user-1: I've read about the jhānas which one can experience from concentration (samatha) meditation. I have been practicing some form of mindfulness, breathing meditation, but have not experienced a state of absorption that is characteristic of the jhānas. I know that the jhānas cannot be experienced only by reading about them, however, the insight from my meditation practice alone has not brought me to them.
From my perspective as a novice practitioner, it makes sense to look to the Zen (derived from the word dhyāna) branch of Buddhism for guidance on the subject of meditation:

In the process of deepening meditation, one can roughly identify three
  distinct stages: the stage of concentration, the stage of meditation,
  and the stage of absorption. 
This dualistic relationship is broken gradually as the practitioner
  moves into the stage of meditation. The ego-conscious activity is
  gradually lessened, and the barriers it set up for itself will
  gradually be removed. When the practitioner enters the stage of
  absorption, the dualistic framing of the mind will be removed such
  that the mind starts structuring itself non-dualistically. There will
  be no separation or distancing between an object of the mind and the
  activity of the mind itself.

I don't have the budget for every book on Zen Buddhism, but I have read all the Zen books on Kindle Unlimited (the content is mostly history, terminology, and sitting accessories). What books are there that go into depth on Zen/samatha meditation practice?

user8044: Right concentration (Samma Samadhi) is what you call jhana.  Right concentration then develop into directed thought and evaluation. Directed thought is when you keep your thoughts on one topic. Thinking of the breath or focusing on the breath, and being inquisitive will lead one to directed thought and evaluation, two of the factors of jhana.  If you would read the article “The Steps of Breath Meditation” you will see that:

...as you let go, say, of the directed thought and evaluation of the first jhana,
  releasing yourself from the burden of those factors as you move into the second
  jhana, and so on through the different levels of concentration. As you do this,
  you begin to see how much those levels of concentration are willed.


user120: I want to add that I disagree that looking into Zen is the best place to learn about the jhānas. 'Jhāna' is a Pāli word and jhānas are most systematically talked about in Theravada traditions. In my admittedly limited experience with Zen, they do not talk about jhāna except to discourage it in some cases.

user7039: There are some great ressources around, some I frequently relay on are:
The Buddha Net file Library on Zen, Chan
The Buddha Net e-book Library on Meditation
Zen Buddhism WWW Virtual Library
But my advise will be, look for a Zen Sangha and learn to practice by a good teacher. 
EDIT:
Since you ask: Which readings would you suggest for the purpose of concentration meditation?, start with;
Wikipedia:
Dhyāna in Chan Buddhism and see also the footnotes,
Commercial, i.e. not free of charge:
Sit: Zen Teachings of Master Taisen Deshimaru
Zen Training: Methods and Philosophy 
A Guide to Zen: Lessons from a Modern Master
The Three Pillars of Zen
Japanese Zen sites of different lineages:
From Rinzai-Obaku Zen

As expressed in the famous description of Zen attributed to
  Bodhidharma, “A special transmission outside the scriptures, not
  relying on words and letters / Direct pointing to the human mind,
  seeing one’s true nature and attaining Buddhahood,” the essential
  standpoint of the Zen school is not to depend on words and letters.
  Nevertheless, it does regard the study and chanting of sacred texts as
  an important part of Zen practice. The texts and sutras most often
  used in Zen temples are the following:
Links to Text

From Soto Zen

Basic Key Terms of Soto Zen Teaching
A great many people are now practicing zazen in the Japanese Soto Zen
  Buddhist tradition outside Japan. But, because the tradition is so new
  in other lands, their practice may not always firmly based on a good
  understanding of the teaching. Before we begin practice, we need some
  clarity about what we are doing and why we are doing it that way. It
  is important to begin the practice with an understanding of the basic
  principles and underlying teaching behind what we will be doing. We
  need accurate background knowledge and a good understanding of ideas
  about the nature of zazen so that we can practice properly and
  skillfully. We should not start blindly. Otherwise our practice would
  likely go astray and get lost.
Links to Text

On both sites find some how to's and or chapters about Practice.

