin lasers with quantum dot material gain at steady state, as the holes density associated with rate equations of the quantum dot spin laser does not increase explicitly. Generally the time developments of the levels occupation probability by electrons in rate equations for spin laser, it is as follows13.

dfwα±/dt=Iα±-Cα±+2/κα Eα±-R∓f “(10)”

dfqα±/dt=κα/2 Cα±-Eα±-R-G±∓f “(11)”

(dfS∓/dt=G±+βR-fS∓)⁄τPh “(12)”

Here is the subscriptα=n,p, which n represents the electrons and p holes, respectively. Iα± =j(1 -f± ) Represents the injection carrier, Cα±=fwα± 1-fα± q)⁄τc the capture carrier andEα± =f± 1-f± ⁄τe in the spin laser. While jα±=(1 ±P ) jα represents the number of injected carriers, in which P=((jα+)-jα-) )⁄(jα++jα )expressive the spin Polarization. The two processes of inductive and spontaneous emission in spin lasers are G±=g(fqn±)+fqp±-1 ) f)and Rη±)=bη fηn± fηp±). In the process of spontaneous emission is η=w,q and the spontaneous recombination rate is denoted bybη. In the spontaneous emission, fηα=fηα+-fηα-⁄τsαη represents the period of the spin relaxation, which τsαηis the spin relaxation time. To calculate and analyze the rate equations in this topic, τC، τe، τsnηs، τspη=0 ، β=0 ، Jα=J و P=Pj are assumed. Occupied electrons are 0≤fw,q≤1 , the probability of polarized spin occupancies is shown in three areas: wetting layer fwα± =n ̅wα± ⁄N/2) ), quantum dot fqα± =n ̅qα± ⁄Nq ،f ), and photon production f =s±⁄Nq .

In this section, we examined the time developments of the occupation probabilities of electrons and holes and photons in the rate equation in both QD and QW lasers for two state conventional (un-polarized) and spin-polarized. In the conventional state according to Figures (5) and (10) and Table.3, we review the time developments of the occupation probabilities electrons in wetting layer (WL) and quantum dot regions, then at end the time developments of the photons production probability. In conventional state, only are electrons and photons occupation probabilities. According to structure of quantum dot lasers, it is expected that the probability of electron occupation in the quantum dot region is higher than the wetting layer region, which according to Figure 10 and study the time developments equations in all three regions, we understand that it is true.

Figure 10. The time developments of the occupation probability in rate equations of spin laser

Figure 10. The time developments of the occupation probability in rate equations of spin laser.

Table 3. Investigation numerical values of the temporal evolution of the occupation probabilities in the RE model for conventional and spin lasers.

t 0 10 20 30 40 50
f_ω 0 0.591 0.591 0.591 0.591 0.591
f_(ωn+) 0 0.593 0.593 0.593 0.593 0.593
f_(ωn-) 0 0.595 0.595 0.595 0.595 0.595
f_(ωp+) 0 0.591 0.591 0.591 0.591 0.591
f_(ωp-) 0 0.580 0.580 0.580 0.580 0.580
f_q 0 0.972 0.972 0.972 0.972 0.972
f_(qn+) 0 0.973 0.973 0.973 0.973 0.973
f_(qn-) 0 0.973 0.973 0.973 0.973 0.973
f_(qp+) 0 0.972 0.972 0.972 0.972 0.972
f_(qp-) 0 0.971 0.971 0.971 0.971 0.971
f_S 0 0 0 0 0 0
f_(S+) 0 0 0 0 0 0
f_(S-) 0 0 0 0 0 0

Table 4. The QD laser parameters are , , , , , and (Ref.10, 16).

QW parameters τ_C=0 τ_C=2ps Unit
ϵ_s 0 1.62×〖10〗^(-14) 〖cm〗^3
ϵ_d(Ref.50) 0 6.39×〖10〗^(-15) 〖cm〗^3
g_0 1.90×〖10〗^(-3) 1.65×〖10〗^(-3) 〖cm〗^3 S^(-1)
n_tran 3.50×〖10〗^16 3.58×〖10〗^16 〖cm〗^3
B 143×〖10〗^(-7) 1.28×〖10〗^(-7) 〖cm〗^3 S^(-1)
τ_Ph 2
Γ 0.03
β 0

Then, the time developments of the occupation probabilities of electrons, holes and photons in the wetting layer and the quantum dots region, according to Figure (10) and Table 3, using the carriers of injection, capture, escape, The excitation and spontaneous emission in the spin state are analyzed and evaluated. Here, we conclude that f_(ωn+) the time developments of the occupation probabilities electrons with spin-up in the wetting layer is 0.002 smaller than the spin- down electrons. But 0.002 is larger than the time developments of the occupation probabilities holes with spin-up in the wetting layer. The time developments of the occupation probabilities holes with spin- down in the wetting layer is close to 0.009 from the spin-up state in the holes and near 0.015 from the of occupancy of the electrons in the time developments of the occupation probabilities electrons with up and down spin in wetting layer. Then, we study the time developments of the occupation probabilities electrons and holes with up and down spin at the quantum dot region. Here we find that time developments of the occupation probabilities electrons and holes in the up and down spin increases more than the time developments of the occupation probabilities electrons and holes in the up and down spin in wetting layer, and it’s like conventional lasers close to one. The time developments of the occupation probabilities electrons with up and down spin in wetting layer Both are the same size, and from the time developments of the occupation probabilities, holes with up spin at quantity 0.001 and with down spin is greater quantity 0.002. Finally, with review the time developments of the photons occupation probabilities, we can see which, like conventional lasers, are zero.