user-1: Zen Training by Katsuki Sekida offers a thorough introduction to the practice of samatha meditation in Zen practice.  He comes at it from a Rinzai perspective.  John Daido Loori presents a very solid introduction to samatha from the Soto school in his Art of Just Sitting.  Neither author ever explicitly describes the jhanas in quite the same way as the Theravada canon does.  Still, even if they aren't explicit, the techniques both authors describe can be employed to reach everything up to and including the second jhana.  After all, there aren't all that many ways you can riff on one-pointed concentraton!  Beyond that, I'd recommend either turning to the Pali suttas, the Visuddhimagga, or any of the many jhana manuals that are currently available.  No Zen teacher or text that I'm aware of discusses techniques like the abandonment of rapture (piti) which is essential for moving into the third jhana.  This isn't to say that such texts don't exist, however.  It just means I haven't read them!  ;-)    
But to be blunt, it doesn't make much sense to me to go digging in Zen for what's extremely well described elsewhere.  The Elder tradition did a great job documenting these states.  Might as well read what they have to say!     

user8157: Ajahn Brahm wrote a book about jhana called 'Mindfulness, Bliss & Beyond'. It is 291 pages long. The first 65 pages are at this link.
The method in this book emphasizes 'letting go' & is thus similar to many Zen teachings, such as the Xinxin Ming.
While I have not read it, a well-known book on Xinxin Ming is Faith in Mind  by Master Sheng-Yen. 

user-1: A true Jhana master, Venerable Pa Auk Sayadaw:

Breakthrough in
SAMATHA MEDITATION
Knowing and Seeing (4th revised edition)

And if you can purchase this book, I highly recommend it:

Practicing the Jhanas: Traditional Concentration Meditation as Presented by the Venerable Pa Auk Sayadaw

It's written by actual students of the Venerable Pa Auk, who actually succeeded in developing the Jhanas: very clear and concise, with practical advice.

-1: <p>I've read about the jhānas which one can experience from concentration (samatha) meditation. I have been practicing some form of mindfulness, breathing meditation, but have not experienced a state of absorption that is characteristic of the jhānas. I know that the jhānas cannot be experienced only by reading about them, however, the insight from my meditation practice alone has not brought me to them.</p>

<p>From my perspective as a novice practitioner, it makes sense to look to the Zen (derived from the word <a href="https://en.wikipedia.org/wiki/Dhy%C4%81na_in_Buddhism" rel="nofollow">dhyāna</a>) branch of Buddhism for guidance on the subject of <a href="http://plato.stanford.edu/entries/japanese-zen/#TheAdjMin" rel="nofollow">meditation</a>:</p>

<blockquote>
  <p>In the process of deepening meditation, one can roughly identify three
  distinct stages: the stage of concentration, the stage of meditation,
  and the stage of absorption. </p>
  
  <p>This dualistic relationship is broken gradually as the practitioner
  moves into the stage of meditation. The ego-conscious activity is
  gradually lessened, and the barriers it set up for itself will
  gradually be removed. When the practitioner enters the stage of
  absorption, the dualistic framing of the mind will be removed such
  that the mind starts structuring itself non-dualistically. There will
  be no separation or distancing between an object of the mind and the
  activity of the mind itself.</p>
</blockquote>

<p>I don't have the budget for every book on Zen Buddhism, but I have read all the Zen books on Kindle Unlimited (the content is mostly history, terminology, and sitting accessories). What books are there that go into depth on Zen/samatha meditation practice?</p>
 8044: <p>Right concentration (Samma Samadhi) is what you call jhana.  Right concentration then develop into directed thought and evaluation. Directed thought is when you keep your thoughts on one topic. Thinking of the breath or focusing on the breath, and being inquisitive will lead one to directed thought and evaluation, two of the factors of jhana.  If you would read the article <a href="http://www.dhammatalks.org/Archive/Writings/CrossIndexed/Published/Meditations/0211n3a1%20M1%20The%20Steps%20of%20Breath%20Meditation.pdf" rel="nofollow">“The Steps of Breath Meditation”</a> you will see that:</p>

<blockquote>
  <p>...as you let go, say, of the directed thought and evaluation of the first jhana,
  releasing yourself from the burden of those factors as you move into the second
  jhana, and so on through the different levels of concentration. As you do this,
  you begin to see how much those levels of concentration are willed.</p>
</blockquote>
 120: <p>I want to add that I disagree that looking into Zen is the best place to learn about the jhānas. 'Jhāna' is a Pāli word and jhānas are most systematically talked about in Theravada traditions. In my admittedly limited experience with Zen, they do not talk about jhāna except to discourage it in some cases.</p>
 7039: <p>There are some great ressources around, some I frequently relay on are:</p>

<p><a href="http://www.buddhanet.net/ftp11.htm" rel="nofollow">The Buddha Net file Library on Zen, Chan</a></p>

<p><a href="http://www.buddhanet.net/ebooks_m.htm" rel="nofollow">The Buddha Net e-book Library on Meditation</a></p>

<p><a href="http://www.ciolek.com/WWWVL-Zen.html" rel="nofollow">Zen Buddhism WWW Virtual Library</a></p>

<p>But my advise will be, look for a Zen Sangha and learn to practice by a good teacher. </p>

<p><strong>EDIT:</strong></p>

<p>Since you ask: <em>Which readings would you suggest for the purpose of concentration meditation?</em>, start with;</p>

<p>Wikipedia:</p>

<p><a href="https://en.wikipedia.org/wiki/Dhy%C4%81na_in_Buddhism#Dhy.C4.81na_in_Chan_Buddhism" rel="nofollow">Dhyāna in Chan Buddhism</a> and see also the footnotes,</p>

<p>Commercial, i.e. not free of charge:</p>

<p><a href="https://www.amazon.de/Sit-Teachings-Master-Taisen-Deshimaru-ebook/dp/B0086V4DUI/ref=sr_1_3?s=books-intl-de&amp;ie=UTF8&amp;qid=1471694375&amp;sr=1-3&amp;keywords=taisen+deshimaru" rel="nofollow">Sit: Zen Teachings of Master Taisen Deshimaru</a></p>

<p><a href="https://www.amazon.de/Zen-Training-Philosophy-Katsuki-Sekida-ebook/dp/B007WVNUUW/ref=sr_1_1?s=books-intl-de&amp;ie=UTF8&amp;qid=1471685804&amp;sr=1-1&amp;keywords=katsuki+sekida" rel="nofollow">Zen Training: Methods and Philosophy </a></p>

<p><a href="https://www.amazon.de/Guide-Zen-Lessons-Modern-Master-ebook/dp/B00AW2P4U2/ref=sr_1_4?s=books-intl-de&amp;ie=UTF8&amp;qid=1471685804&amp;sr=1-4&amp;keywords=katsuki+sekida" rel="nofollow">A Guide to Zen: Lessons from a Modern Master</a></p>

<p><a href="https://www.amazon.de/Three-Pillars-Zen-Roshi-Kapleau-ebook/dp/B00H129592/ref=sr_1_7?s=books-intl-de&amp;ie=UTF8&amp;qid=1471685742&amp;sr=1-7&amp;keywords=philip+kapleau" rel="nofollow">The Three Pillars of Zen</a></p>

<p>Japanese Zen sites of different lineages:</p>

<p>From <a href="http://zen.rinnou.net/whats_zen/sacred_texts.html" rel="nofollow">Rinzai-Obaku Zen</a></p>

<blockquote>
  <p>As expressed in the famous description of Zen attributed to
  Bodhidharma, “A special transmission outside the scriptures, not
  relying on words and letters / Direct pointing to the human mind,
  seeing one’s true nature and attaining Buddhahood,” the essential
  standpoint of the Zen school is not to depend on words and letters.
  Nevertheless, it does regard the study and chanting of sacred texts as
  an important part of Zen practice. The texts and sutras most often
  used in Zen temples are the following:</p>
  
  <p><em>Links to Text</em></p>
</blockquote>

<p>From <a href="http://global.sotozen-net.or.jp/eng/library/key_terms/index.html" rel="nofollow">Soto Zen</a></p>