STEADY-STATE

According to the rate equations in Section 2, we examine the steady state mapping between two quantum dot and quantum well lasers. Here, to check the steady state, we first extract the parameters of quantum well lasers from quantum dot lasers by solving the rate equations in conventional (un-polarized) and polarized spin states. In this state, according to Figure 1-b, the active region laser includes the quantum dot, and the wetting layer is considered as the quantum well. Ideally, the steady state equations are as follows13:

J=K(Nq⁄V)j (13)

n(J)=(Nq⁄V)[2fq (j)+Kfω (j)] (14)

S(J)=2Γ(Nq⁄V) fs (j) (15)

According to equation (13) and table (5), we conclude that when the electrons number in quantum dot region is fixed and j change between zero and ten, J(j) increases, That is, the increase of J(j) is greater than all states when that the number of electron occupation states within the quantum dot region is equal to〖 N〗_q=100, and in the states that〖 N〗_q=10, J(j) has the least increase. That is depicted in Figure (11). Here, by the figure (12) and table (6), we consider variation n according to the temporal evolution REs QW and QD lasers in the wetting layer and quantum dots region. We first consider three constant states for f_q, that by examining these states, we find that in all states, n increases to an equal magnitude. Then we again calculate n variation in the states where f_(w )is constant and f_q variation. According to the calculations done, we find that in all states there is an increase of 0.2. Here we compute the variation in S according to Equation (15), which is shown in Fig. 13, which increases with f_S, S has a unisonous linear gain of 0.003.

 Figure 11. Variation injection J versus number of QDs N_q with number of carriers (electrons) injected j

Figure 11. Variation injection J versus number of QDs N_q with number of carriers (electrons) injected j.

Table 5. Computation numerical values injection in the RE for QW and QD lasers.

J j N_q J N_q j
0 0 10 0 10 0
0 50 5 5
0 100 10 10
4 4 10 0 40 0
20 50 20 5
40 100 40 10
8 8 10 0 70 0
40 50 35 5
80 100 70 10
10 10 0 0
50 50 50 5
100 100 100 10

Figure 12. Variation n versus electron occupation in the wetting layer (WL) and electron occupation in the QD region (QD) .

Figure 12. Variation n versus electron occupation in the wetting layer (WL) and electron occupation in the QD region (QD) .

Table 6. Computation numerical values and in the RE for QW and QD lasers.

fw fq n fq fw n fS S
0 0 0.2 0 0 0 0
1/2 0.52 1/2 0.1
1 1.02 1 0.2 0
0 1/2 0.2 0 1/2 5 1/2
1/2 0.52 1/2 5.1
1 1.02 1 5.2
0 1 0.2 0 1 10 1
1/2 0.52 1/2 10.1 0.003
1 1.02 1 10.2

Figure 13. Variation photon densities (S) versus photon occupation f_S.

Figure 13. Variation photon densities (S) versus photon occupation f_S.

The results of the steady state mapping are that in different τc the injected light and the injected carrier density are equal for the quantum dot and well lasers13,19. When isτc→0, the injected carriers at quantum dot region are captured and at rate equations isϵ_s=0. This means that behavior of both the QW and QD laser is similar in this state. And photon density increases continuous linearly. But when τc is finite, the carrier density is higher the threshold and the photon density stops. The injection current density is normalized at un-polarized threshold value, JT=NT⁄τr ،with NT shows the total electron density is higher the thresholdN_T=n(J≥JT )=(ΓgτPh1 +ntran. The photon density is normalized in ST=JT ΓτPh .

S+/ST =2/3 [J/JT (1+Pj/2)-1] & S/ST =0 , nT=(J/(JT B1⁄2). (16)

Based on Figure 12 analysis, we find that in a quantum dot laser, the finite effect (ie, increasing N and stopping S by injected carriers) corresponds to the effect of the finite gain coefficient on quantum well laser. So in this state, the rate equations of quantum dot lasers can be replaced by the rate equations of the quantum well laser.

Figure14. Carrier (n) and photon densities (S) as functions of injection n and J are normalized to their threshold values n_T0 and JT0 for capture time τc=0Ps (QD lasers) or gain compression factor ϵs=0 (QW lasers), while (S) is normalized to ST0=S(2J_T0 ). Note that QW laser characteristics for ϵs=0 are identical to QD laser characteristics for τc=0Ps. The mapping parameters used throughout this paper are given in Table 6.

CONCLUSIONS

In this paper, we describe the structure of two types of quantum spin laser form type quantum dots and wells according to the bucket model. Then, with according to the structures, we study the laser equations first in the conventional state, then we propagate the spin state to better gain. That is, carriers and photons density, with according to the optical gain in both the conventional and spin states, is ratio to time. In the end, we examine the system in a steady state for a state not captured ( ) and after passing ( ), which is captured and photons produced. We found that the photon density initially that system does work, but no capture yet, the photon does not emit, but after the passage ( ) that dose capture, photons are produced. So the photon density increases in this case. Also, the carrier density increases because we have a number of electrons and holes in the active region and by injecting the carrier’s density into the active region laser, we find that carriers’ density increases in the system.