<blockquote>
  <p>Basic Key Terms of Soto Zen Teaching</p>
  
  <p>A great many people are now practicing zazen in the Japanese Soto Zen
  Buddhist tradition outside Japan. But, because the tradition is so new
  in other lands, their practice may not always firmly based on a good
  understanding of the teaching. Before we begin practice, we need some
  clarity about what we are doing and why we are doing it that way. It
  is important to begin the practice with an understanding of the basic
  principles and underlying teaching behind what we will be doing. We
  need accurate background knowledge and a good understanding of ideas
  about the nature of zazen so that we can practice properly and
  skillfully. We should not start blindly. Otherwise our practice would
  likely go astray and get lost.</p>
  
  <p><em>Links to Text</em></p>
</blockquote>

<p>On <strong>both</strong> sites find some how to's and or chapters about <strong>Practice</strong>.</p>
 -1: <p><a href="http://rads.stackoverflow.com/amzn/click/1590302834" rel="nofollow noreferrer">Zen Training</a> by Katsuki Sekida offers a thorough introduction to the practice of samatha meditation in Zen practice.  He comes at it from a Rinzai perspective.  John Daido Loori presents a very solid introduction to samatha from the Soto school in his <a href="http://rads.stackoverflow.com/amzn/click/086171394X" rel="nofollow noreferrer">Art of Just Sitting</a>.  Neither author ever explicitly describes the jhanas in quite the same way as the Theravada canon does.  Still, even if they aren't explicit, the techniques both authors describe can be employed to reach everything up to and including the second jhana.  After all, there aren't all that many ways you can riff on one-pointed concentraton!  Beyond that, I'd recommend either turning to the Pali suttas, the Visuddhimagga, or any of the many jhana manuals that are currently available.  No Zen teacher or text that I'm aware of discusses techniques like the abandonment of rapture (piti) which is essential for moving into the third jhana.  This isn't to say that such texts don't exist, however.  It just means I haven't read them!  ;-)    </p>

<p>But to be blunt, it doesn't make much sense to me to go digging in Zen for what's extremely well described elsewhere.  The Elder tradition did a great job documenting these states.  Might as well read what they have to say!     </p>
 8157: <p>Ajahn Brahm wrote a book about <em>jhana</em> called 'Mindfulness, Bliss &amp; Beyond'. It is 291 pages long. The first 65 pages are at this <a href="http://www.dhammaloka.org.au/files/pdf/Ajahn_Brahm-Mindfulness_Bliss_and_Beyond-Chapters1-4.pdf" rel="nofollow">link</a>.</p>

<p>The method in this book emphasizes 'letting go' &amp; is thus similar to many Zen teachings, such as the <a href="https://en.wikipedia.org/wiki/Xinxin_Ming" rel="nofollow">Xinxin Ming</a>.</p>

<p>While I have not read it, a well-known book on Xinxin Ming is <a href="http://enlight.lib.ntu.edu.tw/FULLTEXT/JR-MISC/misc38333.pdf" rel="nofollow">Faith in Mind </a> by Master Sheng-Yen. </p>
 -1: <p>A true Jhana master, Venerable Pa Auk Sayadaw:</p>

<ul>
<li><p><a href="http://www.dhammatalks.net/Books14/Pa-Auk-Sayadaw_Breakthrough-in-samatha-meditation-and-vipassana-meditation.pdf" rel="nofollow">Breakthrough in
SAMATHA MEDITATION</a></p></li>
<li><p><a href="http://www.dhammatalks.net/Books13/Pa-Auk-Sayadaw_Knowing-and-Seeing-4th-Ed-2010.pdf" rel="nofollow">Knowing and Seeing (4th revised edition)</a></p></li>
</ul>

<p>And if you can purchase this book, I highly recommend it:</p>

<ul>
<li><em>Practicing the Jhanas: Traditional Concentration Meditation</em> as Presented by the Venerable Pa Auk Sayadaw</li>
</ul>

<p>It's written by actual students of the Venerable Pa Auk, who actually succeeded in developing the Jhanas: very clear and concise, with practical advice.</p>
